博士学位论文 - ustcstaff.ustc.edu.cn/~drzhangx/group-chn/dcc/pdfs/zhanghui.pdf · the analog...

分类号： O157.2 单位代码： 10335

密级：学号： 10706071

博士学位论文

文论文：几类组合编码问题研究

文论文： Combinatorial Constructions forSeveral Classes of Codes

申请人姓名：

指导教师：

合作导师：

专业名称：

研究方向：组合编码

所在学院：

论文日期：二〇一三年月

几类组合编码问题研究

论文作名：

指导教师名：

论文评阅人 1： \ \评阅人 2： \ \评阅人 3： \ \评阅人 4： \ \评阅人 5： \ \

答辩委员会主席： \ \委员 1： \ \委员 2： \ \委员 3： \ \委员 4： \ \委员 5： \ \

答辩日期：二〇一三年五月

Combinatorial Constructions for

Several Classes of Codes

Author’s signature:

Supervisor’s signature:

External Reviewers: Fangwei Fu\Professor\Nankai University

Lijun Ji\Professor\Soochow University

Jianxing Yin\Professor\Soochow University

Yusheng Li\Professor\Tongji University

Yanxun Chang\Professor\Beijing Jiaotong University

Examining Committe Chairperson:Keqin Feng\Professor\Tsinghua University

Examining Committe Members:Keqin Feng\Professor\Tsinghua University

Song Li\Professor\Zhejiang University

Junde Wu\Professor\Zhejiang University

Zhiyi Tan\Professor\Zhejiang University

Gennian Ge\Professor\Zhejiang University

Date of oral defence: May 2013

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Abstract

A fundamental problem in coding theory is to constuct optimal codes. This

problem turns out to be extremely difficult, even for a single code with affir-

matory parameters. In this dissertation, we will concentrate on this topic and

bring some new combinatorial construction methods to obtain several new class-

es of optimal codes. Our results will include constant-weight codes (CWCs),

constant-composition codes (CCCs), constant-weight covering codes (CWCCs),

and authentication codes (ACs).

The first part consists of Chapters 3 and 4. In which, we will construct

optimal ternary or quaternary constant-weight codes with weight 4 and minimum

Hamming distance 5 via generalized Steiner systems. It was the famous Israeli

mathematician Etizon who first raised the idea of constructing optimal q-ary

CWCs with generalized Steiner systems. Later, Yin generalized this idea to

packing designs. Most of works in literature focused on the case of weight 3. For

weight 4, only a few results are known due to limitation of methods.

In Chapter 3, we will study the case of ternary CWCs with weight 4 and

minimum Hamming distance 5. For length n ≡ 1 (mod 3), the existence of

generalized Steiner system was determined by the Euler Medal owner Prof. Zhu

and his students. We will establish a connection between some auxiliary designs

and group divisible codes (GDCs), and construct the optimal codes with other

lengths. The concept of group divisible codes was brought up by Chee et al. It is

the analog of group divisible designs in combinatorial design theory and shows a

great power in the recursive constructions of CWCs and CCCs. With the help of

GDCs, we will construct the optimal ternary CWCs with weight 4 and distance

5 for all but 8 lengths.

In Chapter 4, we will construct the optimal quaternary CWCs with weight

4 and distance 5 for all length n except for 55 values. We will improve an SIP

method in literature and obtain the desired generalized Steiner systems from

group divisible designs and super simple orthogonal arrays. Before our work,

vi Aa|Ü?è¯KïÄ

only some infinite classes of length n ≡ 0, 1 (mod 4) are known.

The second part consists of Chapters 5 and 6, in which, we will investigate

the constructions of optimal ternary or quaternary CWCs with weight 4 and

distance 6 using completely reducible super simple (CRSS) designs. This idea

is based on the work of Chee et al. in 2007, in which they obtained optimal

q-ary CWCs from disjoint designs and large set of designs. In Chapter 5, we will

use CRSS designs together with CRSS group divisible designs to get the optimal

ternary CWCs with weight 4 and distance 6 for all length n, except for n = 17.

In view of the important applications of these two kind of designs with CRSS

property, we will investigate the existence problems related to both designs in

Chapter 6. As a byproduct, a new class of optimal quaternary CWCs with weight

4 and distance 6 is obtained as well.

The third part consists of Chapters 7 and 8. We will exploit two special

classes of optimal q-ary CWCs and CCCs.

The researches on optimal CWCs with large alphabet usually focused on

fixed weight w and distance d and let length n and alphabet size q vary. To

the best of author’s knowledge, the only complete solutions to this problem are

when (d, w) = (3, 2) or (4, 3). In Chapter 7, we will construct optimal CWCs

with (d, w) = (5, 3) for all the length n and alphabet size q ≥ 2. They are the so

called linear size CWCs. Our main tool is the Hanani triple packings (HTPs),

which can be regarded as the generalization of Hanani triple systems raised by

the Euler Medal owner Prof. Colbourn in 1993. The existence of HTPs with all

orders will be determined as well in this chapter.

In Chapter 8, optimal ternary CCCs with weight 4, distance 6, and type

[2, 2] will be constructed. When n 6≡ 5 (mod 6), optimal codes for all but 11

lengths will be obtained. For n ≡ 5 (mod 6), near-optimal lower bounds on the

code size will be derived from GDCs and skew Room frames.

The last part consists of Chapters 9 and 10. We will study the CWCCs and

ACs with splitting property in these two chapters respectively.

Not only do CWCCs find itself having important applications in universal

data compression algorithms, but also share similar structural properties with

ABSTRACT vii

several subjects in combinatorial design theory, such as Turan designs, lottery

schemes, and covering designs. These topics are well investigated by numbers

of mathematicians during the past sixty years. In Chapter 9, we will show a

connection between optimal CWCCs with specific parameters and group divisible

covering designs. Using the H-frame as auxiliary designs, we will obtain optimal

q-ary CWCCs with weight 4 for all n ≥ 4, q ∈ {3, 4} or q = 2m + 1 (m ≥ 2),

excepting (q, n) = (3, 5), such that every word with weight 3 is at distance 1 from

at least one codewords.

In Chapter 10, we will consider the problem of constructing optimal ACs

with splitting property. By the equivalence of splitting ACs and splitting designs

established by Huber, we will generalize the methods in design theory to construct

two new infinite classes of splitting ACs. The (3, 2)-splitting ACs we obtained

are the first known infinite family of (t, c)-splitting ACs with t > 2 and c > 1. We

also prove that a (2, c)-splitting AC with k source states and v messages exists

for all sufficiently large v (with k and c fixed).

Keywords:

q-Ary constant-weight codes, constant-composition codes, constant-weight cover-

ing codes, generalized Steiner Systems, completely reducible super simple designs,

group divisible codes, authentication codes

888 ¹¹¹

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Abstract v

8¹ ix

Chapter 1 XØ 1

Chapter 2 Ä�½ÂÚÎÒ 7

2.1 è . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

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Chapter 4 ^2ÂSteinerX�E�`o�~è 31

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Chapter 5 ^��ü�O�E�`õ�~è 49

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Chapter 6 |��o��ü�OÚ�'�üW¿ 75

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7.3 rHananin�W¿��35 . . . . . . . . . . . . . . . . . . . . . 104

7.4 Aq(n, 5, 3)�(½ . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.5 Hananin�W¿��35 . . . . . . . . . . . . . . . . . . . . . . 113

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Chapter 8 ^�©|è�E�`~EÜè 115

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8.4 (Ø . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

8.5 N¹ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Chapter 9 ^�©|CX�E�`õ�~CXè 131

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9.2 O��£Ú�E�{ . . . . . . . . . . . . . . . . . . . . . . . . . 132

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9.3 �`n�~CXè . . . . . . . . . . . . . . . . . . . . . . . . . 135

9.4 �`o�~CXè . . . . . . . . . . . . . . . . . . . . . . . . . 138

9.5 Z2m+1þ��`~CXè . . . . . . . . . . . . . . . . . . . . . . 145

9.6 (Ø . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Chapter 10 �Eäk©�5��@yè 147

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10.2 O��£ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

10.3 ��35ÚìC(J . . . . . . . . . . . . . . . . . . . . . . . . . 151

10.4 ©�2-�O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

10.5 ©�3-�O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

10.6 (Ø . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

N¹ 173

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é?¿n ≥ 4§Øn ∈ {12, 13, 21, 27, 33, 39, 45, 52} ±�¤k�Ý(½A3(n, 5, 4)�O(�"�ÙSN®uLu,�IEEE Transactions on Infor-

mation Theory"

�´ù«�{�qC��§duI��E��O�|C�§Ã¦JÝ�O

\éõ"éþ�4§��Ç²ål�5��`o�~è§=(n, 5, 4)3è§�

�Ýn ≡ 0, 1 (mod 4) �§¦+3©[74, 75, 150, 165] �¥�Ñ��E�

{Ú��Ã¡a§�´ålù�¯K��)û�é�"314Ù¥§·

�U?�5©Ù¥�SIP�E{§^�©|�OÚ�ü��L��EÑ2

ÂSteinerX§¦�3�E��ëê�è��\k�"$^ù��{§2(Ü�

.Ø��9Ï�O§frame-GS§·�é?¿n ≥ 4§Ø55�Ø(½��Ý

±§(½A4(n, 5, 4)�O(�"�ÙSN®uLu,�IEEE Transactions

on Information Theory"

2¤2007cChee�3©[26]¥JÑ|Ü�O¥�Ø��O!�8��

�`õ�~è��d5§¿é¤k�q§n(½Aq(n, 3, 2)ÚAq(n, 4, 3) �

�[23, 26]"3dÄ:þ§315Ù¥§·�ïá«|��4��

ü�O£completely reducible super simple design¤��`(n, 6, 4)qè�éX"

(Ü�«9Ï�O§��ü�©|�O§ù«�O�±w��Eda

è��©|è§(Ü�©|è�48�E�{§·�é?¿�Ýn§��(

½Øn = 17±�¤kA3(n, 6, 4)�O(�"da~è�c®��(J�

kOstergard ÚSvanstrom3©[115] ¥(½��Ý10±S��"�ÙSN®u

4 Aa|Ü?è¯KïÄ

Lu,�IEEE Transactions on Information Theory"

du��ü�OéA�`õ�~è§ ��ü�©|�O

éA3�Eè�k�A^��©|è§ �üW¿��èkX��éX§3

16Ù¥§·�)û«|��4§�ê�3��ü�O§«|��

�4§�ê�2��ü�©|�OÚ�üW¿��35¯K§¿

��a#��`(n, 6, 4)4 è"�ÙSN®uLu,�Designs, codes and

Cryptography"

3¤é�`õ�~è�ïÄó�§õ´�Ä�½þwÚ��Ç²åld§

4�ÝnÚè�qCÄ�§�`õ�~è��E¯K"8c®�äk��

(J�§�k�ëê(d, w)��(3, 2)Ú(4, 3)��/"3©[24]¥§Chee�é?

¿q ≥ 2§��`(n, 2w − 1, w)qè�ìC(J"¦�¡ù«è´�5��

�§Ï�Aq(n, 2w − 1, w) = O(n)"317Ù¥§·�é¤k�Ýn§q ≥ 2§(

½Aq(n, 5, 3)�O(�"äN�§·�ïáù«è�Hananin�W¿�é

X§ Hananin�W¿´|Ü�O¥Hananin�X�í2§�ö´Í¶|ÜÆ

[Stinson§Colbourn£î.ø¼�ö¤�31993cJÑ¿)û�[140]"3�Ù

¥§·�Ø�ïáÙ�(n, 5, 3)qè�éX§�òÙí2�W¿�¹§¿��

)ûHananin�W¿��35"

~~~EEEÜÜÜèèè

~EÜè£constant-composition code¤´�aAÏ�~è§§�¦è

¥�z�èi��|¤Ñ´�Ó�"äk�~�A^��èÒ´�«~

EÜè"duÙ3õ�Ï&!DNAè!>å�ÏÕ!aªS��Nõ�¡�

A^§þVÊ�c�"§é~EÜèÒkXÚ�ïÄ[14, 17, 133]"<�

�(½~EÜè��U�èi�êÚ\�«��{§XO�Å

|¢�{[16]§W¿�O[38, 49, 50, 93, 144, 153, 154, 157]§¿m�O[158]§õ

�ªÚ��5¼ê[38, 47, 48, 51, 52]§PBD4��{[25, 28] Ú�Ù¦��

{[105, 106, 134, 149] �"

3©[25]§Chee�JÑéÙ�Eå��^��©|è�Vg±9�|

Ü�O¥aq��E�{"318Ù¥§·�ïá�a�©|è�þ�4§

��Ç²ål�6��`n�~EÜè§=(n, 6, [2, 2])3è�éX"·��

Ñ��^�Room frame�Eda�©|è��{§(Ü��Ý��è§

CHAPTER 1 XØ 5

Ú.Ø��©|è��E§·�é�Ýn 6≡ 5 (mod 6)§Ø11��

(½daè�èi�ê§én ≡ 5 (mod 6)§·��ÑÐ�e."

~~~CCCXXXèèè

~CXè£constant weight covering code¤�Ì�A^´êâØ �

{[39, 80]"lêÆ�¡ ó§ÙèiNþe.�(½´�´��Ä��|Ü¯

K"3|Ü�OnØ¥§k�Ú§�d�|Ü(�[41]§XTuran�O!ç¦

�Y!CX�O�§3L��8�õc¥kNõïÄöéÙ?1ïÄ"

-Kq(n,w, t, d)L«�Ý�n§þ�w§z�þ�t�i��èi

��Ç²ål�d�q�~CXè��U�èi�ê"319Ù¥§·

�ïá�a~CXèÚ�©|CX�éX§(Ü�«9Ï�OµH-frame§

·�é?¿�ên ≥ 4§q ∈ {3, 4}½öq = 2m + 1£m ≥ 2¤§Ø(q, n) = (3, 5)

±§��(½Kq(n, 4, 3, 1)��"�ÙSN®uLu,�Designs, codes and

Cryptography"

@@@yyyèèè

3@yè£authentication code¤�IO�.¥[122–124, 127]§��u�

ìI�3��ØS�&�Dx&E��Âì§ ��'<�¯ù�&�¿

�Ð¢�Âì"3��ê�i�Ð¢ôÂ[108]¥§'<3�S�&�¥*

�du�ìuÑ�3�Ó?è5Ke�i �ØÓ�&E§¿\\��®ux

�i�&EØÓ�#�&E§¿F"��Âì@�´�&�"3ù«µee§�

é@yè��°�[Ú�°O�ôÂ�´�ê�0Ú1�Ð¢ôÂ�§®²k

éõïÄó�§, ��êi ≥ 2�§é©�@yè�ïÄ%é�"

3©[92]¥§Huberïá�êi ≥ 2�§©�@yè�©��O�éX"3

110Ù¥§·�ÏL�E©��O5�Eäk©�5��`@yè"äN

�§·�½Â��|Ü�OnØ¥aq�9Ï�O§X©�t-GDD§©�

Ç��O�§¿í2|Ü�O¥�48�{§��#�Ã¡a"�Ù

SN®uLu,�Advances in Mathematics of Communications"

�Ø©¥¤A^��{�9|Ü�OnØ¥�õ«ØÓ��OÚ�E�

{", §|Ü�O^5�Eè��{�~2�§©¥¤^��{=´ÊÚ�

f"3±��ïÄó�¥§�öòUY&¢?ènØ¥�|Ü�O�{"

6 Aa|Ü?è¯KïÄ

�©Â¹�öôÖÆ¬Æ Ïm�Ü©Ø©§uLÚÝv��[�¹�

ë�©¥NL"�u�öY²k�§©¥J�kØ�¸Ø�?§¹�Ã Ø�

1µÚ��"

Chapter 2

ÄÄÄ��½½½ÂÂÂÚÚÚÎÎÎÒÒÒ

3�Ù¥§·�ò0�?ènØÚ|Ü�O¥��Ä�½ÂÎÒÚÄ�

(J"

é�êm ≤ n§-[m,n]aL«8Ü{m,m + a,m + 2a, . . . , n}"�a = 1�§

·�òeI�Ñ"·�r�n�ê�§Z/nZP�Zn§�K�ê8P�Z≥0"é?¿ü�8ÜXÚY§X × YP�§��(k�È§=µX × Y = {(x, y) : x ∈X, y ∈ Y }"é��k�8ÜXÚ�ên ∈ [1, |X|]§P(

X

n

)= {A ⊆ X : |A| = n}"

2.1 èèè

-XÚR´k�8§RXL«��Ý�|X|��þ�8Ü§Ù¥z��þu ∈ RX3R¥��§¿^X¥��IP§=µu = (ux)x∈X§�é?¿x ∈X§ux ∈ R"

��Ý�n�q�èÒ´��8ÜC ⊆ ZXq §Ù¥|X| = n"C¥��¡�èi"��þu ∈ ZXq �Ç²þ½Â�‖u‖ = |{x ∈ X : ux 6= 0}|"ü�èiu, v ∈ ZXq �Ç²ål§P�dH§Ò´dH(u, v) = ‖u− v‖"é?¿�þu ∈ ZXq §½Âu�| 8�supp(u) = {x ∈ X : ux 6= 0}"

·�`��èCäk£��Ç²¤åld§XJ?¿ü��Éèiu, v ∈ C§dH(u, v) ≥ d"Ï�3�©¥·�ïÄ�Ñ´��Ç²ål§�{Bå�§�

·�`��è�ål§Ò´��´§��Ç²ål"XJé?¿èiu ∈ C§‖u‖ = w§@o·�¡Cäk£~¤þw"·�r��Ý�n§ål�d§þ�w�q�~è§P�(n, d, w)qè"��(n, d, w)qè¥�èi�ê�¡�è

��§¿r§��U�èi�êP�Aq(n, d, w)"��Aq(n, d, w)�è

�¡�´�`�"

Svanstrom 3©[133]¥�ÑAq(n, d, w)�þ.µ

8 Aa|Ü?è¯KïÄ

Ún 2.1 (Svanstrom [133]).

Aq(n, d, w) ≤⌊

n

n− wAq(n− 1, d, w)

⌋"

Ún 2.2 (Svanstrom [133]).

Aq(n, d, w) ≤⌊n(q − 1)

wAq(n− 1, d, w − 1)

⌋"

��èiu ∈ C�EÜ´��þw = [w1, · · · , wq−1]§¦�u�¹i ∈Zq \ {0}TÐwig"XJC¥�z�èiÑkEÜw§@oÒ¡q�èCäk~EÜw"��Ý�n§ål�d§äk~EÜw�q�èP�(n, d, w)qè"�

�(n, d, w)qè¥�èi�ê�¡�è��§¿r§��U�èi�ê

P�Aq(n, d, w)"��Aq(n, d, w)�è�¡�´�`�"��§·�b�

3w = [w1, · · · , wq−1]¥§w1 ≥ · · · ≥ wq−1"

Svanstrom�3©Ù[135]¥�ÑAq(n, d, w)��þ."

Ún 2.3 (Svanstrom�[135]).

Aq(n, d, [w1, . . . , wq−1]) ≤⌊n

w1

Aq(n− 1, d, [w1 − 1, . . . , wq−1])

⌋"

2.2 ��OOO

��8ÜXÚ£set system¤´��|(X,B)§Ù¥X´��:8§

B´X��f8x§¡�«|"��8ÜXÚ��êÒ´X¥:��ê"é��K�ê8K§XJé?¿B ∈ B§Ñk|B| ∈ K§@o¡(X,B)�K-��

�"

é��K�ê8ÜK§��¤é²ï�O£(v,K, 1)-PBD¤Ò´��

ê�v�K-��8ÜXÚ(X,B)§¦�X¥�?¿�É:éTÐÑy3B��«|¥"��k ∈ K´“\(�”§P�k?§Ò´`ù�PBDTÐk��

��k�«|"

Ún 2.4 (Ling�[104]). é?¿�êv ≥ 10§v 6∈ [10, 20] ∪ [22, 24] ∪ [27, 29] ∪[32, 34]§Ñ�3��(v, {5, 6, 7, 8, 9}, 1)-PBD"

CHAPTER 2 Ä�½ÂÚÎÒ 9

Ún 2.5 (Ling�[104]§Colbourn�[44]). XJv ≥ 10§v 6∈ [10, 30] ∪ [32, 41] ∪[45, 47] ∪ [93, 95] ∪ [98, 101] ∪ [138, 139] ∪ [142, 150] ∪ [152, 155] ∪ [160, 161] ∪[166, 167] ∪ {185}§@o�3��(v, {6, 7, 8, 9}, 1)-PBD"

Ún 2.6 (Colbourn§Ling [45]). XJv ≥ 11§v 6∈ [11, 56] ∪ [58, 63] ∪ [66, 71] ∪[75, 79] ∪ [101, 109] ∪ [111, 113] ∪ [115, 119] ∪ [126, 127] ∪ [133, 135] ∪ [155, 160] ∪[166, 167] ∪ [173, 231] ∪ {239} ∪ [247, 287] ∪ [290, 295] ∪ [299, 343] ∪ [346, 351] ∪[355, 399]∪ [403, 407]∪ [411, 423]∪ [426, 431]∪ [435, 439]∪ [443, 448]∪ [452, 455]∪[472, 497] ∪ [499, 503] ∪ [507, 511] ∪ [580, 582]§@o�3��(v, {8, 9, 10}, 1)-

PBD"

Ún 2.7 (Rees§Stinson [118]). ��(v, {4, w?}, 1)-PBD§v > w�3��=

�v ≥ 3w + 1§�µ

(i) v ≡ 1½4 (mod 12)§w ≡ 1½4 (mod 12)¶½ö

(ii) v ≡ 7½10 (mod 12)§w ≡ 7½10 (mod 12)"

-(X,B)´��8ÜXÚ§G = {G1, . . . , Gu}´8ÜX��y©§¡�|"@o��n�|(X,G,B)�¡�´��©|�O£group divisible

design§GDD¤§XJX¥Ø3Ó��|�:éTÐÑy3λ�«|¥§�é

?¿B ∈ B§G ∈ G§ |B ∩ G| ≤ 1"XJé?¿B ∈ B§ |B| ∈ K§@o

¡(X,G,B)�(K,λ)-GDD"XJK = {k}§{P�(k, λ)-GDD"XJλ = 1§{

P�K-GDD½k-GDD"

��GDD�.´��õ8{|G| : G ∈ G}"·�Ï~^“�ê”ÎÒ5L

«GDD�.µ.�gu11 gu22 . . . gutt L«é?¿i = 1, 2, . . . , t§kui��gi�|"

��.�1v�(k, λ)-GDD£k < v¤¡��²ïØ��«|�O£balanced

incomplete block design§BIBD¤§P�(v, k, λ)-BIBD"

Ún 2.8 ([66]). ��.�gu�(4, λ)-GDD�3��=�£i¤u ≥ 4¶£ii¤λ(u−1)g ≡ 0 (mod 3)§£iii¤λu(u−1)g2 ≡ 0 (mod 12)§Øü|(½��(g, u, λ) ∈{(2, 4, 1), (6, 4, 1)}"3ùü«�¹e§Ø�3ù��GDD"

��.�nk�(k, λ)-GDD�¡��î��O£transversal design§TD¤§

P�TDλ(k, n)"�λ = 1�§�P�TD(k, n)"

10 Aa|Ü?è¯KïÄ

Ún 2.9 ([3]). -m��ê"@oµ

i) em 6∈ {2, 6}§�3��TD(4,m)¶

ii) em 6∈ {2, 3, 6, 10}§�3��TD(5,m)¶

iii) em 6∈ {2, 3, 4, 6, 10, 14, 18, 22}§�3��TD(6,m)¶

iv) em 6∈ {2, 3, 4, 5, 6, 10, 14, 15, 18, 20, 22, 26, 30, 34, 38, 46, 60}§�3��TD(7,m)¶

v) em 6∈ {2, 3, 4, 5, 6, 10, 12, 14, 15, 18, 20, 21, 22, 26, 28, 30, 33, 34, 35, 38, 39, 42,

44, 46, 51, 52, 54, 58, 60, 62, 66, 68, 74}§�3��TD(8,m)¶

vi) em´��ê�§�3��TD(m+ 1,m)"

��GDD¡��XJ¤k|��Ñ�Ó"«|��3½4��

�GDD��3Ì®²��(½£�[66]¤"

é.�gum1��GDD§�«|��3�§§��35®²��

�Colbourn�)û[42]"�«|��4�§Ge�é§��35��éõïÄ

ó�§X[69, 70, 76–78]�"

ïÄGDD�Ì�óä´Wilson’sÄ��E{£WFC¤£�[41]¤"

�E 2.10. -(X,G,B)´��GDD§w : X → Z+ ∪ {0}´Xþ��\�¼ê"b�é?¿«|B ∈ B§Ñ�3��.�{w(x) : x ∈ B}�K-GDD"@o�3

��.�{∑

x∈Gw(x) : G ∈ G}�K-GDD"

��V�©|�O£DGDD¤´��o�|(X,H,G,B)§Ù¥X´��:

8§HÚGÑ´X�y©£©O¡�ÉÚ|¤§B´X¥��f8x¦�µ

(i) é?¿«|B ∈ B§Ú?¿ÉH ∈ H§|B ∩H| ≤ 1¶

(ii) X¥�Ø3Ó�|¥§�Ø3Ó�É¥�?¿:éÑTÐÑy3��«

|¥§Ù{:éØÑy3?Û«|¥"

CHAPTER 2 Ä�½ÂÚÎÒ 11

z�«|��´K¥��§�kui��gi�|§�z��v�Éhi�

:�DGDDP��.�(g1, hv1)u1(g2, h

v2)u2 . . . (gs, h

vs)us�K-DGDD"��U?

�©|�O§´��.�(g, 1g)u�K-DGDD§P�.�gu�K-MGDD"

-v ≥ k"��(v, k, λ)-W¿´��ê�v�{k}-��8ÜXÚ(X,B)§¦

�X¥�?¿�É:é�õÑy3B�λ�«|¥"W¿êDλ(v, k, 2)Ò´�

�(v, k, λ)-W¿¥��U�«|ê"

Ún 2.11 ([129]). £1�Johnson.¤ Dλ(v, k, 2) ≤ Uλ(v, k, 2)§Ù¥

Uλ(v, k, 2) =

⌊v

k

⌊λ(v − 1)

k − 1

⌋⌋"

��ê�v�{k}-��8ÜXÚ�Ü©²1a£partial parallel class§

PPC¤Ò´�:Ø��«|�8Ü§�¡��XJ§�¹bv/kc�«|£P�MPPC¤§ÄK¡��"XJ§TÐ/¤:8��y

©§@o¡��²1a£parallel class§PC¤"��GDD�¡�´�©)

� £resolvable¤§XJ§�¤k«|�±y©¤²1a�8Ü"��©)

�K-GDD§{P�K-RGDD"

Ún 2.12 ([41]). ��.�hu�{3}-RGDD�3��=�u ≥ 3§h(u − 1)´ó

ê§hu ≡ 0 (mod 3)�(h, u) 6∈ {(2, 3), (2, 6), (6, 3)}"

Chapter 3

^222ÂÂÂSteinerXXX��EEE��`nnn��~~~èèè

3.1 ÚÚÚóóóÚÚÚÌÌÌ��(((JJJ

~è3?ènØ¥k��^[107]"��~è®²�2�ïÄ[19,

128]"�C§du��è�ÃõA^§Xk��°&�[46]!DNAO�¥Ø

��S��O[101, 110]�§õ�~è��EÅìÚå<��À"Ù

¥§�Ü©ó�´8¥3Aq(n, d, w)�(½þ"·�{��Þ�e®k��

ïÄó�µ

(i) ©[26, 56, 60]ïÄ(n, d, w)qè��E"

(ii) ©[15, 59, 115, 131, 133]é��dÚw§ïÄA3(n, d, w)��"

(iii) ©[10, 32, 33, 56, 62–64, 73, 116, 117, 151, 156]ïÄAq(n, 3, 3)"

(iv) ©[67, 74, 75, 97, 150, 152, 162, 165]ïÄAq(n, 5, 4)"

(v) ©[22, 56]ïÄA3(n, 3, 4)"

(vi) ©[23, 26]ïÄAq(n, 3, 2)ÚAq(n, 4, 3)"

(vii) ©[24]ïÄAq(n, 2w − 1, w)"

(viii) ©[159]ïÄA3(n, 6, 4)"

3OstergardÚSvanstrom�©Ù[115]¥§¦��Ñ�(½A3(n, d, w)�

þ.Úe.��{§¿��Ýn ≤ 10�èi�ê�O(�½ö."·�3

L10.18¥�Ñ�ÝnØ�L10�§A3(n, 5, 4)�O(�"

2ÂSteinerXGS(2, k, n, g)´dEtzion3©[56]¥JÑ�§^5�E�Ý

�n§þ�k§ål�2k−3�g+1��`~è"Yin�r2ÂSteinerXí2

�W¿�¹§é��Ýn�E�`õ�~è[97, 156]"<�éGS(2, k, n, g)®

14 Aa|Ü?è¯KïÄ

L 3.1: �n ≤ 10�§A3(n, 5, 4)��n 4 5 6 7 8 9 10

A3(n, 5, 4) 1 2 4 7 13 19 30

²�éõïÄó�§~Xµ [10, 32, 56, 62–64, 67, 73, 75, 97, 116, 117,

151, 156]�"3©[97]¥§Ji§WuÚZhuy²XJn ≥ 10§n ≡ 1 (mod 3)§

n 6∈ {13, 52, 58}§@o�3��GS(2, 4, n, 2)"dd·��µ

Ún 3.1. é?¿n ≥ 10§n ≡ 1 (mod 3)§n 6∈ {13, 52, 58}§�3��n(n−1)

3��`(n, 5, 4)3è"

�©|è§aqu|Ü�O¥��©|�O§´dChee§GeÚLing3

©[25]¥JÑ�"ù«è�±^3~èÚ~EÜè�48�E¥"3�

Ù¥§·�ò^�©|è�E¤k�Ýn��`(n, 5, 4)3è"·�é¤k�

Ýn ≥ 4§Ø8�Ø(½��n ∈ {12, 13, 21, 27, 33, 39, 45, 52}§(½¤k�Ý��`(n, 5, 4)3è�èi�ê"

¯¤±�§éd > 2w§Aq(n, d, w) = 1§ÏddÚn2.2§·��µ

Ún 3.2. A3(n, 5, 4) ≤⌊n2

⌊2(n−1)

3

⌋⌋:= U(n, 3)"

3�Ù¥§·�P�`(n, 5, 4)3è�þ.�U(n, 3)"

ù�Ù�(�Xeµ313.2!¥§·�ò0��Ä�VgÚÄ��E�

{¶313.3!¥§·�ò©�¹�E�`(n, 5, 4)3è¶313.4!¥§òé�Ù

�Ì�(J?1o("

3.2 OOO��£££ÚÚÚÄÄÄ��EEE��{{{

XEtzion[56]ÚYin�[156]3©¥¤ã§��Zg+1þ�(n, d, k)g+1è�±Ï

L�E��.�gn�{k}-GDD§(In × Ig, {{i} × Ig : i ∈ In},B)��§Ù¥Im =

{1, 2, . . . ,m}§d´��è�ål"é?¿«|{(i1, a1), (i2, a2), . . . , (ik, ak)} ∈B§·��±ÏLé?¿1 ≤ j ≤ k§31ij �aj§Ù§ �"��

�Ý�n�èi"XJ��.�gn�{k}-GDD�±/¤��ål�2k − 3�

è§@oÒ¡§��2ÂSteinerX£generalized Steiner system§GS¤§P

�GS(2, k, n, g)"

CHAPTER 3 ^2ÂSTEINERX�E�`n�~è 15

e¡�frame2ÂSteinerX�½Â´dJi§WuÚZhu3©[97]¥�Ñ�"

-n = h1u1 + · · · + htut§P´In��y©§Ù¥kui��hi§i =

1, 2, . . . , t"-(In × Ig, {P × Ig : P ∈ P},B)´��.�(h1g)u1 . . . (htg)ut�{k}-GDD"é?¿��«|{(i1, a1), (i2, a2), . . . , (ik, ak)} ∈ B§-ij� ��aj§1 ≤ j ≤ k§Ù{ ��0§·��Ý�n�èi"XJlù

�{k}-GDD��è��ål2k−3§·�¡��.�hu11 . . . hutt �frame2

ÂSteinerX£½ö{¡�frame¤§P�frame-GS(2, k, (hu11 . . . hutt ), g)"·�¡

ù�{k}-GDD�|�frame�|§P�G§{{i} × Ig : i ∈ In}�frame�É§P

�H"�GDDaq§frame-GS�.Ò´õ8T = {|P | : P ∈ P}§^“�ê”P

�hu11 . . . hutt "w,§��GS(2, k, n, g)Ò´��frame-GS(2, k, (1n), g)"

Chee§GeÚLing3©[25]¥JÑ�©|è�Vg§ù«è3~èÚ~

EÜè�Ì��E¥åX��^"

�½u ∈ ZXq §Y ⊆ X§-u3Yþ��§P�u |Y§´��þv ∈ ZYq§¦�v = (ux)x∈Y"

�é/§-v ∈ ZYq§Y ⊆ X§v3Xþ�*Ü§P�v|X§Ò´��þu ∈ ZXq §¦�µ

ux =

vx, ex ∈ Y¶

0, ex ∈ X \ Y"

�½8ÜC ⊆ ZYq§-C|X = {v|X : v ∈ C}"��ål�d��©|è£Group divisible code§GDC¤Ò´��n�

|(X,G, C)§Ù¥G = {G1, . . . , Gt}´8ÜX£|X| = n¤��y©§C ⊆ZXq ´��Ý�n�q�è§¦�é?¿ü��É�þu, v ∈ C§dH(u, v) ≥d§¿�é?¿�u ∈ C§1 ≤ i ≤ t§‖u|Gi

‖ ≤ 1"G¥��¡�|"XJCäk~þw§·�r��ål�d�GDC(X,G, C)P�w-GDC(d)"XJ

?¿u ∈ C�/ª´w§·�rù�GDCP�w-GDC(d)"GDC(X,G, C)�.´��õ8{|G| : G ∈ G}"�GDD�L«aq§·�^“�ê”L«GDC�

."GDC(X,G, C)��´|C|"5¿��s�(n, d, w)qè�d��

.�1n§��s�w-GDC(d)"

dframe-GS�½Â§·�kXe(Jµ

Ún 3.3. XJ�3��äkb�«|�frame-GS(2, k, (hu11 . . . hutt ), g)§@o�

3��.�hu11 . . . hutt ��b�(g + 1)�k-GDC(2k − 3)"

16 Aa|Ü?è¯KïÄ

íØ 3.4. ©O�3.�35§��60¶.�3561§��120¶.�2u§��

�4u(u−1)3§u ∈ {7, 13}¶.�6u§��12u(u− 1)§u ∈ {4, 5}�4-GDC(5)"

·�éè½öGDC��E´Äu��{§=3��k�+£Ï

~´Zn¤��^e5��è½öGDC�¤kèi"Ïd§·�Ï~=�Ñ�

�Äè�8Ü§,�^\{+½öÙ§gÓ�+5��¤kèi"

�!��m§é��GDC�èiu = (ux)x∈X§·��Ñ8Üsupp(u)§

¿^eI5L«é?¿x ∈ supp(u)§ux��"3e©¥§XJvk²(�Ñ§

·�^�gÓ�Ï~�´�^3Äè�| 8þ§ �±eIØÄ"

~ 3.5. -X = Z20§|8�G = {{i, i + 10} : 0 ≤ i ≤ 9}"@o(X,G, C)Ò´��.�210§��120�n�4-GDC(5)§XJC´de¡èiÌ�£ ��µ

11000200020000000000 10020100000000002000 10010000000000020200

10000012100000000000 10200000010000001000 22000000000020020000

½ö§�d/§·��±`C´de¡èi3Z20¥+1 (mod 20)Ðm �±eIØÄ��µ

(01, 11, 52, 92) (01, 51, 162, 32) (01, 31, 152, 172) (01, 61, 81, 72) (01, 91, 161, 22) (02, 12, 122, 152)

w,§.�210�n�4-GDC(5)�´�`(20, 5, 4)3è§Ï�§�èi�ê

��Ún3.2¥�þ."

~ 3.6. -X = Z16§G = {{i, i + 4, i + 8, i + 12} : 0 ≤ i ≤ 3}"@o(X,G, C)Ò´��.�44§��64�n�4-GDC(5)§XJC´dXeèi3Z16þ+2(mod 16)Ðm��µ

(11, 81, 141, 72) (01, 51, 71, 142) (01, 11, 21, 32) (11, 61, 112, 02)

(12, 32, 42, 142) (11, 41, 152, 62) (11, 71, 22, 42) (01, 132, 62, 72)

~ 3.7. -:8X = {0, 1, . . . , 29}§|G = {{i} : 0 ≤ i ≤ 23}∪{{24, 25, . . . , 29}}§@o(X,G, C)Ò´��.�12461§��276�n�4-GDC(5)§XJC´dXeèidgÓ�+G = 〈(0 2 4 · · · 22)(1 3 5 · · · 23) (24 25 26)(27 28 29)〉Ðm��"

(12, 22, 52, 72) (01, 51, 251, 202) (11, 52, 152, 282) (01, 41, 162, 102) (11, 51, 221, 231)

(01, 261, 92, 12) (01, 61, 211, 132) (11, 81, 102, 112) (11, 21, 132, 202) (11, 141, 161, 242)

(11, 281, 02, 22) (11, 111, 92, 252) (11, 82, 122, 192) (01, 232, 42, 242) (02, 132, 162, 262)

(11, 61, 271, 32) (11, 121, 241, 72) (11, 91, 222, 272) (01, 172, 82, 272) (11, 291, 62, 212)

(01, 141, 291, 52) (11, 251, 42, 182) (01, 161, 142, 282)


Xe'uGDC��E3©[25]¥�Ñ"

�E 3.8 (Chee�[25]). £Ä��E{¤-d ≤ 2(w− 1)§(X,G,B)´��GDD§

ω : X → Z≥0´��\�¼ê"é?¿f8S ⊆ X§-S = ∪x∈S({x} × Zω(x))"b�é?¿B ∈ B§�3��.�{ω(a) : a ∈ B}§��cB�£Ñ\¤q�w-

GDC(d) (B, {{a} : a ∈ B}, CB)"@o(X, {G : G ∈ G},∪B∈B(CB|X))Ò´��

.�{∑

x∈G ω(x) : G ∈ G}§��∑

B∈B cB�q�w-GDC(d)"?�Ú§XJÑ

\GDCk~EÜw§@o��è�k~EÜw"

~ 3.9. lÚn2.9��TD(4, 5) (X,G,B)§¿^Ä��E{éz�:\�4

£=µé?¿x ∈ X§ω(x) = 4¤"é?¿B ∈ B§d~3.6�3��.�44§�

��64�n�4-GDC(5)£·�¡��Ñ\GDC¤"@o·�Ò��.

�204§��1600�n�4-GDC(5)"

�E 3.10 (Chee�[25]). £W|¤ -d ≤ 2(w − 1)"b�(X,G, C)´��a�q�w-GDC(d)"?�Úb�§é?¿G ∈ G§Ñ�3��cG�(|G|, d, w)qèCG"@oC ′ = C∪(∪G∈G(CG|X))Ò´��a+

∑G∈G cG

�(|X|, d, w)qè"AO�§XJCÚCG§G ∈ Gk~EÜw§@o��è�k~EÜw"

~ 3.11. l~3.9��.�204§��1600�n�4-GDC(5)"d~3.5§·�

k��`(20, 5, 4)3è"3.�204�4-GDC(5)�|þW\ù��`è§Ò�

��`(80, 5, 4)3è"

5¿µ�E3.10k��w,�í2§Ò´3GDC�|þW\��GDC§5

��GDC"~X§XJ·�3~3.11¥�.�204�n�4-GDC(5)�|þ

W\.�210§��120�n�4-GDC(5)§·�Ò��.�240§��

�2080�n�4-GDC(5)"

�E 3.12. £)ä{¤ -d ≤ 2(w − 1)"b�(X,G, C)´��c�q�w-

GDC(d)"-X = X × Zm§G = {G × Zm : G ∈ G}"?�Úb�§é?¿èiu ∈ C§�3��TD(w,m)§(supp(u)×Zm, {{x}×Zm : x ∈ supp(u)},Bu)"é?¿B ∈ Bu§�E��èvB ∈ ZXq §(vB)(x,i) = ux§(x, i) ∈ B§Ù§ ��""@o(X, G,∪u∈C ∪B∈Bu vB)Ò´��.�{m|G| : G ∈ G}§��cm2�q�w-

GDC(d)"?�Ú§XJ�©�GDCk~EÜw§@o��GDC�k~E

Üw"

18 Aa|Ü?è¯KïÄ

~ 3.13. dÚn2.9�3��TD(4, 4)"líØ3.4��.�64§��144�

n�4-GDC(5)"d�E3.12§·��.�244§��2304�n�4-

GDC(5)"·��¡ù�L§�é.�64�4-GDC(5)^4)ä"

�E 3.14 (Chee�[25]). £O\y�:¤ -y ∈ Z≥0"b�(X,G, C)´��c�£Ì¤q�w-GDC(d)"-Y´��y�8Ü§¿��XØ��§-X ′ =

X ∪ Y"?�Úb��3Xe�£Ñ\¤èµ

i) é��|G0 ∈ G§�3��cG0�(|G0|+ y, d, w)qèCG0¶

ii) é?¿|G ∈ G \ {G0}§�3��.�1|G|y1§��cG�q�w-GDC(d)

(G ∪ Y, {{x} : x ∈ G} ∪ {Y }, CG)"

@o§(C|X′)∪(CG0|X′)∪(∪G∈G\{G0}(CG|X

′))Ò´��c+cG0+

∑G∈G\{G0} cG

�(|X|+ y, d, w)qè"?�Ú§XJÌèÚÑ\èÑk~EÜw§@o��è

�k~EÜw"

~ 3.15. ùp·��E��`(30, 5, 4)3è"-:8�Z30§¤kèidXe

Äè3Z30þ+2 (mod 30)Ðm��"

(01, 11, 31, 241) (11, 61, 82, 132) (01, 51, 112, 192) (02, 102, 132, 182) (01, 221, 102, 162)

(11, 51, 151, 282) (01, 21, 201, 232) (01, 131, 261, 222) (01, 161, 62, 172) (11, 292, 62, 202)

(01, 231, 42, 82) (11, 71, 272, 92) (11, 112, 252, 42) (11, 02, 22, 192) (11, 21, 131, 172)

(11, 91, 121, 262) (01, 252, 272, 282) (01, 211, 122, 132) (01, 292, 52, 92)

l~3.13��.�244§��2304�n�4-GDC(5)"O\6�Ã¡:§

3cn�|þëÓù6�Ã¡:W\~3.7¥�.�12461§��276�4-

GDC(5)§3��|þëÓÃ¡:W\þ¡��`(30, 5, 4)3è§·�Ò

��`(102, 5, 4)3è"

3e©¥·�ò�Äþ�4§ål�5�n�~è§Ïd§3�Ù¥·

�òrn�4-GDC(5){P�GDC"

3.3 ÌÌÌ��yyy²²²LLL§§§

a. ��GDCÚÚÚ��`èèè

3�!¥§·�ò�E��GDCÚ�`è§ùè·�ò3e¡��

E¥^�"


Ún 3.16. é¤ku ∈ {16, 19, 22, 28}§�3.�2u§��4u(u−1)3�GDC"

y². é?¿u ∈ {16, 19, 22, 28}§-:8�Xu = Z2u§|8�Gu = {{i, i+ u} :

0 ≤ i ≤ u − 1}"@o(Xu,Gu, Cu)´��.�2u§��4u(u−1)3�GDC§X

JCu´d[162, L III]¥�èi3Z2u¥+1 (mod 2u)Ðm��"

Ún 3.17. �3��.�47§��224�GDC"

y². -:8X = Z28§|8�G = {{i, i + 7, i + 14, i + 21} : 0 ≤ i ≤6}"@o(X,G, C)Ò´��.�47§��224�GDC§XJC´dXeèi3Z28¥+1 (mod 28)Ðm��"

(01, 171, 42, 192) (01, 122, 132, 172) (01, 61, 81, 91) (01, 101, 231, 112)

(01, 41, 102, 262) (01, 182, 242, 272) (01, 32, 52, 232) (01, 161, 82, 252)

Ún 3.18. é¤ku ∈ [6, 10] ∪ {19}§�3.�6u§��12u(u− 1)�GDC"

y². é?¿u ∈ {6, 8, 9}§-:8Xu = Z6u§|8�Gu = {{i, i+u, . . . , i+5u} :

0 ≤ i ≤ u − 1}"@o(Xu,Gu, Cu)Ò´.�6u§��12u(u − 1)�GDC§X

JCu´dXeèi3Z6uþ+1 (mod 6u)Ðm��"

u = 6µ

(01, 22, 92, 102) (01, 101, 191, 211) (01, 281, 311, 292) (01, 141, 172, 212) (01, 142, 192, 282)

(01, 52, 82, 252) (01, 132, 152, 262) (01, 321, 162, 312) (01, 131, 291, 42) (01, 11, 232, 332)

u = 8µ

(01, 21, 441, 12) (01, 102, 272, 382) (01, 281, 311, 411) (01, 72, 112, 262) (01, 251, 361, 422)

(01, 261, 271, 92) (01, 122, 182, 192) (01, 291, 341, 142) (01, 222, 252, 352) (01, 391, 342, 362)

(01, 301, 22, 232) (01, 212, 392, 442) (01, 331, 372, 462) (01, 32, 152, 292)

u = 9µ

(01, 52, 72, 82) (01, 101, 511, 342) (01, 262, 312, 422) (01, 201, 421, 532) (01, 261, 212, 412)

(01, 11, 71, 202) (01, 102, 322, 472) (01, 301, 351, 162) (01, 172, 382, 462) (01, 501, 392, 512)

(01, 21, 171, 331) (01, 111, 251, 232) (01, 461, 222, 482) (01, 62, 252, 292) (01, 42, 142, 282)

(01, 32, 442, 502)

éu ∈ {7, 10, 19}§©Oé.�2u�GDC£Ún3.5§íØ3.4ÚÚn3.16¤

^3)ä��¤I�GDC"

Ún 3.19. �3��.�3661§��162�GDC"

20 Aa|Ü?è¯KïÄ

y². -:8�X = {0, 1, . . . , 23}§|8�G = {{i, i + 6, i + 12} : 0 ≤ i ≤ 5} ∪{{18, . . . , 23}}"@o(X,G, C)´��.�3661§��162�GDC§XJC´dXeèidgÓ�+G = 〈(0 2 4 · · · 16)(1 3 5 · · · 17)(18 19 20)〉 Ðm��"

(01, 12, 42, 192) (11, 31, 61, 162) (11, 121, 52, 202) (11, 111, 221, 92) (11, 141, 151, 192)

(01, 221, 72, 82) (11, 81, 62, 222) (12, 92, 142, 191) (02, 32, 102, 182) (11, 112, 152, 212)

(01, 31, 201, 22) (01, 141, 211, 92) (01, 161, 152, 222) (11, 181, 22, 32) (11, 211, 42, 82)

(01, 32, 52, 142) (01, 81, 171, 191) (11, 102, 122, 232)

Ún 3.20. é?¿u ∈ {4, 5}§�3��.�6u31§��12u2�GDC"

y². é?¿u ∈ {4, 5}§-Xu = {0, 1, . . . , 6u + 2}§|8�Gu = {{i, i +

u, . . . , i + 5u} : 0 ≤ i ≤ u − 1} ∪ {{6u, 6u + 1, 6u + 2}}"@o(Xu,Gu, Cu)Ò´.�6u31§��12u2�GDC§XJCu´dXeèidgÓ�+G = 〈(0 1 2 3

· · · 6u− 1)(6u 6u+ 1 6u+ 2)〉Ðm��"u = 4µ

(01, 21, 151, 211) (01, 171, 102, 232) (01, 22, 52, 112) (01, 141, 261, 152)

(01, 32, 132, 262) (01, 251, 212, 222) (01, 72, 92, 142) (01, 11, 192, 252)

u = 5µ

(01, 41, 32, 272) (01, 11, 142, 302) (01, 91, 121, 281) (01, 81, 311, 262) (01, 112, 282, 312)

(01, 42, 62, 222) (01, 301, 22, 212) (01, 122, 162, 192) (01, 71, 241, 12) (01, 82, 92, 172)

Ún 3.21. é?¿u ≥ 4§�3��.�12u§��48u(u− 1)�GDC"

y². �u ≡ 0½1 (mod 4)§u ≥ 4�§dÚn2.7§�3��(3u + 1, {4}, 1)-

PBD"3ù�PBD�:8�K��:��.�3u�{4}-GDD"�u ≡ 2½3

(mod 4)§u ≥ 7�§dÚn2.7�3��(3u+ 1, {4, 7?}, 1)-PBD"lù�PBD�

:8�K��Ø3��7�|¥�:��.�3u�{4, 7?}-GDD"Ïd§é?

¿u ≥ 4§u 6= 6§·�ok��.�3u�{4, 7}-GDD"

éù�.�3u�{4, 7}-GDD§^Ä��E{\�4��.�12u�GDC"ù

p§Ñ\�´.�44Ú47�GDC£~3.6ÚÚn3.17¤"

éu = 6§��.�46�{5}-GDD£�©[71]¤§^Ä��E{\�3��

¤I�GDC"ùp§Ñ\�´.�35�GDC£íØ3.4¤"

Ún 3.22. é?¿u ∈ [4, 9]§�3��.�12u181§��48u(u+ 2)�GDC"


y². é?¿u ∈ [4, 9]§-:8�Xu = {0, 1, . . . , 12u+17}§|8�Gu = {{i, i+u, . . . , i + 11u} : 0 ≤ i ≤ u − 1} ∪ {{12u, . . . , 12u + 17}}"@o(Xu,Gu, Cu)Ò´.�12u181§��48u(u+ 2)�GDC§XJCu´d[162,L IV]¥�èi3Xe

gÓ�+GeÐm��"

�u = 4�§G = 〈(0 2 4 · · · 46)(1 3 5 · · · 47)(48 51 54 · · · 63)(49 52 55 · · ·64)(50 53 56 · · · 65)〉"

�u ∈ {5, 7, 8, 9}�§G = 〈(0 1 2 · · · 12u−1)(12u 12u+1 12u+2 · · · 12u+

5)(12u+ 6 12u+ 7 12u+ 8 · · · 12u+ 17)〉"

�u = 6�§G = 〈(0 1 2 · · · 71)(72 73 74)(75 76 77)(78 79 80)(81 82 83)

(84 85 86)(87 88 89)〉"

Ún 3.23. �3��.�11241§��72�GDC"

y². -:8�X = {0, 1, . . . , 15}§|�G = {{i} : 0 ≤ i ≤ 11}∪{{12, . . . , 15}}"@o(X,G, C)Ò´.�11241§��72�GDC§XJC´dXeèidgÓ�+G = 〈(0 4 8)(1 5 9)(2 6 10)(3 7 11)(12 13 14)(15)〉Ðm��"

(01, 21, 71, 12) (11, 21, 41, 122) (31, 71, 02, 142) (31, 121, 12, 22) (31, 72, 102, 122) (91, 141, 22, 62)

(11, 02, 22, 52) (11, 42, 72, 132) (01, 61, 22, 122) (21, 51, 32, 152) (21, 151, 92, 112) (21, 141, 02, 42)

(21, 62, 72, 82) (11, 61, 92, 142) (01, 51, 111, 72) (21, 31, 61, 131) (12, 72, 112, 141) (51, 11, 31, 141)

(01, 11, 151, 82) (12, 42, 52, 142) (01, 31, 62, 152) (01, 41, 121, 92) (01, 131, 42, 112) (41, 72, 22, 142)

�n ≡ 1 (mod 3)�§·�éA3(n, 5, 4)kXeU?"

Ún 3.24. A3(58, 5, 4) = U(58, 3)§A3(13, 5, 4) ≥ U(13, 3) − 4§A3(52, 5, 4) ≥U(52, 3)− 12"

y². én = 58§-:8�Z58"¤I�èidXeèi3Z58¥+2 (mod 58)Ðm��"

(11, 32, 52, 62) (11, 241, 482, 42) (01, 442, 192, 262) (11, 131, 302, 72) (01, 331, 491, 182)

(01, 41, 472, 52) (11, 372, 82, 172) (01, 572, 362, 422) (01, 311, 391, 522) (12, 102, 272, 482)

(01, 62, 92, 492) (11, 451, 481, 22) (11, 102, 202, 462) (01, 302, 342, 172) (11, 351, 501, 122)

(11, 21, 51, 341) (11, 61, 221, 291) (11, 111, 181, 572) (01, 81, 281, 562) (01, 82, 212, 322)

(01, 191, 532, 32) (11, 71, 552, 322) (11, 121, 271, 392) (01, 271, 412, 222) (11, 332, 382, 452)

(01, 11, 502, 312) (01, 102, 112, 352) (11, 141, 191, 272) (01, 181, 152, 252) (11, 242, 402, 522)

(01, 211, 542, 42) (01, 131, 372, 232) (11, 211, 231, 412) (01, 22, 292, 332) (01, 61, 462, 512)

(01, 21, 141, 241) (01, 171, 142, 162) (11, 232, 292, 512)

22 Aa|Ü?è¯KïÄ

én = 13§-:8�{0, 1, 2, . . . , 12}"¤I�48�èi��EXeµ

(01, 72, 41, 61) (101, 12, 81, 61) (81, 01, 112, 52) (42, 51, 91, 82) (72, 52, 82, 101) (01, 101, 111, 62)

(12, 01, 32, 82) (112, 51, 72, 12) (81, 41, 102, 31) (32, 92, 11, 41) (71, 21, 101, 41) (101, 11, 122, 42)

(12, 02, 91, 71) (112, 61, 11, 22) (81, 62, 121, 42) (31, 52, 21, 111) (61, 21, 102, 91) (102, 111, 11, 82)

(21, 11, 81, 02) (121, 91, 22, 52) (91, 41, 62, 112) (02, 72, 111, 22) (02, 41, 122, 52) (102, 121, 01, 71)

(21, 51, 01, 92) (121, 92, 61, 02) (92, 111, 71, 42) (52, 71, 11, 62) (62, 12, 21, 122) (111, 12, 41, 121)

(22, 12, 42, 31) (11, 31, 72, 121) (32, 91, 81, 111) (82, 92, 22, 62) (122, 32, 112, 71) (42, 02, 102, 112)

(31, 51, 02, 62) (12, 102, 52, 92) (72, 81, 92, 122) (81, 22, 71, 51) (122, 51, 61, 111) (82, 21, 121, 112)

(31, 61, 71, 82) (31, 01, 91, 122) (72, 62, 102, 32) (61, 32, 42, 52) (121, 51, 101, 32) (92, 101, 112, 31)

én = 52§dÚn3.21�3.�124�GDC"O\4�Ã¡:§¿3c3�|

þëÓÃ¡:W\Ún3.23¥�.�11241�GDC§3��|þëÓÃ¡:

W\�`(16, 5, 4)3èÒ��¤I�è"

b. ��ÝÝÝn ≡ 0 (mod 6)��

3�!¥§·�ò(½�n ≡ 0 (mod 6)�§A3(n, 5, 4)��"�n = 12�§

·�kXe�."

Ún 3.25. A3(12, 5, 4) ≥ U(12, 3)− 1"

y². -:8�{0, 1, 2, . . . , 11}§¤I41�èiXeµ

(02, 11, 41, 71) (91, 02, 52, 61) (51, 92, 42, 01) (52, 92, 101, 22) (51, 02, 31, 102) (52, 102, 112, 12)

(11, 52, 81, 01) (91, 22, 31, 11) (51, 91, 72, 12) (62, 102, 42, 81) (112, 11, 61, 92) (42, 91, 112, 101)

(12, 21, 41, 81) (92, 02, 12, 32) (42, 82, 02, 22) (62, 11, 72, 101) (112, 02, 72, 21) (71, 102, 22, 111)

(12, 22, 01, 62) (01, 102, 91, 21) (81, 22, 61, 51) (11, 32, 82, 102) (111, 31, 12, 42) (101, 81, 111, 02)

(31, 81, 92, 71) (01, 112, 31, 82) (62, 92, 21, 82) (112, 62, 51, 71) (72, 92, 102, 41) (111, 21, 51, 11)

(32, 61, 42, 72) (01, 61, 41, 111) (61, 71, 12, 82) (31, 61, 21, 101) (91, 111, 62, 32) (71, 01, 101, 32)

(41, 62, 31, 52) (101, 51, 41, 82) (42, 21, 71, 52) (32, 112, 41, 22) (111, 52, 72, 82)

Ún 3.26. é?¿n ≡ 0 (mod 6)§18 ≤ n ≤ 66½n = 78§A3(n, 5, 4) =

U(n, 3)"

y². én = 30§¤Iè3~3.15¥�E"éÙ§�n§-:8�Zn§¤I�èid[162, L V]¥�èi3Zn¥+2 (mod n)Ðm��"

Ún 3.27. �3.�166181§��2211�GDC"


y². -X = {0, 1, . . . , 83}§|8�G = {{i} : 0 ≤ i ≤ 65}∪{{66, 67, . . . , 83}}"@o(X,G, C)´.�166181§��2211�GDC§XJC´d[162, L VI]¥�

èidgÓ�+G = 〈(0 2 4 · · · 64)(1 3 5 · · · 65)(66 67 68)(69 70 71)

(72 73 74)(75 76 77)(78 79 80)(81 82 83)〉Ðm��"

Ún 3.28. é?¿u ≥ 4§�3��.�18u§��108u(u− 1)�GDC"

y². éu ≥ 4§u 6= 6§y²�Ún3.21�y²aq"ùpÑ\�GDC�.

�64Ú67�GDC£íØ3.4ÚÚn3.18¤"éu = 6§�Ún3.18¥�.�66�GDC§

^3)ä��¤I�GDC"

íØ 3.29. é?¿u ≥ 4§A3(18u, 5, 4) = U(18u, 3)"

y². �Ún3.28¥�.�18u�GDC"3|þW\�`(18, 5, 4)3è£Ún3.26¤§

Òé?¿u ≥ 4��`(18u, 5, 4)3è"

Ún 3.30. é?¿u ∈ {3, 4}§A3(18u+ 30, 5, 4) = U(18u+ 30, 3)"

y². �u = 3§lÚn3.27�.�166181�GDC"3�Ý�18�|þW\�

`(18, 5, 4)3èÒ��¤I�è"éu = 4§¤I�è3~3.15¥�Ñ"

Ún 3.31. é?¿u ≥ 31§m ∈ {24, 30}§�3��.�18um1§��108u(u−1) + 12um�GDC"

y². lÚn2.9��TD(6, 3t)§^Ä��E{éc4�|�¤k:§15�|

�3x�:§��|�y�:\�6§Ù¥x = 0½3 ≤ x ≤ t§y ∈ {1, 2}"Ù{:\�0"ùpÑ\�´.�64§65Ú66�GDC£íØ3.4ÚÚn3.18¤"·�

Ò��.�(18t)4(18x)1(6y)1�GDC"O\18�Ã¡:§3c5�|ëÓÃ¡

:W\.�18t+1½18x+1�GDCÒ��.�184t+x(6y + 18)1�GDC"-u =

4t + x§m = 6y + 18§·�Ò��.�18um1�GDC§Ù¥m ∈ {24, 30}§u = 4t+ x�±�Ø�u31�?Û�"

íØ 3.32. é?¿u ≥ 31§m ∈ {24, 30}§A3(18u+m, 5, 4) = U(18u+m, 3)"

y². �Ún3.31¥�.�18um1�GDC§Ù¥u ≥ 31§m ∈ {24, 30}"3|þW\·��Ý��`è£Ún3.26¤Ò��¤I�è"

24 Aa|Ü?è¯KïÄ

Ún 3.33. é?¿u ≡ 0 (mod 3)§12 ≤ u ≤ 27§m ∈ {24, 30, 42, 48, 60, 66}§�3��.�18um1§��108u(u− 1) + 12um�GDC"

y². lÚn2.9��TD(10, 9)§^Ä��E{éc4�|�¤k:§��

�|�y�:§�{i�|�3xi�:§1 ≤ i ≤ 5§\�6§Ù{:\�0"·�

��.�544(18x1)1 · · · (18x5)

1(6y)1§xi ∈ {0, 3}§y ∈ {1, 2, 4, 5, 7, 8}�GDC"

ùp§Ñ\è�.�6s§s ∈ [4, 10]£íØ3.4ÚÚn3.18¤�GDC"O\18�

Ã¡:§3Ø��6y�¤k|þëÓÃ¡:W\.�184�GDC§·

��.�1812+∑xi(6y + 18)1�GDC§Ù¥xi ∈ {0, 3}§y ∈ {1, 2, 4, 5, 7, 8}"

-u = 12 +∑xi§m = 6y + 18§·�Ò��.�18um1�GDC§Ù¥u ≡ 0

(mod 3)§12 ≤ u ≤ 27§m ∈ {24, 30, 42, 48, 60, 66}"

íØ 3.34. é?¿12 ≤ u ≤ 29§m ∈ {24, 30}§A3(18u + m, 5, 4) = U(18u +

m, 3)"

y². lÚn3.33�.�18um1�GDC§Ù¥u ≡ 0 (mod 3)§12 ≤ u ≤ 27§

m ∈ {24, 30, 42, 48, 60, 66}"3|þW\·��Ý��`è£Ún3.26¤Ò��

¤I�è"

Ún 3.35. XeGDCÑ�3µ

i) .�24u§��192u(u− 1)§u ∈ {4, 5, 7, 8}¶

ii) .�30u§��300u(u− 1)§u ∈ {5, 7, 19}¶

iii) .�24u361§��192u(u+ 2)§u ∈ {4, 5, 7, 8, 22}¶

iv) .�24u181§��96u(2u+ 1)§u ∈ {4, 5, 7}¶

v) .�248301§��14592¶

vi) .�305241§��8400"

y². .�24u§u ∈ {4, 5, 7, 8}�GDC´d.�6u�GDC£íØ3.4ÚÚn3.18¤

^4)ä��"

.�30u§u ∈ {5, 7, 19}�GDC´d.�6u�GDC£íØ3.4ÚÚn3.18¤

^5)ä��"


.�24u361§u ∈ {4, 5, 7, 8, 22}�GDC´é.�6u91�{4}-GDD£�[77, ½

n 1.6]¤^Ä��E{\�4��"

é.�24u181§u ∈ {4, 5, 7}�GDC§lÚn2.9��TD(5, u)§�K��

:��.�4u(u − 1)1�{5, u}-GDD"^Ä��E{é��4�|�¤k:§

��u− 1�|�3�:\�6§Ù{:\�0§Ò��.�24u181�GDC"

é.�248301�GDC§��TD(5, 8)§�K��:��.�4871�{5, 8}-GDD"^Ä��E{é��4�|�¤k:§Ú��7�|�5�:\�6§

Ù{:\�0§Ò��.�248301�GDC"

é.�305241�GDC§��TD(6, 5)§^Ä��E{éc5�|�¤k

:§��|�4�:\�6§Ù{:\�0Ò��.�305241�GDC"

íØ 3.36. é?¿u ∈ [5, 11]∪ {30}§m ∈ {24, 30}½(u,m) = (4, 24)§A3(18u+

m, 5, 4) = U(18u+m, 3)"

y². �Ún3.35¥�GDC"3|þW\·��Ý��`è£Ún3.26¤§Ò�

�I��è"

(ÜÚn3.25§3.26Ú3.30§íØ3.29–3.36�(J§·��µ

½n 3.37. é?¿n ≡ 0 (mod 6)§n ≥ 18§A3(n, 5, 4) = U(n, 3)¶A3(12, 5, 4) ≥U(12, 3)− 1"

c. ��ÝÝÝn ≡ 2 (mod 6)��

3�!¥§·�ò(½�n ≡ 2 (mod 6)�§A3(n, 5, 4)��"w,§XJ

�3��.�2u§��4u(u−1)3§u ≡ 1 (mod 3)�GDC§@oÒ�3��

`(2u, 5, 4)3è"

Ún 3.38. é?¿u ≡ 1 (mod 3)§7 ≤ u ≤ 34½u ∈ {40, 43, 52}§�3��.�2u§��4u(u−1)

3�GDC"

y². éu ∈ {7, 10, 13, 16, 19, 22, 28, 40}§¤IGDC3~3.5§~3.11ÚíØ3.4§

Ún3.16¥©O�Ñ"

éu ∈ {25, 31, 43}§lÚn3.21�.�12s§s ∈ {4, 5, 7}�GDC"O\2�

Ã¡:§¿3|þëÓÃ¡:W\.�27�GDCÒ��¤IGDC"

26 Aa|Ü?è¯KïÄ

éu ∈ {34, 52}§lÚn3.22�.�12s181§s ∈ {4, 7}�GDC"O\2�Ã

¡:§¿3|þëÓÃ¡:W\.�27½210�GDCÒ��¤IGDC"

Ún 3.39. é?¿u ≡ 1 (mod 3)§u ∈ {37, 46, 49}½u ≥ 55§�3��.�2u§

��4u(u−1)3�GDC"

y². �Ún3.28ÚÚn3.31–3.35¥.�gtm1�GDC§Ù¥g ∈ {18, 24, 30}§m ∈ {18, 24, 30, 42, 48, 60, 66}"O\2�Ã¡:§¿3��g½m�|þëÓÃ

¡:W\.�2g2+1½2

m2+1�GDC£Ún3.38¤§Ò��.�2

gt+m2

+1�GDC"

é?¿u = gt+m2

+ 1§¤I�GDC�.Ú5 �3L3.2¥"

L 3.2: Ún3.39¥¤IGDC�.Ú5 u GDC�. 5

9s+ 1§s ≥ 4 18s−1181§s ≥ 4 Ún3.28

9s+ 4½9s+ 7§s ≥ 32 18s−1m1§s ≥ 32§m ∈ {24, 30} Ún3.31

9s+ 4½9s+ 7§s ∈ [13, 30] 18s−1m1§s ≡ 0 (mod 3)§13 ≤ s ≤ 28§ Ún3.33

m ∈ {24, 30, 42, 48, 60, 66}49, 61, 85, 97 24s241§s ∈ {3, 4, 6, 7} Ún3.35

76, 106, 286 30s301§s ∈ {4, 6, 18} Ún3.35

67, 79, 103, 115, 283 24s361§s ∈ {4, 5, 7, 8, 22} Ún3.35

58, 70, 94 24s181§s ∈ {4, 5, 7} Ún3.35

112 248301 Ún3.35

88 305241 Ún3.35

(ÜÚn3.38Ú3.39§·��Xe(Jµ

½n 3.40. é?¿�ên ≡ 2 (mod 6)§n ≥ 14§A3(n, 5, 4) = U(n, 3)"

d. ��ÝÝÝn ≡ 5 (mod 6)��

3�!¥§·�ò(½n ≡ 5 (mod 6)�§A3(n, 5, 4)��"

Ún 3.41. A3(11, 5, 4) = U(11, 3)"

y². -:8�{0, 1, 2, . . . , 10}§¤I�33�èiXeµ

(91, 52, 81, 101) (01, 81, 22, 71) (32, 81, 92, 72) (02, 21, 101, 11) (31, 52, 01, 72) (21, 62, 91, 72)

(01, 102, 82, 32) (02, 51, 81, 12) (51, 62, 82, 71) (101, 41, 62, 01) (31, 61, 82, 21) (91, 42, 82, 02)

(81, 21, 102, 41) (11, 32, 51, 91) (61, 01, 91, 12) (101, 51, 31, 22) (92, 12, 41, 52) (81, 31, 42, 11)

(72, 12, 101, 82) (11, 72, 61, 41) (61, 22, 42, 92) (11, 71, 92, 102) (92, 01, 51, 21) (31, 41, 91, 71)

(61, 71, 101, 32) (11, 82, 52, 22) (62, 31, 02, 92) (51, 72, 102, 42) (22, 02, 32, 41) (52, 102, 02, 61)

(62, 12, 22, 102) (21, 42, 71, 12) (62, 32, 52, 42)


Ún 3.42. é?¿n ≡ 5 (mod 6)§17 ≤ n ≤ 71½n ∈ {83, 89}§A3(n, 5, 4) =

U(n, 3)"

y². -:8�Zn§¤Ièid[162, L VII]¥�èi3Zn¥+1 (mod n)Ðm

��"

Ún 3.43. é?¿u ∈ {9, 12, 15}§�3��.�2u51§��4u(u+4)3�GDC"

y². é?¿u ∈ {9, 12, 15}§-:8Xu = {0, 1, . . . , 2u + 4}§|8�Gu =

{{i, i + u} : 0 ≤ i ≤ u − 1} ∪ {{2u, 2u + 1, . . . , 2u + 4}}"@o(Xu,Gu, Cu)Ò´.�2u51§��4u(u+4)

3�GDC§XJC´d[162, L VIII]¥�èidgÓ�

+G = 〈(0 3 6 · · · 2u− 3)(1 4 7 · · · 2u− 2)(2 5 8 · · · 2u− 1)(2u)(2u+ 1)(2u+

2)(2u+ 3)(2u+ 4)〉Ðm��"

Ún 3.44. é?¿u = 12½u ≥ 15§A3(6u+ 5, 5, 4) = U(6u+ 5, 3)"

y². éu = 17§lÚn2.9��TD(5, 4)§^Ä��E{éc4�|�¤k

:§��|�1�:\�6§Ù{:\�0§��.�24461�GDC"

O\5�Ã¡:§¿3��24�|þëÓÃ¡:W\.�21251�GDC§3�

��6�|þëÓÃ¡:W\�`(11, 5, 4)3è��¤I�è"

éu ∈ {12, 15, 16}½u ≥ 18§y²�Ún3.39¥�y²aq"�Ún3.28§

Ún3.31–3.35¥�E�.�gtm1�GDC§Ù¥g ∈ {18, 24, 30}§m ∈ {18, 24,

30, 42, 48, 60, 66}"O\5�Ã¡:§¿3��g�|þëÓÃ¡:W\.

�2g2 51�GDC£Ún3.43¤§3��m�|þëÓÃ¡:W\�Ý�m + 5�

�`è£Ún3.42¤§Ò��Ý�gt+m+ 5��`è"

(ÜÚn3.41§3.42Ú3.44§·��Xe(Jµ

½n 3.45. é?¿��ên ≡ 5 (mod 6)§n ≥ 11§A3(n, 5, 4) = U(n, 3)"

e. ��ÝÝÝn ≡ 3 (mod 6)��

3�!¥§·�ò(½�n ≡ 3 (mod 6)�§A3(n, 5, 4)��"

Ún 3.46. A3(15, 5, 4) = U(15, 3)"

28 Aa|Ü?è¯KïÄ

y². -:8�Z15§·�¤I��67�èidüÜ©|¤"1�Ü©�¹ü�

èi(01, 32, 61, 121)§(02, 62, 91, 122)"1�Ü©�¹Xe65�èi"Xe13�è

i¥�z��ò3fs§s ∈ [0, 4]��^e)¤5�èi§Ù¥µ

fs(ai) = b1+bj/15c§

b ∈ [0, 14]§ b ≡ a+ 6s (mod 15)§

j ∈ [0, 29]§ j ≡ a+ 15i+ 6s (mod 30)"

(01, 11, 41, 72) (31, 82, 112, 01) (31, 132, 41, 142) (21, 32, 71, 112) (01, 91, 111, 131)

(31, 42, 21, 12) (41, 61, 111, 82) (41, 122, 21, 141) (11, 21, 51, 102) (21, 111, 121, 132)

(01, 101, 62, 21) (01, 102, 122, 71) (51, 131, 142, 62)

Ún 3.47. �3��.�142151§��987�GDC"

y². -:8X = {0, 1, . . . , 56}§|G = {{i} : 0 ≤ i ≤ 41}∪{{42, 43, . . . , 56}}"@o(X,G, C)Ò´.�142151§��987�GDC§XJC´d[162, L IX]¥�

èidgÓ�+G = 〈(0 2 4 · · · 40)(1 3 5 · · · 41)(42 43 44)(45 46 47)(48 49 50)

(51 52 53)(54 55 56)〉Ðm��"

Ún 3.48. é¤kn ∈ {15, 21, 27, 33, 39, 45}§�3.�1n−331§�� (n−3)(2n+3)6

�GDC§ÏdA3(n, 5, 4) ≥ U(n, 3)− 1"

y². é?¿n ∈ {15, 21, 27, 33, 39, 45}§-:8�Xn = {0, 1, . . . , n− 1}§|8�Gn = {{i} : 0 ≤ i ≤ n− 4} ∪ {{n− 3, n− 2, n− 1}}"@o(Xn,Gn, Cn)Ò´.

�1n−331§�� (n−3)(2n+3)6

�GDC§XJCn´d[162, L X]¥�èidXeg

Ó�+GÐm��"

�n = 15�§G = 〈(0 4 8)(1 5 9)(2 6 10)(3 7 11)(12 13 14)〉"�n ∈ {21, 27,

33, 39, 45}�§G = 〈(0 2 4 · · · n− 5)(1 3 5 · · · n− 4)(n− 3 n− 2 n− 1)〉"

Ún 3.49. é?¿n ≡ 3 (mod 12)§n ≥ 51§A3(n, 5, 4) = U(n, 3)"

y². é?¿n§lÚn3.21��.�12u�GDC§O\3�Ã¡:§¿3

cu − 1�|ëÓÃ¡:W\.�11231�GDC§3��|ëÓÃ¡:W

\�`(15, 5, 4)3èÒ��¤I�è"

Ún 3.50. é?¿n ≡ 9 (mod 12)§n ≥ 57§A3(n, 5, 4) = U(n, 3)"


y². én = 57§lÚn3.47��.�142151�GDC§3�Ý�15�|þW\

�`(15, 5, 4)3èÒ��¤I�è"

é?¿n ∈ {69, 81, 93, 105, 117, 129}§lÚn3.22��.�12u181§4 ≤u ≤ 9�GDC"O\3�Ã¡:§3��12�|þëÓÃ¡:W\�

`(15, 5, 4)3è§3Ù§�|þëÓÃ¡:W\.�11231½11831�GDCÒ��

¤I�è"

én = 141§��.�2951�{4}-GDD£�©[69]¤§^Ä��E{é

¤k:\�6Ò��.�129301�GDC"O\3�Ã¡:§3��

�12�|þëÓÃ¡:W\�`(15, 5, 4)3è§¿3Ù§|þëÓÃ¡:W\.

�11231½13031�GDCÒ��¤I�è"

én = 177§��.�285181�{4}-GDD£�©[66]¤§^Ä��E{é

¤k:\�6��.�128301481�GDC"O\3�Ã¡:§¿3��

�48�|ëÓÃ¡:W\�`(51, 5, 4)3è§3Ù§|þëÓÃ¡:W\.

�11231½13031�GDC��¤I�è"

én = 189§lÚn2.9��TD(7, 8)"^Ä��E{éc5�|�¤k

:Ú16�|�6�:\�3§é��|�¤k:\�6§Ù{:\�0Ò

��.�245181481�GDC"ùp§Ñ\�´.�3561Ú3661�GDC£í

Ø3.4ÚÚn3.19¤"O\3�Ã¡:§3��48�|þëÓÃ¡:W\�

`(51, 5, 4)3è§3Ù{�|þW\.�12431½11831�GDCÒ��¤I�è"

én ∈ {153, 165}½n ≥ 201§lÚn2.9��TD(6, t)"^Ä��E{é

c4�|�¤k:§15�|�x�:§��|�2�:\�6§é��

|��:\�3§Ù{:\�0§·�Ò��.�(6t)4(6x)1151�GDC§

Ù¥x = 0½3 ≤ x ≤ t"3z�|þW\�Ý�6t§6x£½n3.37¤½15£Ú

n3.46¤��`èÒ��Ý�6(4t+x) + 15��`è§ùp4t+x�±��23§

25½?ÛØ�u31�Ûê"

nÜXþÚn§·��µ

½n 3.51. é?¿n = 15½n ≡ 3 (mod 6)§n ≥ 51§A3(n, 5, 4) = U(n, 3)¶é

?¿n ∈ {21, 27, 33, 39, 45}§A3(n, 5, 4) ≥ U(n, 3)− 1"

30 Aa|Ü?è¯KïÄ

3.4 (((ØØØ

3�Ù¥§·�A��(½þ�4§��ål�5��`n�~è

�èi�ê"·�rÌ�(Jo(Xeµ

½n 3.52. é?¿��ên ≥ 4§

A3(n, 5, 4) =

1§ �n = 4�

2§ �n = 5�

4§ �n = 6�

7§ �n = 7�

13§ �n = 8�

19§ �n = 9�⌊n2

⌊2(n−1)

3

⌋⌋§ �n ≥ 10§n 6∈ {12, 13, 21, 27, 33, 39, 45, 52}�

A3(n, 5, 4) ∈[⌊n

2

⌊2(n− 1)

3

⌋⌋− 1§

⌊n

2

⌊2(n− 1)

3

⌋⌋]§

é?¿n ∈ {12, 21, 27, 33, 39, 45}¶A3(13, 5, 4) ∈ [48, 52]¶A3(52, 5, 4) ∈ [872, 884]"

Chapter 4

^222ÂÂÂSteinerXXX��EEE��`ooo��~~~èèè


Etzion3©[56]¥ÄkJÑ2ÂSteinerXGS(2, k, n, g)�Vg§¿^5�

E�Ý�n§þ�k§ål�2k − 3��`g + 1�~è"

'uGS(2, k, n, g)��Ü©ó�Ñ8¥3k = 3��¹£�©[10, 32, 33, 56,

62–64, 73, 116, 117, 151, 156]¤§éÙ§k�(J%Øõ"3c�Ù¥§·�y²

�k = 4§g = 2�§GS(2, 4, n, 2)�7�^�Øn ∈ {7, 13, 52}Ñ´¿©�£��©[67, 97, 152, 162]¤"�k = 4§g = 3�§GS(2, 4, n, 3)�k�ïÄ(J

£�©[74, 75, 150, 165]¤"¦+©Ù¥®²��GS(2, 4, n, 3)��E�{

ÚÃ¡a§�´ålù�¯K��)û�é�"©[75]¥�ÑGS(2, 4, n, 3)�

3�7�^�´n ≥ 8§n ≡ 0, 1 (mod 4)"

Ún 4.1 (Ge§Wu [75]). ¤k�ên ≡ 1 (mod 4)§n > 13§�3GS(2, 4, n, 3)"

Ún 4.2 (Ge§Wu [74]). XJm ≡ 0, 1 (mod 4)§m ≥ 8§n ≡ 1 (mod 4)´�

ê§n > 13§@o�3��GS(2, 4,mn, 3)Ú��GS(2, 4,m(n− 1) + 1, 3)"

Ún 4.3 (Zhu§Ge [165]). XJm ≥ 14§2n+ 1 ≡ 1 (mod 4)´�ê§2n+ 1 >

13§@o�3��GS(2, 4, 2mn+ 1, 3)Ú��GS(2, 4,m(2n+ 1), 3)"

3�Ù¥§·�òé¤k�Ý�n§ïÄ�`(n, 5, 4)4è��E"·�Ø

55�Ø(½��Ý§(½¤k�Ý��`(n, 5, 4)4è�èi�ê"AO

�§�n ≡ 0, 1 (mod 4)�§·��k7�Ø(½��§�Ò´`Ø7�Ø(½

��±§GS(2, 4, n, 3)�3�7�^��´¿©�"

dÚn2.2§·��µ

íØ 4.4. A4(n, 5, 4) ≤⌊3n(n−1)

4

⌋:= U(n, 4)"

32 Aa|Ü?è¯KïÄ

ù�Ù�(�Xeµ314.2!¥§·�ò0��Ä�VgÚÄ��E�

{¶314.3!¥§·�ò�Ñ��SDP�E�{¶314.4!¥§·�ò©�

¹�E�`(n, 5, 4)4 è¶314.5!¥§òé�Ù�Ì�(J?1o("


-N = In × Ig"3e¡�E¥§·�ò¦^þ�Ùframe½Â¥�ÎÒ"

Ji�3©[97]¥JÑXe�E�{"

�E 4.5 (Ji�[97]).£W|¤b�(N,G,H,B)´��frame-GS(2, k, T , g)"?�

Úb�é?¿P ∈ P§Ñ�3��GS(2, k, |P |, g) (P×Ig, {{i}×Ig : i ∈ P},BP )"

@o§(N, {{i} × Ig : i ∈ In},B ∪ (∪P∈PBP ))´��GS(2, k, n, g)"

�E 4.6 (Ji�[97]).£O\s�:¤b�(N,G,H,B)´��frame-GS(2, k, T , g)"

-S´��s��InØ��8Ü"?�Úb�µ

i) é?¿P0 ∈ P§Ñ�3��GS(2, k, |P0| + s, g) ((P0 ∪ S) × Ig, {{i} × Ig :

i ∈ P0 ∪ S},BP0)¶

ii) é?¿P ∈ P \ {P0}§Ñ�3��frame-GS(2, k, (1|P |s1), g)§((P ∪ S) ×Ig, {{i} × Ig : i ∈ P} ∪ {S × Ig}, {{i} × Ig : i ∈ P ∪ S},BP )"

@o§((In∪S)×Ig, {{i}×Ig : i ∈ In∪S}, B∪(∪P∈PBP ))´��GS(2, k, n+s, g)"

�E 4.7 (Ji�[97]). £Ä��E{¤ -(X,G,B)´��GDD§ω : X → Z≥0´��\�¼ê"é?¿S ⊆ X§-S = ∪x∈S(({x} × Zω(x)) × Ig)"b�é

?¿B ∈ B§Ñ�3��frame-GS(2, k, {ω(x) : x ∈ B}, g) (B, {{x} : x ∈B}, {{(x, y)}×Ig : (x, y) ∈ B×Zω(x)},BB)"@o(X, {G : G ∈ G}, {{(x, y)}×Ig :

(x, y) ∈ X × Zω(x)},∪B∈BBB)´��frame-GS(2, k, {∑

x∈G ω(x) : G ∈ G}, g)"

�E 4.8 (Ji�[97]). £)ä{¤ b�(N,G,H,B)´��frame-GS(2, k, T , g)"

-N = (In × Zm) × Ig§ G = {(P × Zm) × Ig : P ∈ P}§H = {{(i, l)} ×Ig : (i, l) ∈ In × Zm}"?�Úb��3TD(k,m)§({(i, l, α) : (i, α) ∈ B, l ∈Zm}, {{(i, l, α) : l ∈ Zm} : (i, α) ∈ B},BB)§B ∈ B"@o(N , G, H,∪B∈BBB)´

��frame-GS(2, k, (mT ), g)"

CHAPTER 4 ^2ÂSTEINERX�E�`o�~è 33

·�`��K-GDDäk“(”5�§P�K-*GDD§XJ?¿ü�«|�

õ�uü�ú��|"K-*GDD�½ÂÄk3©[33]¥JÑ§�^5�E2

ÂSteinerX"

Ún 4.9 (Ge§Wu [74]). é?¿v ≡ 0, 1 (mod 4)§v ≥ 8§�3��.�3v�{4}-*GDD"

(ÜÚn2.5ÚÚn2.6¥�(J§·��µ

íØ 4.10. é?¿v ≥ 11§v 6∈ [11, 30] ∪ [32, 41] ∪ [45, 47] ∪ {101, 155, 160,

166, 167, 185}§�3��(v, {6, 7, 8, 9, 10}, 1)-PBD"

4.3 ��SDP��EEE{{{

3©[74]¥§GeÚWu�Ñ�SIP�E§¿32ÂSteinerX��E¥u

��^"3�!¥§·�ò�Ñ��[74, �ESIP-3]aq��E�{"

Äk§·�I��ü��L�Vg"

-X = {1, 2, . . . , v}§L´��Xþ�v2 × kÝ"·�¡L´��L§P�OA(k, v)§XJéL�?¿v2 × 2fÝ§X × X¥�?¿kS:éÑT

ÐÑy�g"b�L = (eij)´��OA(k, v)§1 ≤ i ≤ v2§1 ≤ j ≤ k"Ri =

(ei1, . . . , eik)¡�L��þ"b�L1, L2, . . . , Lr´�Ó8Üþ�r�OA(k, v)"

ùr�OA(k, v)�¡�´�ü�§XJL1, L2, . . . , Lr¥�?¿ü��þÑ�õk

ü� ��Ó"

Ún 4.11 (Ge [65]). é?¿w ≥ 4§w 6≡ 2 (mod 4)§�3w��ü�OA(4, w)"

�E 4.12. -g§w§r§h§uÚs��K�ê§1 ≤ r ≤ w"b�Xe�OÑ�

3µ

(1) ��.�(gh)u�{4}-GDD§¿�¤k�«|�±©¤r�8ÜS0§S1§

. . .§Sr−1§z�|�±©¤h��g�f|§¦�é?¿0 ≤ a ≤r − 1§dSa��è3f|þ��ålÑ´5¶

(2) r��ü�OA(4, w)"

@o�3��frame-GS(2, 4, ((hw)u), g)"e��3��frame-GS(2, 4, (1hws1), g)

Ú��GS(2, 4, hw + s, g)§@o�3��GS(2, 4, uhw + s, g)"

34 Aa|Ü?è¯KïÄ

y². -(X0,G0,B0)´��÷vXþ^��.�(gh)u�{4}-GDD§Ù¥

X0 = (Zu × Zh)× Zg§

G0 = {({i} × Zh)× Zg : i ∈ Zu}§B0 =r−1⋃a=0

Sa"

G0�f|´H0 = {{(i, j)} × Zg : (i, j) ∈ Zu × Zh}"é?¿«|B =

{[i1, j1, α1], [i2, j2, α2], [i3, j3, α3], [i4, j4, α4]} ∈ B0§Ø�b�i1 < i2 < i3 < i4"

-L0§L1§. . .§Lr−1´Zwþ�r��ü�OA(4, w)"y3§·�^Xe�

ª�E#�«|"-B =⋃r−1a=0 Va§Ù¥µ

Va = {{[i1, j1, l1, α1], [i2, j2, l2, α2], . . . , [i4, j4, l4, α4]} :

{[i1, j1, α1], . . . , [i4, j4, α4]} ∈ Sa, (l1, . . . , l4) ∈ La}"

-

X = (Zu × (Zh × Zw))× Zg§G = {({i} × (Zh × Zw))× Zg : i ∈ Zu}"

@o§ØJwÑ(X,G,B)´��.�(hwg)u�{4}-GDD"

e¡§·�ò`²(X,G,H,B)�´��É8�

H = {{(i, j, l)} × Zg : (i, j, l) ∈ Zu × (Zh × Zw)}

�frame-GS(2, 4, (hw)u), g)"

ÄK§b��3ü�«|A,A′ ∈ B�ål�u5"b�A = {[i1, j1, l1, α1],

. . . , [i4, j4, l4, α4]}§A′ = {[i′1, j′1, l′1, α′1], . . . , [i′4, j′4, l′4, α′4]}"XJAÚA′�ål�u5§@o��÷ve¡ü«�¹��µ

(1) AÚA′�k��ú�:§��u3�ú��f|"@o|{[i1, j1, l1], . . . ,[i4, j4, l4]} ∪ {[i′1, j′1, l′1], . . . , [i′4, j′4, l′4]}| ≥ 3"·��±b�A ÚA′�cn

�I©O�{[i1, j1, l1], [i2, j2, l2], [i3, j3, l3], [i4, j4, l4]}§{[i1, j1, l1], [i2, j2, l2],[i3, j3, l3], [i

′4, j′4, l′4]}§�α1 = α′1"·��{[i1, j1, α1], [i2, j2, α2], [i3, j3, α3],

[i4, j4, α4]}Ú{[i1, j1, α1], [i2, j2, α′2], [i3, j3, α

′3], [i

′4, j′4, α

′4]}ål�õ�4§§

��½´^ØÓ�OA)ä"Ïd§·�kli = l′i§i = 1, 2, 3§ù��

ü^��gñ"


(2) AÚA′vkú�:§�´�u4�ú��f|"{[i1, j1, l1], . . . , [i4, j4, l4]} =

{[i′1, j′1, l′1], . . . , [i′4, j′4, l′4]}"Ï�i1 < i2 < i3 < i4§i′1 < i′2 < i′3 < i′4§·�

ké?¿1 ≤ i ≤ 4§li = l′i", §{[i1, j1, α1], . . . , [i4, j4, α4]}Ú{[i1, j1, α′1],. . . , [i4, j4, α

′4]}�ål�4§§��½´©O^ØÓ�OA)ä"�Ò´

`(l1, . . . , l4)Ú(l′1, . . . , l′4) �½´ü�ØÓOA��þ§ù�1 ≤ i ≤ 4§

li = l′igñ"

-S = {∞1,∞2, . . . ,∞s}"b�(Xi,Gi,Hi,Bi)§i ∈ {0, 1 . . . , u−2}´frame-

GS(2, 4, (1hws1), g)§Ù¥µXi = (({i} × (Zh × Zw)) × Zg) ∪ (S × Zg)§Gi =

{{(i, j, l)} × Zg : (j, l) ∈ Zh × Zw} ∪ {S × Zg}§Hi = {{(i, j, l)} × Zg : (j, l) ∈Zh × Zw} ∪ {{m} × Zg : m ∈ S}"

2b�(Xu−1,Gu−1,Bu−1)´��GS(2, 4, hw + s, g)§Ù¥µXu−1 = (({u −1} × (Zh × Zw)) × Zg) ∪ (S × Zg)§Gu−1 = {{(u − 1, j, l)} × Zg : (j, l) ∈ Zh ×Zw} ∪ {{m}×Zg : m ∈ S}"@o(X ′,G ′,B′)´��GS(2, 4, uhw+ s, g)§Ù¥µ

X ′ = ((Zu × (Zh × Zw))× Zg) ∪ (S × Zg)§

G ′ = {{(i, j, l)} × Zg : (i, j, l) ∈ Zu × (Zh × Zw)} ∪ {{m} × Zg : m ∈ S}§

B′ = B ∪ (⋃i∈Zu

Bi)"

ùp�GDD´��|.�§y²��E4.12�Ó§�´Lã�E,§¤

±·�re¡�E�y²�Ñ"

�E 4.13. �g§w§r§sÚgi§ui§hi§i = 1, 2, . . . , t´�K�ê§¦�é?

¿i = 1, 2, . . . , t§1 ≤ r ≤ w§gi = hig"b�Xe�OÑ�3µ

(1) ��.�gu11 gu22 . . . gutt �{4}-GDDäkXe5�§¤k«|�±y©¤r�

8ÜS0§S1§. . .§Sr−1§z��gi�|�±y©¤hi��g �f

|§¦�é?¿0 ≤ a ≤ r − 1§Sa��è3f|þ��ålÑ´5¶

(2) r��ü�OA(4, w)"

36 Aa|Ü?è¯KïÄ

@o�3��frame-GS(2, 4, ((h1w)u1(h2w)u2 . . . (htw)ut), g)"?�Ú§XJé?

¿1 ≤ i ≤ t−1§Ñ�3��frame-GS(2, 4, (1hiws1), g)§��3��GS(2, 4, htw+

s, g)§@oÒ�3��GS(2, 4,∑t

i=1 uihiw + s, g)"

3�E4.12�h = 1§·��Xe(Jµ

½n 4.14. -g§w§r§sÚuÑ´�K�ê§1 ≤ r ≤ w"b�e��OÑ�

3µ

(1) ��.�gu�{4}-GDDäkXe5�µ¤k�«|�±y©¤r�8

ÜS0§S1§. . .§Sr−1§¦�é?¿0 ≤ a ≤ r − 1§Sa��è��å

lÑ´5¶

(2) r��ü�OA(4, w)"

@o�3��frame-GS(2, 4, (wu), g)"?�Ú§e�3frame-GS(2, 4, (1ws1), g)

ÚGS(2, 4, w + s, g)§@oÒ�3GS(2, 4, uw + s, g)"


a. ��frame-GSÚÚÚ��`èèè

Ún 4.15. é?¿u ∈ [6, 10]§�3��frame-GS(2, 4, (4u), 3)"

y². é?¿u ∈ {6, 7, 10}§:8�{0, 1, 2, . . . , 12u− 1}§|8�{{0, u, 2u, . . . ,11u}+ i : 0 ≤ i ≤ u− 1}§É8�{{0, 4u, 8u}+ i : 0 ≤ i ≤ 4u− 1}"¤I«|de¡�Ä«|ÏLgÓ�+G = 〈(0 1 2 · · · 4u− 1)(4u 4u + 1 4u + 2 · · · 8u−1)(8u 8u+ 1 8u+ 2 · · · 12u− 1)〉Ðm��"u = 6µ

{0, 4, 5, 69} {0, 7, 9, 53} {0, 13, 28, 50} {0, 55, 56, 59} {0, 25, 32, 51} {24, 63, 65, 56}{0, 63, 71, 52} {0, 3, 46, 47} {0, 8, 41, 57} {0, 27, 31, 58} {24, 32, 34, 69} {0, 37, 34, 62}{0, 29, 38, 67} {24, 35, 68, 58} {0, 10, 26, 45}

u = 7µ

{0, 1, 6, 52} {0, 11, 15, 59} {0, 53, 62, 64} {0, 2, 29, 67} {0, 16, 33, 38} {28, 68, 76, 79}{0, 54, 30, 48} {0, 3, 82, 60} {0, 36, 61, 80} {28, 31, 48, 74} {0, 8, 74, 58} {0, 44, 71, 75}{0, 39, 68, 69} {28, 40, 41, 73} {0, 9, 41, 43} {0, 40, 31, 81} {28, 62, 75, 80} {0, 10, 47, 83}


u = 10µ

{0, 1, 7, 94} {0, 2, 55, 111} {40, 44, 56, 93} {40, 43, 116, 91} {0, 69, 74, 56} {40, 108, 112, 86}{0, 5, 77, 46} {0, 78, 64, 99} {40, 61, 63, 69} {0, 66, 105, 107} {0, 45, 52, 119} {0, 83, 95, 106}{0, 9, 23, 58} {0, 44, 82, 89} {0, 43, 112, 85} {0, 16, 108, 102} {0, 4, 65, 117} {0, 13, 101, 104}{0, 3, 57, 42} {0, 22, 73, 97} {0, 15, 63, 118} {40, 106, 97, 98} {0, 62, 116, 81} {40, 41, 84, 103}{0, 8, 29, 76} {0, 12, 71, 96} {0, 67, 98, 114}

éu ∈ {8, 9}§�.�3u�{4}-*GDD£Ún4.9¤§Ún��ü�OA(4, 4)

£Ún4.11¤§¿A^½n4.14§Ò�±��¤I�frame-GS(2, 4, (4u), 3)"

Ún 4.16. é?¿m ∈ {6, 8}§�3��frame-GS(2, 4, (45m1), 3)"

y². -:8�{0, 1, 2, . . . , 59 + 3m}§|8�{{(20 +m)k, (20 +m)k + 5, (20 +

m)k + 10, (20 + m)k + 15 : 0 ≤ k ≤ 2} + i : 0 ≤ i ≤ 4} ∪ {{20 + (20 +

m)k, 20 + (20 + m)k + 1, . . . , 20 + (20 + m)k + m − 1 : 0 ≤ k ≤ 2}}§É8�{{0, 20 +m, 40 + 2m}+ i : 0 ≤ i ≤ 20 +m− 1}"¤I«|de¡Ä«|ÏLXegÓ�+GÐm��"

m = 6µG = 〈(0 3 6 9 12 15 18 27 30 33 36 39 42 45 54 57 60 63 66 69)(1 4 7 10 13 16 19 28 31 34 37 40 43

52 55 58 61 64 67 70)(2 5 8 11 14 17 26 29 32 35 38 41 44 53 56 59 62 65 68 71)(20 21 22 23 24)(25)(46 47

48 49 50)(51)(72 73 74 75 76)(77)〉"

{0, 1, 7, 14} {0, 11, 18, 46} {1, 10, 43, 75} {0, 2, 6, 49} {0, 13, 23, 27} {2, 46, 10, 14}{1, 26, 46, 61} {0, 22, 4, 28} {0, 16, 44, 56} {1, 33, 41, 74} {1, 4, 23, 59} {0, 37, 38, 77}{0, 17, 34, 63} {1, 47, 52, 54} {2, 3, 25, 58} {0, 29, 58, 76} {2, 20, 11, 41} {2, 5, 29, 72}{2, 4, 15, 51} {0, 30, 33, 72} {2, 21, 27, 39}

m = 8µG = 〈(0 3 6 9 12 15 18 29 32 35 38 41 44 47 58 61 64 67 70 73)(1 4 7 10 13 16 19 30 33 36 39 42 45

56 59 62 65 68 71 74)(2 5 8 11 14 17 28 31 34 37 40 43 46 57 60 63 66 69 72 75)(20 21 22 23 24)(25 26)(27)

(48 49 50 51 52)(53 54)(55)(76 77 78 79 80)(81 82)(83)〉"

{0, 7, 59, 77} {0, 19, 26, 74} {1, 38, 46, 83} {1, 2, 24, 61} {0, 23, 62, 70} {2, 43, 57, 76}{1, 47, 48, 59} {1, 4, 71, 77} {0, 29, 51, 73} {2, 10, 49, 59} {1, 5, 21, 65} {1, 22, 40, 66}{0, 34, 36, 82} {2, 11, 26, 47} {2, 9, 51, 66} {0, 42, 72, 81} {2, 16, 52, 75} {0, 18, 67, 80}{0, 11, 12, 21} {0, 45, 55, 60} {2, 34, 64, 77} {0, 13, 54, 63} {1, 12, 14, 27} {2, 41, 42, 54}

Ún 4.17. é?¿m ∈ {6, 8, 10}§�3��frame-GS(2, 4, (46m1), 3)"

y². -:8�{0, 1, 2 . . . , 71 + 3m}§|8�{{(24 +m)k, (24 +m)k + 6, (24 +

m)k + 12, (24 + m)k + 18 : 0 ≤ k ≤ 2} + i : 0 ≤ i ≤ 5} ∪ {{24 + (24 +

38 Aa|Ü?è¯KïÄ

m)k, 24 + (24 + m)k + 1, . . . , 24 + (24 + m)k + m − 1 : 0 ≤ k ≤ 2}}§É8�{{0, 24 +m, 48 + 2m}+ i : 0 ≤ i ≤ 24 +m− 1}"¤I«|de¡Ä«|ÏLXegÓ�+GÐm��"

m = 6µG = 〈(0 1 2 · · · 23)(24 25)(26 27)(28 29)(30 31 32 · · · 53)(54 55)(56 57)(58 59)(60 61 62 · · · 83)(84 85)(86 87)(88 89)〉"

{0, 1, 62, 86} {0, 33, 47, 49} {0, 74, 65, 89} {0, 3, 37, 58} {0, 35, 39, 79} {30, 58, 75, 62}{30, 25, 82, 63} {0, 4, 32, 63} {0, 38, 31, 54} {30, 27, 33, 68} {0, 5, 24, 46} {0, 11, 57, 75}{0, 40, 45, 88} {30, 28, 64, 67} {0, 7, 29, 50} {0, 51, 76, 80} {30, 31, 56, 77} {0, 70, 77, 85}{0, 8, 10, 81} {0, 53, 44, 84} {30, 41, 80, 87} {0, 9, 27, 67} {0, 55, 68, 69} {30, 71, 73, 81}

m = 8µG = 〈(0 1 2 · · · 23)(24 25 26 27 28 29)(30 31)(32 33 34 · · · 55)(56 57 58 59 60 61)

(62 63)(64 65 66 · · · 87)(88 89 90 91 92 93)(94 95)〉"

{0, 8, 9, 77} {0, 30, 66, 71} {0, 75, 84, 92} {0, 2, 45, 57} {0, 31, 35, 48} {0, 52, 47, 89}{0, 87, 74, 94} {0, 3, 10, 29} {0, 33, 86, 90} {32, 24, 33, 80} {0, 4, 53, 91} {0, 28, 39, 67}{0, 34, 36, 56} {32, 26, 36, 87} {0, 5, 58, 83} {0, 37, 40, 95} {32, 40, 49, 65} {64, 58, 66, 80}{0, 13, 54, 62} {0, 42, 79, 88} {32, 46, 57, 68} {0, 24, 55, 72} {0, 46, 85, 65} {32, 66, 67, 91}{32, 63, 74, 77} {0, 27, 73, 80} {0, 51, 60, 81}

m = 10µG = 〈(0 1 2 · · · 23)(24 25 26 27 28 29)(30 31 32 33)(34 35 36 · · · 57)(58 59 60 61 62 63)(64 65

66 67)(68 69 70 · · · 91)(92 93 94 95 96 97)(98 99 100 101)〉"

{0, 2, 3, 66} {0, 28, 37, 48} {0, 63, 78, 71} {34, 36, 61, 83} {0, 31, 45, 75} {0, 53, 73, 93}{34, 24, 69, 89} {0, 9, 29, 79} {0, 33, 87, 88} {34, 30, 37, 79} {0, 10, 32, 41} {68, 67, 73, 83}{0, 35, 59, 72} {0, 7, 51, 62} {0, 11, 76, 98} {0, 36, 69, 96} {0, 27, 56, 82} {34, 42, 90, 95}{0, 16, 39, 95} {0, 43, 65, 84} {34, 43, 44, 67} {0, 20, 81, 94} {0, 5, 38, 101} {0, 60, 90, 77}{34, 70, 73, 98} {0, 24, 49, 42} {0, 50, 54, 58} {34, 85, 87, 97} {0, 25, 83, 91} {34, 39, 77, 100}

Ún 4.18. �3��frame-GS(2, 4, (47101), 3)"

y². -:8�{0, 1, 2 . . . , 113}§|8�{{38k, 38k + 7, 38k + 14, 38k + 21 : 0 ≤k ≤ 2} + i : 0 ≤ i ≤ 6} ∪ {{28 + 38k, 29 + 38k, . . . , 37 + 38k : 0 ≤ k ≤ 2}}§É8�{{0, 38, 76} + i : 0 ≤ i ≤ 37}"¤I«|de¡Ä«|ÏLgÓ�+G =〈(0 1 2 · · · 27)(28 29 30 31 32 33 34)(35 36)(37)(38 39 40 · · · 65)(66 67 68 69 70 7172)(73 74)(75)(76 77 78 · · · 103)(104 105 106 107 108 109 110)(111 112)(113)〉Ðm��"

{0, 1, 87, 92} {0, 5, 39, 109} {0, 3, 63, 111} {38, 67, 89, 92} {0, 50, 51, 79} {38, 77, 88, 111}{0, 6, 28, 54} {38, 29, 53, 80} {0, 56, 64, 96} {38, 32, 85, 86} {0, 68, 95, 82} {42, 76, 78, 109}{0, 8, 71, 88} {38, 36, 43, 84} {0, 11, 15, 66} {0, 31, 101, 81} {38, 48, 64, 71} {0, 73, 84, 103}{0, 2, 12, 32} {0, 37, 65, 100} {0, 19, 36, 89} {0, 94, 78, 105} {0, 41, 85, 106} {0, 55, 58, 110}{0, 33, 40, 44} {38, 72, 81, 91} {0, 47, 75, 77} {0, 53, 102, 113} {38, 30, 99, 103} {0, 93, 99, 108}{0, 43, 49, 67} {0, 57, 46, 107} {0, 42, 61, 74}


b. ��ÝÝÝn ≡ 0, 1 (mod 4)��

ÏLO�Å|¢�±éN´(½�`(5, 5, 4)4è�k3�èi§~Xµ

{30122, 12011, 21230}"

Ún 4.19. é?¿n ∈ {17, 29, 37, 41, 53, 61, 73, 89, 101, 109, 149, 157}§�3��GS(2, 4, n, 3)"

y². ¤I�OdÚn4.1��"

Ún 4.20. é?¿n ∈ [16, 60]4 ∪ {72, 92}§�3��GS(2, 4, n, 3)"

y². -:8�{0, 1, 2, . . . , 3n− 1}§|8�{{0, n, 2n}+ i : 0 ≤ i ≤ n− 1}"¤I«|d[161,L I]¥�Ä«|ÏLgÓ�+G = 〈(0 1 2 · · · n−2)(n−1)(n n+

1 n+ 2 · · · 2n−2)(2n−1)(2n 2n+ 1 2n+ 2 · · · 3n−2)(3n−1)〉Ðm��"

Ún 4.21. é?¿n ∈ {21, 25, 33, 45, 49, 57, 69}§�3��GS(2, 4, n, 3)"

y². -:8�{0, 1, 2, . . . , 3n− 1}§|8�{{0, n, 2n}+ i : 0 ≤ i ≤ n− 1}"¤I«|d[161,L II]¥�Ä«|ÏLgÓ�+G = 〈(0 1 2 · · · n− 1)(n n+ 1 n+

2 · · · 2n− 1)(2n 2n+ 1 2n+ 2 · · · 3n− 1)〉Ðm��"

w,§�s ∈ {0, 1}�§��frame-GS(2, 4, (1ws1), 3)Ò´GS(2, 4, w + s, 3)"

A^½n4.14ÚÚn4.11§·��Xe(J"

Ún 4.22. é?¿n ∈ {64, 68, 76, 80, 84, 100, 116, 148, 156} ∪ {65, 77, 81, 85, 93}§�3��GS(2, 4, n, 3)"

y². Ï��.34�{4}-GDDk9�«|§��.�35�{4}-GDDk15�«

|§·��±r«|8y©¤z�Ü©��«|"�g = 3§ÚL4.1¥�ë

ê(u,w, s)§�±��A�GS(2, 4, n, 3)"

Ún 4.23. e�frame-GSþ�3µ

i) frame-GS(2, 4, (16u), 3)§u ∈ [6, 10]¶

ii) frame-GS(2, 4, (20u), 3)§u ∈ {7, 9}¶

40 Aa|Ü?è¯KïÄ

L 4.1: Ún4.22¥�ëê(u,w, s)n (u,w, s) n (u,w, s) n (u,w, s) n (u,w, s)

64 (4, 16, 0) 65 (4, 16, 1) 68 (4, 17, 0) 76 (5, 15, 1)

77 (4, 19, 1) 80 (5, 16, 0) 81 (4, 20, 1) 84 (4, 21, 0)

85 (5, 17, 0) 93 (4, 23, 1) 100 (5, 20, 0) 116 (5, 23, 1)

148 (4, 37, 0) 156 (5, 31, 1)

iii) frame-GS(2, 4, (165241), 3)¶

iv) frame-GS(2, 4, (167401), 3)¶

v) frame-GS(2, 4, (4010), 3)"

y². i)§lÚn4.15�frame-GS(2, 4, (4u), 3)§u ∈ [6, 10]^4)ä§��I�

�frame-GS"ii)§�frame-GS(2, 4, (4u), 3)§u ∈ {7, 9}§^5)ä��¤I�O"

iii)§lÚn4.16�frame-GS(2, 4, (4561), 3)¿^4)ä" iv)§lÚn4.18�frame-

GS(2, 4, (47101), 3)¿^4)ä"v)�frame-GS(2, 4, (410), 3)¿^10)ä"

íØ 4.24. é?¿n ∈ {96, 104, 112, 128, 140, 144, 152, 160, 180, 400}§©O�3��GS(2, 4, n, 3)Ú��GS(2, 4, n+ 1, 3)"

y². �Ún4.23¥�frame-GS"XJ3|þW\GS(2, 4,m, 3)§m ∈ {16, 20,

24, 40}£Ún4.20¤§·��I��GS(2, 4, n, 3)"XJO\��Ã¡:§¿

3|þëÓÃ¡:W\GS(2, 4,m + 1, 3)£Ún4.19Ú4.21¤§·��I�

�GS(2, 4, n+ 1, 3)"

Ún 4.25. é?¿n ∈ {124, 132, 136, 176, 184}§�3��GS(2, 4, n, 3)Ú�

�GS(2, 4, n+ 1, 3)"

y². lÚn2.9¥��TD(6, t)§t ∈ {5, 7}§^Ä��E{éc5�|�¤k

:§��|�x�:\�4§��|�y�:\�8§Ù¥x+y = t"ùp

Ñ\�O�frame-GS(2, 4, (46), 3)Úframe-GS(2, 4, (4581), 3)£Ún4.15Ú4.16¤"

·��frame-GS(2, 4, ((4t)5(4x+ 8y)1), 3)"

XJ3ùframe-GS�|þW\GS(2, 4, 4t, 3)ÚGS(2, 4, 4x + 8y, 3)£Ú

n4.20¤§·��GS(2, 4, n, 3)§Ù¥n = 20t + 4x + 8y"e(t, x, y) =

(5, 4, 1)§·�kGS(2, 4, 124, 3)¶e(t, x, y) = (5, 2, 3)§·�kGS(2, 4, 132, 3)¶


e(t, x, y) = (5, 1, 4)§·�kGS(2, 4, 136, 3)¶e(t, x, y) = (7, 5, 2)§·��

�GS(2, 4, 176, 3)¶e(t, x, y) = (7, 3, 4)§·�kGS(2, 4, 184, 3)"

XJO\��Ã¡:§¿3|þëÓÃ¡:W\GS(2, 4, 4t+1, 3)ÚGS(2, 4,

4x+ 8y + 1, 3)£Ún4.19Ú4.21¤§�þ¡�Ó�n�|(t, x, y)§·��±��

�A�GS(2, 4, n+ 1, 3)"

Ún 4.26. é?¿n ∈ {616, 636, 660, 664, 736}§�3��GS(2, 4, n, 3)Ú�

�GS(2, 4, n+ 1, 3)"

y². lÚn2.9��TD(8, 24)§^Ä��E{éc6�|�¤k:§17�

|�x�:§��|�y�:\�4§Ù{:\�0"ùpÑ\�O�frame-

GS(2, 4, (4u), 3)§u ∈ {6, 7, 8}"·��frame-GS(2, 4, (966(4x)1(4y)1), 3)"

XJ·�3ùframe-GS�|þ©OW\GS(2, 4, 96, 3)§GS(2, 4, 4x, 3)½

öGS(2, 4, 4y, 3)£Ún4.20§Ún4.22§íØ4.24¤§·��GS(2, 4, 576 +

4x+ 4y, 3)"e(x, y) = (10, 0)§·�kGS(2, 4, 616, 3)¶e(x, y) = (15, 0)§·�

kGS(2, 4, 636, 3)¶e(x, y) = (15, 6)§·�kGS(2, 4, 660, 3)¶e(x, y) = (15, 7)§

·�kGS(2, 4, 664, 3)¶e(x, y) = (24, 16)§·�kGS(2, 4, 736, 3)"

XJO\��Ã¡:§¿3|þëÓÃ¡:W\GS(2, 4, 97, 3)§GS(2, 4,

4x + 1, 3)½öGS(2, 4, 4y + 1, 3) £Ún4.19§4.20Ú4.22¤§�þ¡�Ó�n�

|(t, x, y)§·��±��A�GS(2, 4, n+ 1, 3)"

-P = [10, 29] ∪ [31, 40] ∪ [44, 46] ∪ {100, 154, 159, 165, 166, 184}"

Ún 4.27. é?¿t ≥ 10§t 6∈ P§�3��GS(2, 4, 4t, 3)Ú��GS(2, 4, 4t +

1, 3)"

y². é?¿t ≥ 10§t 6∈ P§dÚn4.10§�3��(t + 1, {6, 7, 8, 9, 10}, 1)-

PBD"lù�PBD�:8�K��:§��.�5i6j7k8l9m�{6, 7, 8, 9, 10}-GDD§Ù¥5i + 6j + 7k + 8l + 9m = t"^Ä��E{\�4��frame-

GS(2, 4, (20i24j28k32l36m), 3)"ùp§Ñ\�O�frame-GS(2, 4, (4u), 3)§u ∈[6, 10]£Ún4.15¤"

XJ3ùframe-GS�|þW\GS(2, 4,m, 3)§m ∈ {20, 24, 28, 32, 36}£Ún4.20¤§·��¤IGS(2, 4, 4t, 3)"XJO\��Ã¡:¿3ùframe-

GS�|þëÓÃ¡:W\GS(2, 4,m + 1, 3)£Ún4.19 Ú4.21¤§·��¤

IGS(2, 4, 4t+ 1, 3)"

42 Aa|Ü?è¯KïÄ

nÜÚn4.19–4.22§íØ4.24ÚÚn4.25–4.27¥�(J§·��µ

½n 4.28. é?¿n ≡ 0, 1 (mod 4)§n ≥ 8§n 6∈ {8, 9, 12, 13, 88, 108, 117}§�3��GS(2, 4, n, 3)"

c. ��ÝÝÝn ≡ 2, 3 (mod 4)��

-gÚn��ê"��ÉW¿£holey packing¤§P�.�gn�K-HP§

´��n�|(X,G,B)§Ù¥X´��gn�8Ü£:8¤"G´��X�y©§¡�É£½ö|¤§òXy©¤n�Ü©§zÜ©g�:"B´X��f8x£«|¤§é?¿B ∈ B§|B| ∈ K§¿�?¿ü�ØÓ|�:é�õ�¹3��«|¥§Ó��|�:éØ�¹3?Û«|¥"

-PN(2, k, n, g)�W¿ê§=µ.�gn�{k}-HP��U�«|�ê"

©[156]¥�ÑPN(2, k, n, g) ≤ BN(2, k, n, g)§Ù¥µ

BN(2, k, n, g) =

⌊ngk

⌊(n−1)gk−1

⌋⌋− 1§e(n− 1)g ≡ 0 (mod k − 1)§

n(n− 1)g2 6≡ 0 (mod k(k − 1))¶⌊ngk

⌊(n−1)gk−1

⌋⌋§ÄK"

w,§en ≡ 2, 3 (mod 4)§A4(n, 5, 4) ≤ PN(2, 4, n, 3) ≤ BN(2, 4, n, 3) =

U(n, 4)−1"Ïd§XJk��frame-GS(2, 4, (1n−221), 3)§Òk�`(n, 5, 4)4è"

ù�!¥§·�ò�Eframe-GS(2, 4, (1n−221), 3)§n ≡ 2, 3 (mod 4)"

Ún 4.29. A4(6, 5, 4) = 9§A4(7, 5, 4) ∈ [15, 21]"

y². dÚn2.1§A4(6, 5, 4) ≤ 9"ÏLO�Å|¢§·�é��`(6, 5, 4)4èµ

{333300, 310033, 031011, 003132, 100321, 201203, 122002, 012220, 220110}"

dÚn2.1§A4(7, 5, 4) ≤ 21"ÏLO�Å|¢§·�é�k15�èi

�(7, 5, 4)4èµ{2102003, 3001201, 0120301, 0211002, 1300102, 1010031, 2013300,

2320010, 0033013, 0200233, 0101110, 0002322, 0332200, 3030120, 1203020}"

Ún 4.30. é?¿n ∈ [19, 83]4∪{91, 99}§�3��frame-GS(2, 4, (1n−221), 3)"

y². -:8�{0, 1, 2, . . . , 3n − 1}§|8�{{0, n, 2n} + i : 0 ≤ i ≤ n − 3} ∪{{n− 2, n− 1, 2n− 2, 2n− 1, 3n− 2, 3n− 1}}§É8�{{0, n, 2n} + i : 0 ≤ i ≤


n− 1}"¤I�O�«|�±d[161, L IV]Ú[161, L V]¥�Ä«|ÏLgÓ�

+G = 〈(0 1 2 · · · n− 3)(n− 2)(n− 1)(n n+ 1 n+ 2 · · · 2n− 3)(2n− 2)(2n−1)(2n 2n+ 1 2n+ 2 · · · 3n− 3)(3n− 2)(3n− 1)〉Ðm��"

éu ∈ {4, 5}§·�r.�3u�{4}-*GDD�«|8y©¤z�Ü©�k

��«|"éu ∈ {8, 9}§dÚn4.9�3��.�3u�{4}-*GDD"Ï�.

�gu�{4}-*GDD�«|8�±�õy©¤g�Ü©§¦�z�Ü©��è�

��ål´5"3½n4.14¥§-g = 3§s = 2§�Ún4.22 �y²aq§·�

�±��Xe(Jµ

Ún 4.31. é?¿n ∈ {87, 107, 127, 147, 155, 167} ∪ {70, 86, 102, 118, 134, 138,

150, 166, 170, 182, 198, 202, 206, 214, 230, 234, 246, 254, 262, 294, 298, 302, 402}§�3��frame-GS(2, 4, (1n−221, 3)"

y². én ∈ {87, 107, 127, 147, 167}§©O�u = 5§w ∈ {17, 21, 25, 29, 33}"én = 155§�u = 9§w = 17"én ∈ {70, 86, 102, 118, 134, 150, 166, 182, 198,

214, 230, 246, 262, 294}§©O�u = 4§w ∈ {17, 21, 25, 29, 33, 37, 41, 45, 49, 53,

57, 61, 65, 73}"én ∈ {138, 170, 202, 234, 298}§©O�u = 8§w ∈ {17, 21, 25,

29, 37}"én ∈ {206, 254, 302}§©O�u = 12§w ∈ {17, 21, 25}"én = 402§

�u = 16§w = 25"

Ún 4.32. é?¿n ∈ {123, 131, 135, 139, 171, 175, 179, 183}§�3��frame-

GS(2, 4, (1n−221), 3)"

y². lÚn2.9��TD(6, t)§t ∈ {5, 7}§^Ä��E{éc5�|�¤

k:\�4§é��|�x, y, z�:©O\�4, 6, 8§Ù¥x + y + z = t"

ùpÑ\�O�frame-GS(2, 4, (46), 3)£Ún4.15¤Úframe-GS(2, 4, (45m1), 3)§

m ∈ {6, 8}£Ún4.16¤"·��frame-GS(2, 4, ((4t)5(4x+6y+8z)1), 3)"O

\��Ã¡:§3dframe-GS�|þëÓÃ¡:W\GS(2, 4, 4t+ 1, 3)Úframe-

GS(2, 4, (14x+6y+8z−121), 3)§Ò��frame-GS(2, 4, (120t+4x+6y+8z−121), 3)"-n =

20t+ 4x+ 6y + 8z + 1"�En¤I�ëê(t, x, y, z)�3L4.2¥"

-Q = {10, 14, 15, 18, 20, 22, 26, 30, 34, 38, 46, 60} ∪ {29}"

Ún 4.33. é?¿n ≡ 3 (mod 4)§n ≥ 187§�3��frame-GS(2, 4, (1n−221), 3)"

44 Aa|Ü?è¯KïÄ

L 4.2: ½n 4.32¥�ëên (t, x, y, z) n (t, x, y, z) n (t, x, y, z) n (t, x, y, z)

123 (5, 4, 1, 0) 131 (5, 2, 1, 2) 135 (5, 1, 1, 3) 139 (5, 0, 1, 4)

171 (7, 6, 1, 0) 175 (7, 5, 1, 1) 179 (7, 3, 3, 1) 183 (7, 3, 1, 3)

y². lÚn2.9¥��TD(7, t)§^Ä��E{éc6�|�¤k:\�4§

é��|�x, y, z�:©O\�4, 6, 8"Ù{:\�0"ùpÑ\�O

�frame-GS(2, 4, (4u), 3)§u ∈ {6, 7}£Ún4.15¤§frame-GS(2, 4, (46m1), 3)§m ∈{6, 8}£Ún4.17¤"·��frame-GS(2, 4, ((4t)6(4x+ 6y + 8z)1), 3)"

a. é?¿k ≥ 7§k 6∈ Q§dÚn2.9�3��TD(7, k)"éN´�y�±��

Ü·�n�|(x, y, z)§¦�4x+ 6y + 8z ∈ [18, 38]4"

b. é?¿k ∈ Q \ {15, 30}§dÚn2.9�3��TD(7, k − 1)"éN´�y�

±��Ü·�n�|(x, y, z)§¦�4x+ 6y + 8z ∈ [42, 62]4"

c. é?¿k ∈ {15, 30}§dÚn2.9�3��TD(7, k − 2)"éN´�y�±�

�Ü·�n�|(x, y, z)§¦�4x+ 6y + 8z ∈ [66, 86]4"

O\��Ã¡:§,�3|þëÓÃ¡:W\GS(2, 4, 4t+1, 3)£½n4.28¤

Úframe-GS(2, 4, (14x+6y+8z−121), 3)£Ún4.30ÚÚn4.31¤§·�Ò��frame-

GS(2, 4, (124t+4x+6y+8z−121), 3)"-n = 24t + 4x + 6y + 8z + 1"·�Ò��

frame-GS(2, 4, (1n−221), 3)§Ù¥n�±��Xe«mµ

a’. é?¿k ≥ 7§k 6∈ Q§n ∈ [24k + 19, 24k + 39]4"

b’. é?¿k ∈ Q \ {15, 30}§n ∈ [24(k − 1) + 42 + 1, 24(k − 1) + 62 + 1]4 =

[24k + 19, 24k + 39]4"

c’. é?¿k ∈ {15, 30}§n ∈ [24(k − 2) + 66 + 1, 24(k − 2) + 86 + 1]4 =

[24k + 19, 24k + 39]4"

nÜþã(J§·��±��frame-GS(2, 4, (1n−221), 3)§Ù¥n�±

��Ø�u24× 7 + 19 = 187 �?Û�"

nÜÚn4.30§4.31§4.32Ú4.33¥�(J§·��µ


½n 4.34. é?¿n ≥ 7§n ≡ 3 (mod 4)§n 6∈ {7, 11, 15, 95, 103, 111, 115,

119, 143, 151, 159, 163}§�3��frame-GS(2, 4, (1n−221), 3)"

½n 4.35. b�e��OÑ�3µ

(1) ��.�34m+161�{4}-GDD§¿äkXe5�µ¤k�«|�±©¤r�

Ü©§¿��6�|�±©¤��3�f|§¦�z�Ü©��

è'uf|��ålÑ´5¶

(2) r��ü�OA(4, w)¶

(3) ��GS(2, 4, w + 1, 3)¶

(4) ��frame-GS(2, 4, (12w−121), 3)"

@o�3��frame-GS(2, 4, (1(4m+1)w+2w−121), 3)"

y². d^�(1)Ú(2)§·�d�E4.13��frame-GS(2, 4, (w4m+1(2w)1), 3)"

O\��Ã¡:§3��w�|þëÓÃ¡:W\�½�GS(2, 4, w + 1, 3)§

3��2w�|ëÓÃ¡:W\�½�frame-GS(2, 4, (12w−1)21, 3)"·�Ò�

��frame-GS(2, 4, (1(4m+1)w+2w−121), 3)"

íØ 4.36. é?¿n ∈ {106, 162, 190, 210, 218, 226}§�3��frame-GS(2, 4,

(1n−221), 3)"

y². éN´é�.�3561§3961§31361§�«|8�õ�±y©¤15�Ü©

�{4}-GDD"¤±ùp·��ÑäN�E"

é?¿w ∈ {15, 19, 23, 27, 31}§dÚn4.11�315��ü�OA(4, w)§��

3GS(2, 4, w + 1, 3)£½n4.28¤§frame-GS(2, 4, (12w−121), 3)£Ún4.30¤"

A^þã½n§·��±��I��frame-GS(2, 4, (1n−221), 3)"äN/§

�n ∈ {106, 162, 190, 218}�§©O�m = 1§w ∈ {15, 23, 27, 31}¶�n = 210

�§�m = 2§w = 19¶�n = 226�§�m = 3§w = 15"

Ún 4.37. é?¿n ∈ {250} ∪ [266, 290]4 ∪ [306, 398]4 ∪ [406, 946]4§�3�

�frame-GS(2, 4, (1n−121), 3)"

46 Aa|Ü?è¯KïÄ

L 4.3: ½n4.37¥�ëên t a (x, y, z) 4x+ 6y + 8z

250 9 0 (0, 1, 8) 70

290 11 0 (4, 1, 6) 70

[266, 286]4 9 [4, 9] (0, 1, 8) 70

[306, 334]4 11 [4, 11] (0, 9, 2) 70

[338, 358]4 12 [7, 12] (1, 11, 0) 70

[362, 398]4 13 [4, 13] (0, 9, 4) 86

[406, 454]4 16 [4, 16] (13, 3, 0) 70

[458, 510]4 17 [4, 17] (1, 15, 1) 102

[514, 574]4 19 [4, 19] (0, 17, 2) 118

[578, 642]4 21 [5, 21] (0, 15, 6) 138

[646, 714]4 24 [4, 21] (0, 21, 3) 150

[718, 782]4 28 [5, 21] (15, 13, 0) 138

[786, 854]4 31 [4, 21] (18, 13, 0) 150

[858, 926]4 33 [4, 21] (11, 19, 3) 182

[930, 946]4 36 [15, 19] (33, 3, 0) 150

y². lÚn2.9��TD(7, t)§^Ä��E{éc5�|�¤k:§16�|

�a�:\�4"é��|�x, y, z�:©O\�4, 6, 8§Ù¥x + y + z = t"

Ù{:\�0"ùpÑ\�O�frame-GS(2, 4, (4u), 3)§u ∈ {6, 7}£Ún4.15¤

Úframe-GS(2, 4, (4um1), 3)§u ∈ {5, 6}§m ∈ {6, 8}£Ún4.16Ú4.17¤"·�

��frame-GS(2, 4, ((4t)5(4a)1(4x+ 6y + 8z)1), 3)"3|þW\GS(2, 4, 4t, 3)§

GS(2, 4, 4a, 3)£½n4.28¤Úframe-GS(2, 4, (14x+6y+8z−221), 3)£Ún4.31¤"·

��frame-GS(2, 4, (120t+4a+4x+6y+8z−221), 3)"-n = 20t+ 4a+ 4x+ 6y + 8z"

·��±�L4.3¥�ëêt§aÚ(x, y, z)5��I��n"

-R = {10, 12, 14, 15, 18, 20, 21, 22, 26, 28, 30, 33, 34, 35, 38, 39, 42, 44, 46, 51,

52, 54, 58, 60, 62, 66, 68, 74} ∪ {22, 27}"

Ún 4.38. é?¿n ∈ {238, 258}½n ≡ 2 (mod 4)§n ≥ 950§�3��frame-

GS(2, 4, (1n−221), 3)"

y². lÚn2.9��TD(8, t)"^Ä��E{éc6�|�¤k:§17�|

�x�:\�4§é��|�7�:\�10§Ù{:\�0"ùpÑ\�O

�frame-GS(2, 4, (4u), 3)§u ∈ {6, 7}£Ún4.15¤§Úframe-GS(2, 4, (4u101), 3)§

u ∈ {6, 7}£Ún4.17 Ú4.18¤"·��frame-GS(2, 4, ((4t)6(4x)1701), 3)"


a. ék = 7§dÚn2.9�3��TD(8, 7)"�x ∈ {0, 5}"

b. é?¿k ≥ 36§k 6∈ R§dÚn2.9�3��TD(8, k)"�x ∈ [4, 9]"

c. é?¿k ∈ R\{39, 52}§dÚn2.9�3��TD(8, k−1)§�x ∈ [10, 15]"

d. é?¿k ∈ {39, 52}§dÚn2.9�3��TD(8, k − 2)"�x ∈ [16, 21]"

3��frame-GS�|þW\GS(2, 4, 4t, 3)§GS(2, 4, 4x, 3)£½n4.28¤

Úframe-GS(2, 4, (16821), 3)£Ún4.31¤Ò��frame-GS(2, 4, (124t+4x+6821), 3)"

-n = 24t+ 4x+ 70"é?¿n ∈ {238, 258}½nØ�u24× 36 + 16 + 70 = 950§

·��I��frame-GS(2, 4, (1n−221), 3)"

nÜÚn4.31§íØ4.36§Ún4.37ÚÚn4.38¥�(J§·��µ

½n 4.39. é?¿n ≥ 6§n ≡ 2 (mod 4)§n 6∈ [6, 66]4∪{74, 78, 82, 90, 94, 98, 110,

114, 122, 126, 130, 142, 146, 154, 158, 174, 178, 186, 194, 222, 242}§�3��frame-

GS(2, 4, (1n−121), 3)"

4.5 (((ØØØ

3�Ù¥§·�é?¿�Ýn§ïÄ�`(n, 5, 4)4è��E"·�é?

¿n ≥ 4§Ø55��§(½¤kA4(n, 5, 4)��"AO/§·�y²

Ø7�Ø(½��§GS(2, 4, n, 3)�3��=�n ≥ 4§n ≡ 0, 1 (mod 4)"

½n 4.40. é?¿��ên ≥ 4§A4(n, 5, 4) = U(n, 4)§Øn ∈ {4, 5, 6, 7}§ÚXe�U�~µ

1) n ≡ 0, 1 (mod 4)§n ∈ {8, 9, 12, 13, 88, 108, 117}¶

2) n ≡ 2 (mod 4)§n ∈ [10, 66]4 ∪{74, 78, 82, 90, 94, 98, 110, 114, 122, 126, 130,

142, 146, 154, 158, 174, 178, 186, 194, 222, 242}¶

3) n ≡ 3 (mod 4), n ∈ {11, 15, 95, 103, 111, 115, 119, 143, 151, 159, 163}"

Chapter 5

^��üüü��OOO��EEE��`õõõ��~~~èèè


3CheeÚLing�©Ù[26]¥§¦�Ú\�«#��{5�E�`õ�~

è§=ÏLïá|Ü�OnØ¥��«Ø��O§�)W¿! t-�O!

�©|�O��è�éX5�E�`õ�~è"|^ù��{§¦�éA

a(n, d, w)qè§(½Aq(n, d, w)�O(�"Chee��3©[23]¥í2ù��

{§¿��(½Aq(n, 4, 3)�O(�"

Äuù�g�§·�ïá��ü�O��`(n, 6, 4)qè�m�

éX"3�Ù¥§·�òÏLïÄ��ü�O§¿Ú\�«9Ï�

Oµ��ü�©|�O5�E�`(n, 6, 4)3è"·�é¤k�Ýn§Ø

n = 17�§(½A3(n, 6, 4)��"3d�c§ù�¯K�)ûn ≤ 10�

�¹[115]§�L5.1"

L 5.1: �n ≤ 10�§A3(n, 6, 4)��n 4 5 6 7 8 9 10

A3(n, 6, 4) 1 1 3 3 5 9 15

�Ù¥§·�¤^��E�{´Äu|Ü�OnØ¥'uGDD�E

�Wilson’sÄ��E{£WFC¤§Ú13Ù¥'u~èÚ~EÜè�E

��©|è£GDC¤�Ì��E�{"Ï�ùp�¤kGDCÑ´þ�4§å

l�6�n�GDC§Ïd3�Ù¥§·�òþrn�4-GDC(6){P�GDC"

¯¤±�§A3(n, 2w,w) = bn/wc"(ÜÚn2.2§�±��A3(n, 6, 4)��

�þ."

íØ 5.1. A3(n, 6, 4) ≤⌊n2

⌊n−13

⌋⌋=: U(n, 3)"

ù�Ù�(�Xeµ315.2!¥§·�ò0��Ä�Vg§¿ïá�

��ü�O��`(n, 6, 4)qè�éX¶315.3!¥§·�ò©�¹�E�

50 Aa|Ü?è¯KïÄ

`(n, 6, 4)3è¶315.4!¥§òé�Ù�Ì�(J?1o("

5.2 OOO��£££

��OXJvkE�«|§Ò¡�´ü�£simple¤"��O�¡�

´�ü�£super-simple§SS¤§XJ?¿ü�«|�õ�uü�:"�k = 3

�§��ü�OÒ´ü�"�λ = 1�§�O�½´�ü�"

��3:8Xþ§«|8�B§�ê�λ��OD§XJ§�«|8B�±�¤B =

⋃λi=1 Bi§¦�é?¿�1 ≤ i ≤ λ§Bi/¤�Dëê�Ó�´�ê�1�

�O§KD¡��ü£completely reducible super-simple§CRSS¤�

O"ùp§CRSS�½Â�±^u�«�O§¿^5{zÎÒ"·�r�

�:8��v§«|��k§�ê�λ��üBIBD�¤(v, k, λ)-

CRSS�O§r��«|��k§�ê�λ��üGDD{P�(k, λ)-

CRSSGDD"

Ún 5.2 (Adams�[5]). �v ≡ 1½4 (mod 12)§v ≥ 13�§�3��(v, 4, 2)-

CRSS�O"�3.�35�(4, 2)-CRSSGDD"

-m��ê§X´��:8"-H = {H1, H2, . . . , Hm}´X��y©"H��î�Ò´��z�Hi�õ�u��:�X�f8"�

�H(m, g, k, t)�OÒ´��n�|(X,H,B)§Ù¥|X| = mg§HrXy©¤m��g�Ø��f8§B´H �k�î��¤�8Ü£¡�«|¤§¦�?¿t�î�ÑTÐ�¹3��«|¥"

��H(m, g, 4, 3)�O�¡�´s-fan�§XJ§�«|8B�±y©¤Ø��f8B1,B2, · · · ,BsÚA§¦�é?¿1 ≤ i ≤ s§BiÑ´��H(m, g, 4, 2)�

«|8§ �A = B\⋃si=1 Bi"

Ún 5.3 (Ge [65]). �g ≥ 4�g 6≡ 2 (mod 4)�§�3��g-fan H(4, g, 4, 3)"

Ïd§·��(4, g)-CRSSGDD"

Ún 5.4. �g ≥ 4�g 6≡ 2 (mod 4)�§�3��.�g4�(4, g)-CRSSGDD"

Ún 5.5. XJ�3��äkb�«|�(n, 4, q)-CRSS�O§@o�3��

��b�(n, 6, 4)q+1è¶XJ�3��.�gt11 · · · gtss §äkg�«|�(4, q)-

CRSSGDD§@o�3��.�gt11 · · · gtss §��g�(q + 1)�GDC"

CHAPTER 5 ^��ü�O�E�`õ�~è 51

y². ùp·��I��Ñ1��·K�y²"1��·K�±aqy²"

db�§·�k��«|8�B =⋃qi=1 Bi�(n, 4, q)-CRSS�O§|B| = b"

é?¿�1 ≤ i, j ≤ q§Bi ∩ Bj = ∅§ �Bi/¤(n, 4, 1)-BIBD�«|8"

lq�(n, 4, 1)-BIBD�«|8§·��±��q�Ø��(n, 6, 4)2èCi§1 ≤i ≤ q"�©[26]¥��{aq§éCi�z��èi¥�1^iO�Ò��(q +

1)�èC ′i"l C ′ =⋃qi=1 C ′iÒ´��b§þ�4§ål�6�(q + 1)�

è"

(ÜÚn5.2Ú5.5§·��Xe(Jµ

½n 5.6. �v ≡ 1½ö4 (mod 12)§v ≥ 13�§�3��kU(v, 3)�èi��

`�(v, 4, 2)3è"

(ÜÚn5.4Ú5.5§·��Xe(Jµ

½n 5.7. �g ≥ 4§g 6≡ 2 (mod 4)�§�3��.�g4�g�GDC"

½n 5.8. �v ≡ 0½3 (mod 12)§v ≥ 12�§�3��kU(v, 3)�èi��

`(v, 4, 2)3è"

y². é?¿v§-(X,A)´½n5.6¥��(v + 1, 4, 2)-CRSS�O"�K��

:x ∈ X��(v, 4, 2)-W¿(Y,B)§Ù¥Y = X \ {x}§B = A\ {A ∈ A : x ∈A}"�Ún5.5�y²aq§·��`(v, 6, 4)3è"

��SteinerXS(t, k, v)´��H(1, v, k, t)" S(2, 3, v)�¡��Steinern

�X£{P�STS(v)¤§S(3, 4, v)�¡�Steinero�X£{P�SQS(v)¤"�

�SteinerXS(t, k, v)�¡�´i-�©)�§XJ§�«|8�±y©¤S(i, k, v)§

0 ≤ i ≤ t"é2-�©)SQS�ïÄ(Jé�"

Ún 5.9 (Baker [8]§Teirlinck [137]). �32-�©)SQS(v)§XJv = 4n½v =

2 · pn + 2§p ∈ {7, 31, 127}§n��ê"

w,§��2-�©)SQS(v)�´��(v, 4, v2− 1)-CRSS�OµXJ�3�

�2-�©)SQS(v)§@oé?¿q ∈ [1, v2− 1]Ñ�3��(v, 4, q)-CRSS�O"d

Ún5.5§·��Xe(Jµ

52 Aa|Ü?è¯KïÄ

Ún 5.10. XJ�3��2-�©)SQS(v)§@oé?¿�q ∈ [1, v2− 1]§Ñ�3

��k⌊(q−1)v

4

⌊v−13

⌋⌋�èi��`�(v, 6, 4)q+1è"

nÜÚn5.9Ú5.10§·��Xe(Jµ

½n 5.11. �v = 4n½v = 2 · pn + 2§p ∈ {7, 31, 127}§n��ê§é?¿q ∈ [1, v

2− 1]�§�3��k

⌊(q−1)v

4

⌊v−13

⌋⌋�èi��`(v, 6, 4)q+1è"


a. ��GDCÚÚÚ��`èèè

3�!¥§·�ò��E�(4, 2)-CRSSGDD"Ó�dÚn5.5§·�

��GDC"

��/§·�©O�E:8Ú|8�Ó§�ê�1�ü�{4}-GDD§¦�

?¿ü�«|�õ�uü�:",�rùü�{4}-GDD|Üå5Ò��·�

I��(4, 2)-CRSSGDD"ùp��EÑÄu·�Ù��{§=µÏ~�

�GDD�«|´d��k�+£õ´��+Zu¤Ðm"¤±·�Ï~��Ñ�Ä«|§ ¤k�«|Ò�±ÏL^\{+½Ù¦�gÓ�+Ðm��"

3�Ã¡:��O¥§x0 ∈ {x} × Zn�eI´3��+Zu��n�f+¥Ðm"

Ún 5.12. �3��.�47�(4, 2)-CRSSGDD"

y². -:8�Z28§|8�{{0, 7, 14, 21} + i : 0 ≤ i ≤ 6}"éXeÄ«|3Z28þ+4 (mod 28)ÐmÒ��·�I��O"1��GDD�Ä«|

�(0, 11, 24, 27)§(1, 3, 7, 9)§(0, 1, 12, 25)§(3, 18, 22, 23)§(2, 5, 10, 21)§(0, 2, 18,

20)§(0, 9, 19, 22)§(1, 2, 19, 24)"1��GDD�Ä«|´d1��GDD�Ä«

|¦±¦f5��"

Ún 5.13. é?¿u ∈ {7, 10, 13, 16, 19, 22, 28, 34, 52, 58}§�3.�2u�(4, 2)-

CRSSGDD"

y². -:8�Z2u§|8�{{0, u} + i : 0 ≤ i ≤ u − 1}"éu ∈ {7, 13, 19}§^[162, N¹]¥�Ä«|+1 (mod 2u)ÐmÒ��·�I��O"éu =

10§^[162, N¹]¥�Ä«|+4 (mod 20)ÐmÒ��·�I��O"


éu ∈ {16, 22, 28, 34, 52, 58}§^[162, N¹]¥�Ä«|+2 (mod 2u)ÐmÒ�

�·�I��O"

Ún 5.14. é?¿u ∈ [5, 13]§�3��.�6u�(4, 2)-CRSSGDD"

y². -:8�Z6u§|8�{{0, u, 2u, . . . , 5u} + i : 0 ≤ i ≤ u − 1}"éu ∈{5, 6, 8, 12}§^[162, N¹]¥�Ä«|+2 (mod 6u)ÐmÒ��·�I��

�O"éu ∈ {9, 11}§^[162, N¹]¥�Ä«|+1 (mod 6u)ÐmÒ��·

�I��O"éu ∈ {7, 10, 13}§¤I�O©Od.�27§210½213�(4, 2)-

CRSSGDD^3)ä��"

Ún 5.15. é?¿u ∈ {4, 5, 6}§�3��.�6u31�(4, 2)-CRSSGDD"

y². 6431µ-:8�Z24 ∪ ({a} × Z3)§|�{{0, 4, 8, . . . , 20} + i : 0 ≤ i ≤3}∪{{a}×Z3}"¤I�OdXeÄ«|+2 (mod 24)Ðm��"1��GDD�

Ä«|�(1, 6, 11, a0)§(2, 3, 4, a0)§(0, 7, 9, 18)§(0, 3, 10, 21)"1��GDD�Ä

«|´d1��GDD�Ä«|¦±¦f5��"

6531µ-:8�Z30 ∪ ({a, b, c} × Z1)§|�{{0, 5, 10, . . . , 25} + i : 0 ≤ i ≤4} ∪ {{a, b, c} × Z1}"¤I�OdXeÄ«|+6 (mod 30)Ðm��"

1��GDD�Ä«|µ

(2, 1, 15, a0) (5, 12, 28, a0) (5, 22, 26, b0) (0, 13, 21, b0) (1, 3, 14, c0)

(1, 23, 29, 12) (1, 24, 27, 0) (5, 21, 27, 23) (4, 27, 28, 15) (1, 28, 10, 7)

(2, 29, 25, 13) (5, 16, 14, 2) (2, 26, 3, 24) (5, 6, 4, c0) (0, 4, 12, 26)


(3, 25, 22, a0) (2, 24, 23, a0) (0, 2, 4, b0) (5, 7, 9, b0) (0, 22, 26, c0)

(5, 8, 17, 24) (2, 21, 15, 14) (0, 1, 18, 7) (5, 2, 13, 1) (0, 16, 17, 28)

(2, 25, 16, 8) (3, 15, 12, 6) (5, 22, 21, 29) (1, 15, 17, c0) (1, 4, 10, 27)

6631µ-:8�Z36 ∪ ({a} × Z3)§|�{{0, 6, 12, . . . , 30} + i : 0 ≤ i ≤ 5} ∪{{a} × Z3}"¤I�OdXeÄ«|+1 (mod 36)Ðm��"1��GDD�Ä

«|�(0, 4, 5, a0)§(0, 11, 14, 27)§(0, 2, 10, 17)"1��GDD�Ä«|´d1�

�GDD�Ä«|¦±¦f11��"

Ún 5.16. é?¿u ∈ {4, 6}§�3��.�6u91�(4, 2)-CRSSGDD"

54 Aa|Ü?è¯KïÄ

y². 6491µ-:8�Z24∪ ({a, b}×Z4)∪ ({c}×Z1)§|�{{0, 4, 8, . . . , 20}+ i :

0 ≤ i ≤ 3}∪{({a, b}×Z4)∪({c}×Z1)}"¤I�OdXeÄ«|+3 (mod 24)Ð

m��"


(2, 15, 20, a0) (9, 10, 23, a0) (5, 6, 12, a0) (7, 13, 16, a0) (4, 15, 18, b0)

(0, 7, 9, b0) (1, 2, 23, b0) (8, 17, 22, b0) (0, 2, 19, c0)


(4, 15, 21, a0) (1, 10, 11, b0) (5, 12, 15, b0) (0, 5, 22, a0) (0, 10, 23, c0)

(1, 6, 7, a0) (2, 8, 11, a0) (2, 4, 7, b0) (6, 8, 21, b0)

6691µ-:8�Z36 ∪ ({a, b, c} × Z3)§|�{{0, 6, 12, . . . , 30} + i : 0 ≤ i ≤5} ∪ {{a, b, c} × Z3}"¤I�OdXeÄ«|+2 (mod 36)Ðm��"


(0, 1, 2, a0) (3, 10, 35, a0) (0, 7, 33, b0) (2, 10, 23, b0)

(4, 20, 31, c0) (0, 22, 26, 31) (0, 15, 23, c0) (0, 3, 17, 19)


(0, 3, 4, a0) (1, 17, 20, a0) (0, 1, 28, b0) (3, 11, 20, b0)

(1, 12, 23, 33) (3, 5, 34, c0) (0, 13, 26, c0) (0, 2, 16, 31)

Ún 5.17. é?¿u ∈ {5, 6, 7}§�3��.�6u121�(4, 2)-CRSSGDD"

y². 65121µ-:8�Z30∪({a, b}×Z5)∪({c}×Z2)§|�{{0, 5, 10, . . . , 25}+i :

0 ≤ i ≤ 4}∪{({a, b}×Z5)∪({c}×Z2)}"¤I�OdXeÄ«|+3 (mod 30)Ð

m��"


(6, 24, 25, a0) (5, 11, 12, a0) (8, 19, 22, a0) (1, 3, 29, a0) (0, 13, 17, a0) (1, 22, 23, c0)

(13, 19, 26, b0) (3, 7, 25, b0) (6, 12, 14, b0) (2, 5, 23, b0) (1, 15, 24, b0) (2, 18, 21, c0)


(12, 19, 23, a0) (2, 13, 24, a0) (3, 5, 21, a0) (1, 7, 10, a0) (11, 14, 15, a0) (1, 8, 17, c0)

(0, 17, 24, b0) (3, 4, 6, b0) (14, 20, 22, b0) (1, 13, 27, b0) (8, 25, 26, b0) (4, 12, 21, c0)

66121µ-:8�Z36∪ ({a, b}×Z3)∪ ({c}×Z6)§|�{{0, 6, 12, . . . , 30}+ i : 0 ≤i ≤ 5} ∪ {({a, b} × Z3) ∪ ({c} × Z6)}"¤I�OdXeÄ«|+2 (mod 36)Ðm

��"



(1, 12, 29, a0) (2, 9, 16, a0) (5, 7, 8, b0) (0, 4, 27, b0) (2, 4, 12, c0)

(10, 13, 29, c0) (6, 15, 19, c0) (8, 9, 23, c0) (0, 5, 20, 31)


(0, 19, 21, a0) (2, 5, 10, a0) (2, 13, 34, b0) (5, 15, 18, b0) (0, 20, 22, c0)

(4, 17, 21, c0) (11, 18, 25, c0) (3, 14, 19, c0) (0, 1, 9, 10)

67121µ-:8�Z42∪({a}×Z7)∪({b, c}×Z2)∪({d}×Z1)§|�{{0, 7, . . . , 35}+i : 0 ≤ i ≤ 6} ∪ {({a} × Z7) ∪ ({b, c} × Z2) ∪ ({d} × Z1)}"¤I�OdXeÄ«|+3 (mod 42)Ðm��"


(10, 40, 1, a0) (13, 39, 7, a0) (11, 0, 23, a0) (15, 6, 35, a0) (5, 29, 16, a0)

(17, 9, 33, a0) (3, 20, 4, a0) (0, 40, 15, b0) (1, 20, 35, b0) (4, 31, 26, c0)

(0, 3, 41, c0) (1, 21, 26, d0) (0, 37, 6, 19) (2, 34, 38, 35) (0, 2, 4, 12)


(4, 1, 37, a0) (6, 12, 24, a0) (7, 9, 19, a0) (14, 39, 34, a0) (2, 8, 20, a0)

(0, 31, 5, a0) (11, 15, 38, a0) (5, 6, 15, b0) (4, 14, 31, b0) (0, 19, 27, c0)

(2, 5, 10, c0) (0, 8, 4, d0) (0, 11, 20, 22) (1, 20, 19, 18) (0, 29, 3, 16)

b. ��ÝÝÝn ≡ 2 (mod 6)��

3�!¥§·�ò(½A3(6t + 2, 6, 4)��"dÚn5.5§XJ�3��.

�23t+1�(4, 2)-CRSSGDD§·�ò��.�23t+1�GDC§�Ò´��

`(6t+ 2, 6, 4)3è"

Ún 5.18. é?¿t ≥ 4§�3��.�12t�(4, 2)-CRSSGDD"

y². �t ≡ 0½1 (mod 4)§t ≥ 4�§dÚn2.7�3��(3t + 1, {4}, 1)-PBD"

lù�PBD�:8�K��:�±��.�3t�{4}-GDD"�t ≡ 2½3

(mod 4)§t ≥ 7�§dÚn2.7�3��(3t + 1, {4, 7?}, 1)-PBD"lù�PBD�

:8�K��Ø3��7�«|¥�:§�±��.�3t�{4, 7?}-GDD"

Ïd§é?¿t ≥ 4§t 6= 6§·�Ñk��.�3t�{4, 7}-GDD"

r��O^WFC\�4§·�Òé?¿t ≥ 4§t 6= 6§��.

�12t�(4, 2)-CRSSGDD"ùp¤^�Ñ\�O´.�44£Ún5.4¤Ú47 £Ú

n5.12¤�(4, 2)-CRSSGDD"

56 Aa|Ü?è¯KïÄ

ét = 6§��.�46�{5}-GDD£�[71]¤§^WFC\�3��

.�126�(4, 2)-CRSSGDD"ùpÑ\�O´.�35�(4, 2)-CRSSGDD £Ú

n5.2¤"

Ún 5.19. é?¿t ≥ 1§�3��.�26t+1�(4, 2)-CRSSGDD"

y². ét ∈ {1, 2, 3}§¤I�O®²3Ún5.13¥�E��"é?¿t ≥ 4§l

Ún5.18��.�12t�(4, 2)-CRSSGDD§O\2�Ã¡:§¿3|þëÓÃ

¡:W\.�27�(4, 2)-CRSSGDD§��.�26t+1�(4, 2)-CRSSGDD"

Ún 5.20. é?¿�t ≥ 1§�3��.�26t+4�(4, 2)-CRSSGDD"

y². ét ∈ {1, 2, 3, 4, 5, 8, 9}§¤I�O®²3Ún5.13¥�E��"ét ∈{6, 10}§©O�.�s4§s ∈ {20, 32}�(4, 2)-CRSSGDD£Ún5.4¤§2©O3

|þW\.�210½216�(4, 2)-CRSSGDD"ét ∈ {7, 13}§é.�65½69�(4, 2)-

CRSSGDD^3)ä§Ò��.�185½189�(4, 2)-CRSSGDD"O\2�Ã¡

:§¿3|þëÓÃ¡:W\.�210�(4, 2)-CRSSGDD"ét ∈ {11, 16}§©O�.�27½210�(4, 2)-CRSSGDD§^10)ä��.�207½2010�(4, 2)-

CRSSGDD§23|þ©OW\.�210�(4, 2)-CRSSGDDÒ��¤I�

O"ét = 15§��.�3871�{5}-GDD£ÏLÖ�.�38�{4}-RGDD�

�§�[71]¤"^WFC\�6§O\2�Ã¡:§©O3|þëÓÃ¡:W\.

�210½222�(4, 2)-CRSSGDD"

ét ∈ {12, 14}½t ≥ 17§lÚn2.4��(v, {5, 6, 7, 8, 9}, 1)-PBD"l

ù�PBD�:8¥�K��:§��.�4i5j6k7l8m�{5, 6, 7, 8, 9}-GDD§Ù

¥4i+ 5j+ 6k+ 7l+ 8m = v− 1"^WFCéù�GDD\�6¿Ñ\.�6u§u ∈{5, 6, 7, 8, 9}�(4, 2)-CRSSGDD£Ún5.14¤§��.�24i30j36k42l48m�(4, 2)-

CRSSGDD"O\2�Ã¡:§¿3|þëÓÃ¡:©OW\.�2s§s ∈ {13,

16, 19, 22, 25}�(4, 2)-CRSSGDD§·�Òé?¿óêv ∈ {26, 30}½v ≥ 36��

.�23(v−1)+1�(4, 2)-CRSSGDD"

nÜÚn5.19Ú5.20¥�(J§·��µ

½n 5.21. é?¿t ≥ 2§�3��.�23t+1�(4, 2)-CRSSGDD"Ïd§é

?¿t ≥ 2§�3��.�23t+1�GDC"=µé?¿t ≥ 2§A3(6t + 2, 6, 4) =

U(6t+ 2, 3)"


c. ��ÝÝÝn ≡ 6, 7 (mod 12)��

3�!¥§·�ò(½A3(12t+6, 6, 4)ÚA3(12t+7, 6, 4)��"én ∈ {6, 7}§L5.1¥®²(½A3(6, 6, 4)ÚA3(7, 6, 4)��"

Ún 5.22. é?¿n ∈ {18, 19, 31, 43, 55}§A3(n, 6, 4) = U(n, 3)"

y². ¤IèÏL�E.�[2, 2]�~EÜè��"-:8�Zn"én = 18§

¤IèdXeèi+2 (mod 18)��"én ∈ {19, 31, 43, 55}§¤IèdXeèi+1 (mod n)��"

n = 18µ〈0, 1, 2, 3〉〈0, 6, 4, 13〉〈0, 9, 5, 17〉〈0, 11, 8, 14〉〈1, 7, 5, 12〉

n = 19µ〈0, 3, 18, 4〉〈0, 17, 10, 7〉〈0, 5, 11, 13〉

n = 31µ〈0, 3, 1, 13〉〈0, 6, 2, 26〉〈0, 7, 15, 18〉〈0, 12, 4, 21〉〈0, 14, 5, 30〉

n = 43µ〈0, 2, 26, 36〉〈0, 10, 25, 38〉〈0, 3, 8, 9〉〈0, 1, 18, 22〉〈0, 4, 11, 31〉〈0, 13, 29, 32〉〈0, 20, 12, 14〉

n = 55µ〈0, 5, 13, 22〉〈0, 7, 19, 33〉〈0, 4, 10, 28〉〈0, 11, 31, 32〉〈0, 14, 49, 53〉〈0, 15, 38, 45〉〈0, 1, 3, 43〉

〈0, 9, 25, 36〉〈0, 18, 47, 52〉

Ún 5.23. é?¿u ∈ [4, 8]§�3��.�12u181�(4, 2)-CRSSGDD"

y². -:8�Z12u∪({a}×Z6)∪({b}×Z12)§|8�{{0, u, 2u, . . . , 11u}+i : 0 ≤i ≤ u−1}∪{({a}×Z6)∪ ({b}×Z12)}"¤I�Ode¡Ä«|+1 (mod 12u)Ð

m��§Ù¥x0 ∈ {x} × Zn�eIdZ12u¥��n�f+Ðm"

124181µ

1��GDD�Ä«|µ(0, 25, 27, a0) (2, 11, 16, a0) (9, 20, 35, b0) (3, 4, 34, b0) (2, 5, 43, b0)

(0, 13, 42, b0)

1��GDD�Ä«|dþ¡Ä«|¦±¦f7��"

125181µ


(3, 6, 52, a0) (5, 37, 38, a0) (0, 16, 37, b0) (2, 56, 58, b0)

(0, 7, 19, 36) (9, 18, 31, b0) (3, 11, 29, b0)

1��GDD�Ä«|dþ¡Ä«|¦±¦f7��"

126181µ


(2, 40, 69, a0) (0, 1, 65, a0) (2, 23, 55, b0) (8, 30, 33, b0)

(0, 4, 14, 49) (5, 46, 61, b0) (4, 24, 63, b0) (0, 9, 11, 55)

58 Aa|Ü?è¯KïÄ

1��GDD�Ä«|dþ¡Ä«|¦±¦f5��"

127181µ


(0, 33, 67, a0) (2, 10, 29, a0) (0, 4, 9, b0) (10, 62, 63, b0) (8, 11, 66, b0)

(0, 39, 41, 54) (0, 25, 61, 72) (0, 18, 38, 62) (5, 73, 79, b0)

1��GDD�Ä«|dþ¡Ä«|¦±¦f5��"

128181µ


(0, 3, 55, a0) (4, 77, 86, a0) (10, 21, 40, b0) (1, 19, 39, b0) (8, 54, 59, b0)

(0, 42, 59, 71) (0, 47, 60, 62) (0, 1, 69, 90) (0, 4, 26, 65) (2, 12, 65, b0)

1��GDD�Ä«|dþ¡Ä«|¦±¦f5��"

Ún 5.24. é?¿u ∈ [4, 8]∪{16}∪[20, 22]½u ≥ 24§�3��.�12u181�(4, 2)

-CRSSGDD"

y². éu ∈ {4, 5, 6, 7, 8}§¤I�O®²3Ún5.23¥�E"éu ∈ {21, 33}§©O�.�124151½127151�{4}-GDD£�[69, ½n3.16]¤§^WFC\�4§O

\18�Ã¡:§¿3|þW\.�124181½125181�(4, 2)-CRSSGDD§Ò�

�¤I�O"éu ∈ {16, 28, 32}§©O�.�s4§s ∈ {4, 7, 8}£Ún5.4¤

�{4}-GDD§^WFC\�12§O\18�Ã¡:§¿3|þëÓÃ¡:W\

.�12s181�(4, 2)-CRSSGDD"éu ∈ {22, 26}§©O�.�6491½6591�{4}-GDD£�[77, ½n1.6]¤§^WFC\�8§O\18�Ã¡:§¿3|þëÓÃ

¡:W\.�124181½126181�(4, 2)-CRSSGDD"éu = 31§�.�4671�{4}-GDD£�[69, Ún3.17]¤§^WFC\�12§O\18�Ã¡:¿3|þëÓÃ¡

:W\.�124181½127181�(4, 2)-CRSSGDD"éu = 27§�.�154211�{4}-GDD£�[69, ½n4.1]¤§^WFC\�4§O\18�Ã¡:§¿3|þëÓÃ¡

:W\.�125181½127181�(4, 2)-CRSSGDD"

éu ∈ {20, 24, 25, 29, 30}½u ≥ 34§lÚn2.4��(u+1, {5, 6, 7, 8, 9}, 1)-

PBD"lù�PBD�:8�K��:��.�4i5j6k7l8m �{5, 6, 7, 8, 9}-GDD§Ù¥4i+5j+6k+7l+8m = u"^WFC\�12§Ñ\�O´.�12t§t ∈{5, 6, 7, 8, 9}�(4, 2)-CRSSGDD£Ún5.18¤§��.�48i60j72k84l96m�(4, 2)-

CRSSGDD"O\18�Ã¡:§¿3|þëÓÃ¡:©OW\.�12s181§


s ∈ {4, 5, 6, 7, 8}�(4, 2)-CRSSGDD£Ún5.23)§·�Ò��.�12u181§

u ∈ {20, 24, 25, 29, 30}½u ≥ 34�(4, 2)-CRSSGDD"

Ún 5.25. �r ∈ {6, 7}�§é?¿t ∈ [2, 9] ∪ {17} ∪ [21, 23]½t ≥ 25§A3(12t+

r, 6, 4) = U(12t+ r, 3)"

y². é?¿t ∈ {2, 3, 4}§��.�62t+1�GDC£Ún5.14¤"¤I��Ý

�12t + 6�è�±ÏL3GDC�¤k|¥W\�`(6, 6, 4)3è��"¤I��

Ý�12t+ 7�è3Ún5.22¥�E��"

dÚn5.24§é?¿u ∈ [4, 8] ∪ {16} ∪ [20, 22]½u ≥ 24§·�Ñk.

�12u181�GDC"XJ3ù�GDC�z�|þW\�Ý�12½18��`è§é

?¿t ∈ [5, 9] ∪ {17} ∪ [21, 23]½t ≥ 25§·�Ò��`(12t+ 6, 6, 4)3è"X

JO\��Ã¡:§¿3ù�GDC�|þëÓù�Ã¡:W\�Ý�13½19�

�`è§·�Ò��A��`(12t+ 7, 6, 4)3è"

Ún 5.26. �r ∈ {6, 7}�§é?¿t ∈ [10, 16]∪[18, 20]∪{24}§A3(12t+r, 6, 4) =

U(12t+ r, 3)"

y². Äk§·��E.�36um1§Ù¥(u,m) ∈ {(5, 42), (6, 18), (6, 30), (6, 78)}�(4, 2)-CRSSGDD"��TD(6, 7)§�K��«|¥�5�:��.

�6571�{5, 6}-GDD§^WFC\�6§Ò��.�365421�(4, 2)-CRSSGDD"

��TD(7, 7)§�K��«|�6�:��.�6671�{6, 7}-GDD"^WFCr

��6�|�¤k:\�6§��|�x, y, z�:©O\�0, 6, 12"ù

pÑ\�O´.�65§66Ú67�(4, 2)-CRSSGDD§Ú.�65121Ú66121�(4, 2)-

CRSSGDD"-(x, y, z) = (0, 1, 6)§·�Ò��.�366781�(4, 2)-CRSSGDD¶

-(x, y, z) = (2, 5, 0)§��.�366301�(4, 2)-CRSSGDD¶-(x, y, z) = (4, 3, 0)§

Ò��.��366181�(4, 2)-CRSSGDD"

,�§·��E.�24um1§Ù¥(u,m) ∈ {(4, 30), (5, 18), (5, 30), (6, 18),

(6, 30), (7, 18), (7, 30)}�(4, 2)-CRSSGDD"��.�6ua1�(4, 2)-CRSSGDD§

�K|��a¥�:§2^WFC\�4"ùp§Ñ\�O´.�44�{4}-MGDD§Ú.�43��©){3}-MGDD"·�Ò��ü�äkCRSS5�

�.�(24, 64)u�{3, 4}-DGDD§�¤k��3�«|/¤3a�²1a"O

\3a�Ã¡:Ö�²1a§¿W\.�6uw1�(4, 2)-CRSSGDD§Ò��.

�24u(3a+ w)1�(4, 2)-CRSSGDD"

60 Aa|Ü?è¯KïÄ

du·�k.�6491Ú6431�(4, 2)-CRSSGDD§-(a, w) = (9, 3)§Ò�

�.�244301�(4, 2)-CRSSGDD"du·�k.�6u61§u ∈ {5, 6, 7}§Ú.�6u121§u ∈ {5, 6, 7}�(4, 2)-CRSSGDD"-(a, w) = (6, 0)§Ò��.�24u181§

u ∈ {5, 6, 7}�(4, 2)-CRSSGDD¶-(a, w) = (6, 12)§Ò��.�24u301§u ∈{5, 6, 7}�(4, 2)-CRSSGDD"

dÚn5.5§·��.�36um1§(u,m) ∈ {(5, 42), (6, 18), (6, 30), (6, 78)}Ú24um1§(u,m) ∈ {(4, 30), (5, 18), (5, 30), (6, 18), (6, 30), (7, 18), (7, 30)}�GDC"

3ùGDC�|þW\�A��`è§·�Ò��`(12t+ 6, 6, 4)3è§Ù

¥t ∈ [10, 16] ∪ [18, 21] ∪ {24}"Ó�§XJO\��Ã¡:§¿3¤kGDC�

|þëÓÃ¡:W\�A��`è§·�Ò��`(12t+ 7, 6, 4)3è"

nÜþã(J§·��µ

½n 5.27. é?¿t ≥ 0§A3(12t + 6, 6, 4) = U(12t + 6, 3)¶é?¿t ≥ 1§

A3(12t+ 7, 6, 4) = U(12t+ 7, 3)"

d. ��ÝÝÝn ≡ 9, 10 (mod 12)��

3�!¥§·�ò(½A3(12t + 9, 6, 4)ÚA3(12t + 10, 6, 4)��"én ∈{9, 10}§A3(9, 6, 4)ÚA3(10, 6, 4)��®²3L5.1¥(½"

Ún 5.28. é?¿n ∈ {21, 22, 33, 34}§A3(n, 6, 4) = U(n, 3)"

y². én = 21§¤I�èÏL�E�`(21, 6, [2, 2])3è��"-:8�Z21§

¤I�èdèi〈0, 1, 3, 11〉§〈0, 8, 6, 15〉§〈0, 9, 4, 5〉 +1 (mod 21)Ðm��"

én ∈ {22, 34}§-:8�Zn§¤I�èÏLdXeèi+2 (mod n)Ðm

��"

(22, 6, 4)3èµ 〈92, 52, 182, 151〉〈12, 161, 212, 151〉〈212, 02, 21, 62〉〈212, 42, 71, 111〉〈181, 121, 51, 71〉

〈01, 101, 11, 182〉〈12, 02, 201, 122〉

(34, 6, 4)3èµ〈12, 42, 32, 72〉〈02, 62, 301, 101〉〈282, 32, 112, 212〉〈271, 01, 221, 161〉〈301, 311, 211, 221〉

〈262, 152, 32, 242〉〈202, 282, 12, 62〉〈271, 81, 101, 51〉〈22, 212, 202, 72〉〈31, 311, 161, 291〉〈91, 251, 51, 21〉

én = 33§¤I�èÏL3.�6491�GDC�|þW\�Ý�6½9��`

è��"

Ún 5.29. é?¿u ∈ [4, 8]§�3��.�12u91�(4, 2)-CRSSGDD"


y². -:8�Z12u ∪ ({a, b, c} × Z3)§|8�{{0, u, 2u, . . . , 11u} + i : 0 ≤ i ≤u− 1} ∪ {{a, b, c} × Z3}"¤I�Ode¡Ä«|+2 (mod 12u)Ðm��"

12491µ


(0, 7, 14, a0) (3, 17, 22, a0) (0, 37, 39, b0) (2, 11, 28, b0) (3, 13, 26, c0)

(5, 30, 40, c0) (0, 1, 2, 19) (0, 3, 30, 45) (0, 6, 11, 33)

1��GDD�Ä«|´dXþÄ«|¦±¦f19��"

12591µ


(1, 3, 52, a0) (5, 18, 32, a0) (0, 56, 57, b0) (5, 13, 46, b0) (1, 2, 23, c0) (0, 13, 32, 49)

(0, 51, 58, c0) (0, 3, 7, 24) (1, 7, 30, 38) (1, 17, 29, 43) (0, 12, 38, 54)

1��GDD�Ä«|´dXþÄ«|¦±¦f7��"

12691µ


(0, 14, 43, a0) (3, 5, 10, a0) (5, 16, 32, b0) (1, 21, 54, b0) (0, 4, 9, c0)

(5, 26, 37, c0) (0, 33, 41, 55) (0, 2, 34, 71) (0, 1, 17, 26) (0, 27, 50, 53)

(1, 38, 48, 69) (0, 13, 23, 57) (0, 7, 20, 64)

1��GDD�Ä«|´dXþÄ«|¦±¦f5��"

12791µ


(4, 36, 83, a0) (1, 14, 75, a0) (3, 18, 43, b0) (4, 5, 38, b0) (4, 37, 57, c0)

(5, 72, 80, c0) (1, 20, 33, 63) (0, 46, 48, 58) (1, 26, 30, 48) (1, 61, 77, 79)

(1, 4, 10, 49) (1, 44, 73, 74) (1, 28, 47, 51) (1, 18, 58, 82) (0, 5, 16, 31)

1��GDD�Ä«|´dXþÄ«|¦±¦f11��"

12891µ


(2, 4, 47, a0) (3, 42, 79, a0) (5, 24, 63, b0) (2, 40, 61, b0) (0, 3, 95, c0) (0, 4, 69, 91)

(2, 49, 64, c0) (0, 55, 67, 73) (1, 48, 62, 63) (0, 6, 66, 76) (1, 11, 18, 71) (0, 41, 44, 71)

(1, 72, 84, 95) (1, 12, 29, 43) (1, 22, 68, 90) (0, 19, 54, 63) (0, 5, 18, 51)

1��GDD�Ä«|´dXþÄ«|¦±¦f19��"

Ún 5.30. é?¿u ∈ [4, 8]∪{16}∪[20, 22]½u ≥ 24§�3��.�12u91�(4, 2)-

CRSSGDD"

62 Aa|Ü?è¯KïÄ

y². y²�Ún5.24�y²aq"ùp§·�I�O\9�Ã¡:§,�W\

.�12u91§u ∈ {4, 5, 6, 7, 8}�(4, 2)-CRSSGDD£Ún5.29¤"

Ún 5.31. �r ∈ {9, 10}�§é?¿t ∈ [4, 8]∪{16}∪ [20, 22]½t ≥ 24§A3(12t+

r, 6, 4) = U(12t+ r, 3)"

y². dÚn5.30§·�k.�12t91§t ∈ [4, 8]∪{16}∪[20, 22]½u ≥ 24�GDC"

3|þW\�Ý�12½9��`è§Ò��`(12t + 9, 6, 4)3è"XJO\

��Ã¡:§3|þëÓÃ¡:W\�Ý�13½10��`è§Ò��

`(12t+ 10, 6, 4)3è"

Ún 5.32. �r ∈ {9, 10}�§é?¿t ∈ [9, 15] ∪ [17, 19] ∪ {3, 23}§A3(12t +

r, 6, 4) = U(12t+ r, 3)"

y². Äk§·��E.�36um1§(u,m) ∈ {(5, 33), (6, 21), (6, 69)}�(4, 2)-

CRSSGDD"��TD(6, 7)§�K��«|¥�5�:��.�6571�{5, 6}-GDD"^WFCé��6�|�¤k:\�6§��7�|¥�4�:\�6§

3�:\�3"ùp§Ñ\�O´.�65§66§6431Ú6531�(4, 2)-CRSSGDD"

Ïd§·��.�365331�(4, 2)-CRSSGDD"l��TD(7, 7)Ñu§�

K��«|¥�6�:��.�6671�{6, 7}-GDD"^WFCé��6�|

�¤k:\�6§é��|�x, y, z�:©O\�3, 6, 12"-(x, y, z) =

(7, 0, 0)§Ò��.�366211�(4, 2)-CRSSGDD¶-(x, y, z) = (1, 1, 5)§Ò��

.�366691�(4, 2)-CRSSGDD"

,�§·��E.�24um1§(u,m) ∈ {(5, 9), (5, 21), (6, 21), (6, 33)}�(4, 2)-

CRSSGDD"�E�Ún5.26¥�E.�24um1§(u,m) ∈ {(4, 30), (5, 18), (5, 30),

(6, 18), (6, 30), (7, 18), (7, 30)}�(4, 2)-CRSSGDD��Eaq"·�k.�6531§

6561§6661Ú6691�(4, 2)-CRSSGDD"-u = 5§ (a, w) = (3, 0)§Ò��.

�24591�(4, 2)-CRSSGDD¶-u = 5§(a, w) = (6, 3)§Òk.�245211�(4, 2)-

CRSSGDD¶-u = 6§(a, w) = (6, 3)§Ò��.�246211�(4, 2)-CRSSGDD¶

-u = 6§(a, w) = (9, 6)§Ò��.�246331�(4, 2)-CRSSGDD"

��§·��E.�9u§u ∈ {5, 13, 17, 21, 25}�(4, 2)-CRSSGDD"é��

.�35�(4, 2)-CRSSGDD^3)ä§��.�95�(4, 2)-CRSSGDD"©O�.

�34½44�{4}-GDD§½��TD(5, 4)§^WFC\�9§O\9�Ã¡:§W\

.�95�(4, 2)-CRSSGDDÒ©O��.�913§917½921�(4, 2)-CRSSGDD"


��TD(5, 5)§^WFC\�9§W\.�95�(4, 2)-CRSSGDDÒ��.

�925�(4, 2)-CRSSGDD"

dÚn5.5§·��.�36um1§(u,m) ∈ {(5, 33), (6, 21), (6, 69)}§.�24um1§(u,m) ∈ {(5, 9), (5, 21), (6, 21), (6, 33)}§Ú.�9u§u ∈ {5, 13, 17, 21,

25}�GDC"XJ3ùGDC�|þW\·��Ý��`è§·�Ò��

�`(12t + 9, 6, 4)3è§Ù¥t ∈ [9, 15] ∪ [17, 19] ∪ {3, 23}"XJO\��Ã¡:§¿3¤k|þëÓÃ¡:W\·��Ý��`è§·�Ò��A��

`(12t+ 10, 6, 4)3è"


½n 5.33. é?¿t ≥ 0§A3(12t + 9, 6, 4) = U(12t + 9, 3)¶é?¿t ≥ 0§

A3(12t+ 10, 6, 4) = U(12t+ 10, 3)"

e. ��ÝÝÝn ≡ 5 (mod 6)��

3�!¥§·�ò(½A3(6t+ 5, 6, 4)��"3ù�a§·�é��Ý�

��`è´23§�´én ∈ {11, 17}§·�kXee."

Ún 5.34. A3(11, 6, 4) ≥ U(11, 3)− 1¶A3(17, 6, 4) ≥ U(17, 3)− 2"

y². én = 11§·��E15�èi�(11, 6, [2, 2])3èµ

〈1, 4, 6, 9〉〈6, 9, 0, 10〉〈1, 8, 2, 5〉〈2, 4, 8, 10〉〈6, 7, 1, 8〉〈8, 9, 4, 7〉〈0, 10, 1, 4〉〈0, 3, 8, 9〉〈4, 7, 0, 3〉〈8, 10, 3, 6〉〈2, 9, 1, 3〉〈7, 10, 5, 9〉〈3, 6, 4, 5〉〈0, 2, 6, 7〉〈1, 3, 7, 10〉

én = 17§¤I�èÏL�Eü��üW¿��§z��üW¿�±�

��(17, 6, 4)2è"�Ún5.5�y²aq§·��±��(17, 6, 4)3è"

1��W¿�«|µ

(7, 4, 10, 1) (13, 12, 2, 5) (10, 14, 6, 13) (8, 4, 11, 14) (10, 9, 8, 2) (7, 2, 14, 15) (2, 6, 0, 4)

(8, 16, 1, 6) (9, 14, 1, 12) (0, 10, 12, 11) (15, 11, 9, 6) (13, 3, 9, 4) (5, 3, 15, 10) (0, 5, 7, 9)

(7, 6, 3, 12) (0, 14, 3, 16) (4, 16, 15, 12) (0, 13, 8, 15) (1, 11, 2, 3) (7, 11, 13, 16)

1��W¿�«|µ

(9, 7, 13, 1) (1, 3, 6, 15) (15, 12, 10, 2) (3, 8, 16, 2) (11, 3, 10, 9) (14, 16, 6, 12) (1, 5, 2, 0)

(4, 6, 5, 10) (2, 11, 6, 7) (10, 16, 13, 0) (0, 12, 3, 4) (5, 15, 16, 9) (11, 0, 15, 14) (6, 9, 0, 8)

(14, 3, 7, 5) (2, 9, 4, 14) (11, 13, 8, 12) (4, 7, 8, 15) (10, 14, 1, 8) (4, 1, 11, 16)

64 Aa|Ü?è¯KïÄ

Ún 5.35. �3��.�2951�(4, 2)-CRSSGDD"

y². -:8�Z18 ∪ ({a} × Z3) ∪ ({b, c} × Z1)§|8�{{0, 9} + i : 0 ≤ i ≤8} ∪ {({a} × Z3) ∪ ({b, c} × Z1)}"¤I�O´dXeÄ«|+6 (mod 18)Ðm

��"


(1, 4, 12, a0) (3, 5, 7, b0) (0, 13, 14, 17) (0, 3, 6, a0) (5, 9, 17, a0) (0, 5, 8, 16) (10, 11, 16, a0)

(2, 7, 13, a0) (0, 2, 4, b0) (8, 14, 15, a0) (3, 4, 8, c0) (0, 1, 11, c0) (4, 7, 9, 15)


(1, 2, 8, a0) (0, 12, 14, a0) (4, 7, 10, a0) (3, 5, 15, a0) (6, 9, 16, a0) (2, 3, 4, 17) (11, 13, 17, a0)

(0, 4, 8, 11) (0, 5, 13, b0) (3, 8, 16, b0) (0, 16, 17, c0) (1, 3, 14, c0) (0, 1, 7, 15)

Ún 5.36. é?¿u ∈ {12, 15, 18, 21, 24, 36}§�3.�2u51�(4, 2)-CRSSGDD"

y². -:8�Z2u ∪ ({a, b, c, d, e} × Z1)§|8�{{0, u} + i : 0 ≤ i ≤ u −1} ∪ {{a, b, c, d, e} × Z1}"éu ∈ {12, 18, 24, 36}§¤I�O´dXeÄ«|+3

(mod 2u)Ðm��"éu ∈ {15, 21}§¤I�OdXeÄ«|+6 (mod 2u)Ðm

��"

21251µ


(2, 18, 22, a0) (0, 1, 2, b0) (2, 9, 16, c0) (2, 7, 21, d0)

(0, 16, 23, e0) (0, 13, 19, 22) (2, 8, 10, 23) (0, 3, 14, 18)


(1, 9, 11, a0) (2, 3, 10, b0) (1, 5, 15, c0) (0, 1, 8, d0)

(1, 2, 21, e0) (1, 7, 20, 22) (2, 5, 9, 20) (0, 3, 9, 22)

21851µ


(2, 7, 33, a0) (0, 7, 29, b0) (2, 24, 25, c0) (1, 2, 12, d0) (0, 22, 32, e0) (0, 6, 8, 23)

(1, 4, 8, 20) (0, 11, 12, 27) (0, 4, 13, 34) (1, 13, 18, 21) (2, 8, 10, 35)


(2, 12, 34, a0) (1, 12, 32, b0) (2, 3, 13, c0) (0, 11, 31, d0) (2, 19, 21, e0) (2, 26, 28, 29)

(1, 4, 21, 33) (0, 1, 13, 28) (0, 3, 5, 9) (1, 8, 23, 31) (2, 9, 24, 32)


22451µ


(0, 1, 8, a0) (0, 4, 29, b0) (0, 11, 40, c0) (2, 13, 30, d0) (2, 6, 16, e0) (2, 20, 23, 36)

(0, 16, 38, 47) (0, 6, 18, 39) (0, 3, 5, 37) (0, 7, 19, 46) (0, 17, 22, 23) (1, 16, 21, 46)

(2, 17, 24, 37) (2, 4, 10, 14)


(0, 13, 32, a0) (1, 30, 32, b0) (1, 17, 18, c0) (0, 25, 26, d0) (0, 1, 38, e0) (2, 5, 17, 44)

(1, 28, 35, 43) (0, 3, 18, 23) (2, 15, 22, 25) (2, 4, 36, 40) (1, 3, 14, 44) (2, 6, 28, 46)

(0, 28, 37, 42) (0, 9, 17, 36)

23651µ


(1, 32, 57, a0) (0, 17, 46, b0) (2, 12, 52, c0) (0, 11, 13, d0) (0, 23, 37, e0) (0, 14, 22, 51)

(0, 12, 15, 20) (2, 3, 9, 36) (2, 17, 22, 45) (2, 15, 41, 71) (1, 20, 31, 42) (0, 32, 63, 67)

(0, 34, 55, 68) (0, 1, 2, 25) (2, 8, 53, 62) (0, 10, 30, 54) (0, 19, 58, 64) (2, 26, 58, 70)

(1, 8, 55, 70) (1, 3, 10, 56)


(0, 1, 50, a0) (2, 36, 46, b0) (0, 46, 65, c0) (1, 6, 59, d0) (0, 2, 28, e0) (2, 8, 11, 59)

(1, 53, 58, 71) (1, 13, 24, 64) (0, 17, 21, 64) (2, 12, 15, 60) (1, 15, 21, 46) (1, 23, 39, 62)

(2, 14, 54, 70) (2, 31, 32, 64) (0, 4, 7, 57) (0, 11, 42, 54) (2, 3, 29, 66) (1, 32, 49, 67)

(0, 5, 33, 70) (1, 35, 43, 60)

21551µ


(21, 17, 8, a0) (28, 6, 25, a0) (0, 14, 5, b0) (21, 19, 16, b0) (5, 1, 8, c0) (5, 12, 15, 18)

(0, 21, 28, c0) (21, 12, 22, d0) (7, 20, 23, d0) (0, 26, 25, e0) (21, 5, 10, e0) (14, 3, 8, 15)

(0, 12, 20, 16) (0, 1, 29, 11) (14, 22, 2, 16) (0, 7, 13, 2) (7, 3, 25, 16) (5, 22, 4, 11)

(21, 1, 15, 23)


(16, 12, 25, b0) (27, 29, 26, b0) (0, 8, 5, e0) (27, 13, 22, e0) (5, 7, 26, c0) (5, 24, 15, 6)

(0, 27, 16, c0) (19, 20, 11, a0) (27, 24, 4, a0) (0, 2, 25, d0) (27, 5, 10, d0) (8, 21, 26, 15)

(0, 7, 23, 17) (8, 4, 14, 22) (0, 19, 1, 14) (0, 24, 20, 22) (19, 21, 25, 22) (5, 4, 28, 17)

(27, 7, 15, 11)

22151µ


(0, 13, 17, a0) (2, 3, 22, a0) (0, 38, 40, b0) (5, 15, 25, b0) (0, 2, 16, c0) (4, 21, 34, 37)

(3, 5, 7, c0) (0, 25, 27, d0) (2, 28, 41, d0) (2, 21, 24, e0) (4, 11, 19, e0) (2, 12, 17, 26)

(1, 6, 25, 36) (0, 26, 34, 35) (1, 7, 21, 38) (1, 17, 31, 34) (0, 1, 8, 11) (1, 2, 9, 14)

(0, 3, 9, 29) (0, 15, 23, 24) (4, 8, 14, 31) (5, 9, 20, 33) (3, 4, 10, 15) (4, 23, 35, 41)

(0, 4, 22, 36)

66 Aa|Ü?è¯KïÄ


(0, 17, 19, b0) (22, 33, 32, b0) (0, 40, 20, c0) (13, 39, 23, c0) (0, 22, 8, a0) (2, 21, 38, 29)

(33, 13, 35, a0) (0, 23, 3, d0) (22, 14, 31, d0) (22, 21, 12, e0) (2, 37, 41, e0) (22, 6, 19, 34)

(11, 24, 23, 18) (0, 34, 38, 7) (11, 35, 21, 40) (11, 19, 5, 38) (0, 11, 4, 37) (11, 22, 15, 28)

(0, 33, 15, 25) (0, 39, 1, 12) (2, 4, 28, 5) (13, 15, 10, 27) (33, 2, 26, 39) (2, 1, 7, 31)

(0, 2, 32, 18)

Ún 5.37. é?¿u ∈ {30, 33, 39, 42, 51}§�3��.�2u51�(4, 2)-CRSSGDD"

y². ùp¤^�{�©[68]¥�{aq"-:8�Z2u ∪ {∞1,∞2, . . . ,∞5}§|8�{{0, u} + i : 0 ≤ i ≤ u− 1} ∪ {∞1,∞2, . . . ,∞5}"-2u = 3x"�O¤I

«|düÜ©|¤"1�Ü©´dXeÄ«|+3 (mod 2u)Ðm��"

1��GDD�1�Ü©Ä«|µ

��x ≡ 1 (mod 3)��

0 x x+ 1 ∞1

2x+ 1 2x+ 2 2x ∞2

x+ 2 4 x+ 4 ∞3

3 1 2x+ 3 ∞4

2x+ 4 x+ 3 2 ∞5

��x ≡ 2 (mod 3)��

0 2x 2x+ 1 ∞1

x+ 1 x+ 2 x ∞2

2x+ 2 4 2x+ 4 ∞3

3 1 x+ 3 ∞4

x+ 4 2x+ 3 2 ∞5



��x ≡ 1 (mod 3)��

x+ 2 x 2x ∞1

3 2x+ 2 x+ 4 ∞2

2x+ 4 4 2x+ 3 ∞3

0 1 2 ∞4

2x+ 1 x+ 3 x+ 1 ∞5

��x ≡ 2 (mod 3)��

2x+ 2 2x x ∞1

3 x+ 2 2x+ 4 ∞2

x+ 4 4 x+ 3 ∞3

0 1 2 ∞4

x+ 1 2x+ 3 2x+ 1 ∞5

éu ∈ {30, 42}§¤I�O�1�Ü©«|´dXeÄ«|+1 (mod 2u)Ð

m��"

23051µ

1��GDD�1�Ü©Ä«|µ(0, 29, 35, 52) (0, 15, 39, 49) (0, 5, 18, 32) (0, 4, 48, 57)

1��GDD�1�Ü©Ä«|µ(0, 8, 11, 36) (0, 5, 17, 26) (0, 4, 18, 33) (0, 6, 16, 53)

24251µ

1��GDD�1�Ü©Ä«|µ(0, 12, 22, 81) (0, 31, 38, 70) (0, 60, 65, 78) (0, 9, 49, 57)

(0, 30, 41, 64) (0, 4, 21, 37)

1��GDD�1�Ü©Ä«|µ(0, 5, 23, 54) (0, 14, 47, 59) (0, 41, 50, 63) (0, 17, 57, 81)

(0, 65, 73, 80) (0, 36, 68, 74)

éu ∈ {33, 39, 51}§1�Ü©´dXeÄ«|+2 (mod 2u)Ðm��"

23351µ


(0, 24, 49, 63) (1, 7, 41, 54) (1, 8, 13, 38) (0, 31, 35, 40) (1, 20, 37, 52)

(1, 40, 43, 51) (0, 8, 14, 18) (0, 7, 45, 55) (0, 9, 38, 54)

68 Aa|Ü?è¯KïÄ


(0, 26, 50, 62) (1, 11, 15, 36) (0, 5, 8, 55) (0, 9, 37, 49) (0, 3, 18, 56)

(0, 6, 17, 35) (0, 7, 14, 39) (0, 15, 21, 57) (0, 19, 27, 32)

23951µ


(1, 47, 60, 70) (1, 5, 19, 43) (1, 12, 57, 73) (1, 13, 16, 21) (0, 8, 29, 38) (0, 13, 20, 34)

(0, 6, 23, 33) (0, 32, 43, 73) (1, 28, 32, 44) (1, 30, 48, 72) (0, 3, 22, 37)


(1, 5, 50, 59) (0, 5, 13, 49) (0, 40, 58, 74) (0, 3, 67, 70) (1, 33, 38, 52) (0, 17, 35, 48)

(1, 17, 58, 73) (0, 31, 61, 71) (0, 51, 63, 72) (0, 24, 36, 43) (0, 10, 32, 55)

25151µ


(1, 5, 49, 77) (0, 43, 66, 93) (0, 17, 64, 78) (0, 3, 19, 58) (1, 13, 18, 95) (1, 43, 65, 89)

(0, 45, 86, 90) (0, 21, 31, 84) (0, 52, 59, 92) (0, 33, 60, 73) (0, 46, 76, 99) (0, 5, 71, 82)

(0, 54, 65, 83) (1, 66, 88, 94) (1, 8, 16, 97)


(1, 4, 67, 88) (0, 27, 38, 90) (1, 6, 39, 59) (0, 23, 76, 80) (1, 7, 25, 84) (0, 13, 30, 58)

(0, 47, 55, 95) (1, 72, 77, 81) (1, 31, 44, 47) (0, 7, 17, 36) (1, 32, 42, 53) (0, 37, 65, 79)

(0, 14, 54, 96) (0, 46, 75, 87) (0, 69, 78, 94)

Ún 5.38. �3��.�22751�(4, 2)-CRSSGDD"

y². -:8�{Z3 × Z18} ∪ ({a, b, c, d, e} × Z1)§|8�{{(i, 2j), (i, 2j + 1)} :

0 ≤ i ≤ 2, 0 ≤ j ≤ 8} ∪ {{a, b, c, d, e} × Z1}"Xe�Ã¡:�Ä«|(−,+1

(mod 18))Ðm§Ù¦�Ä«|(+1 (mod 3),+2 (mod 18))Ðm§Ò��·�I

��O"


((0, 0), (1, 0), (2, 1), a0) ((0, 1), (1, 2), (2, 0), b0) ((0, 2), (1, 4), (2, 4), c0)

((0, 3), (1, 1), (2, 3), d0) ((0, 4), (1, 3), (2, 2), e0) ((0, 1), (0, 3), (0, 13), (1, 8))

((0, 0), (0, 9), (0, 10), (1, 15)) ((0, 1), (0, 16), (1, 10), (2, 6)) ((0, 1), (1, 4), (2, 11), (2, 15))

((0, 1), (1, 11), (1, 16), (2, 7)) ((0, 0), (0, 5), (0, 12), (0, 16)) ((0, 0), (0, 15), (1, 8), (2, 12))



((0, 0), (1, 1), (2, 2), d0) ((0, 2), (1, 0), (2, 0), a0) ((0, 4), (1, 4), (2, 3), c0)

((0, 1), (1, 3), (2, 1), e0) ((0, 3), (1, 2), (2, 4), b0) ((0, 1), (0, 16), (1, 8), (1, 12))

((0, 1), (0, 5), (0, 14), (2, 9)) ((0, 0), (0, 11), (0, 13), (2, 5)) ((0, 0), (0, 6), (0, 16), (1, 13))

((0, 0), (1, 4), (2, 9), (2, 12)) ((0, 0), (1, 12), (2, 3), (2, 15)) ((0, 0), (0, 7), (0, 17), (1, 11))

Ún 5.39. é?¿t ≥ 3§�3.�23t51�(4, 2)-CRSSGDD"

y². ét ∈ [3, 14] ∪ {17}§¤I�O3Ún 5.35–5.38¥�E��"ét ∈{15, 18, 21, 27, 33}§�.�6u§u ∈ {5, 6, 7, 9, 11}�(4, 2)-CRSSGDD¿^3)

ä"O\5�Ã¡:§3|þëÓÃ¡:W\.�2951�(4, 2)-CRSSGDD�

�¤I�O"ét = 16§�.�84�(4, 2)-CRSSGDD¿^3)ä§O\5�Ã

¡:§3|þëÓÃ¡:W\.�21251�(4, 2)-CRSSGDD��¤I�O"

ét ∈ {28, 32}§�.�6u§u ∈ {7, 8}�{4}-GDD§^WFC\�4§O\5�

Ã¡:§3|þëÓÃ¡:W\.�21251�(4, 2)-CRSSGDD"ét ∈ {22, 26}§�.�6u91§u ∈ {4, 5}�{4}-GDD£�[77,½n1.6]¤§^WFC\�4§O\5�

Ã¡:§3|þëÓÃ¡:W\.�21251½21851�(4, 2)-CRSSGDD��¤I

�O"

ét = 31§�.�3871�{5}-GDD§^WFC\�6§O\5�Ã¡:§3|

þëÓÃ¡:W\.�2951½22151�(4, 2)-CRSSGDD��¤I�O"ét = 23§

��TD(5, 5)§^WFCéco�|�¤k:§Ú��|��:\�6§

Ù{:\�3��.�304181�(4, 2)-CRSSGDD"O\5�Ã¡:§3|þëÓ

Ã¡:W\.�21551½2951�(4, 2)-CRSSGDD��¤I�O"ét = 19§��

�TD(5, 4)§^WFCéco�|�¤k:§9��|�ü�:\�6§Ù

{:\�3§��.�244181�(4, 2)-CRSSGDD"O\5�Ã¡:§3|þëÓ

Ã¡:W\.�21251½2951�(4, 2)-CRSSGDD��¤I�O"

ét ∈ {20, 24, 25, 29, 30}½t ≥ 34§lÚn2.4��(t + 1, {5, 6, 7, 8, 9}, 1)-

PBD"�K��:��.�4i5j6k7l8m�{5, 6, 7, 8, 9}-GDD§Ù¥4i+ 5j+ 6k+

7l+8m = t"^WFC\�6§Ñ\.�6u§u ∈ {5, 6, 7, 8, 9}£Ún5.14¤�(4, 2)-

CRSSGDD��.�24i30j36k42l48m�(4, 2)-CRSSGDD"O\5�Ã¡:§3

|þëÓÃ¡:W\.�23s51§s ∈ {4, 5, 6, 7, 8}�(4, 2)-CRSSGDD"·�Ò�

�.�23t51§t ∈ {20, 24, 25, 29, 30}½t ≥ 34�(4, 2)-CRSSGDD"

70 Aa|Ü?è¯KïÄ

L 5.2: Ún5.40¥¤I��n ��

23 (0,17,15,11,12,1,20,18,5,19,4,9,8,14,3,21,6,7,10,16,22,2,13)

29 (0,28,12)(1,2,16,3,8,7,21,27,13,4,11,14,15,24,6)(5,23,20,10,17,26,9)(18)(19)(22,25)

35 (0,33,18,24,11,14,30,8,1,31,2,20,13,21,4,27,5,29,3,32,12,17,9,23,15,6,25,22,10,19,28,26,34,7)(16)

41 (0,32,14,39,26,16,30,23,29,34,11,40,5,13,2,25,18,3,28,27,37)(1,33,4,36,19,22,6,8,31,12,17,35,7,

38,10,24,9,21,15)(20)

47 (0,25,20,26,38,43,8,11,4,3,28,46,5,44,40,27,30,45,23,1,32,6,18,2,35,21,24,14,31,22,39,37,16,36,9,

7)(10,29,33,13,15,42)(12)(17,19)(34,41)

53 (0,8,26,18,2,20,36)(1,31,38,32,46,43,15,44,34,24,23,13,30,12,45,22,28,11,7,29,40,50,19,14,6,27,

39,47,48,41,37,49,25)(3,42,17,16)(4,51)(5,9,21,10,52,33)(35)

59 (0,46,41,5,38)(1,6,18,10,21,17,27,45,58,42,22,23,51,52)(2,44,12,30,49,15,29,19,54)(3,31,8,13,16,

56,53,33,37,36,11,55,26,25,14,32,40,34,24,4,7)(9,57,28,20)(35,39)(43,50,48,47)

65 (0,46,13,55,32,19,59,23,54,53,33,60,50,25,51,47,20,5,42,11,57,1)(2,35,8,18,56,22,10,64,27,62)(3,

48,58,17,34,49,41,7,31,12,43,40,24,9)(4,61,16,15,26,14,30,45,38,29,6,44,36,39,52)(21)(28,63)(37)

77 (0,46,60,72,55,52,36,63,33,62,9,75,64,47,39,53,67,11,31,35,34,45,24,48,2,30,25,19,56,50,61,68,28,

16,65,1,22,17,4,27,51,69,32,5,10,71,42,58,41,12,7,44,37,57,21)(3,8,70,26,23,54,20,73,18,15)(6,13,

14,59,43,74,66,76)(29,38,49)(40)

89 (0,46,44,49,58,10,70,68,43,16,54,39,62,67,41,34,38,84,88,6,81,26,69,40,86,66,74,47,12,82,55,2,27,

78,11,13,71,80,76,7,48,51,24)(1,22,56,83,42,77,53,60,29,19,65,50,37,35,25,17,30,23,73,9,72,33,79,

5,85,64,52,21,4,63,20,15,32,59,36,45,28,61,87,18,3,8,75)(14,31,57)

Ún 5.40. é?¿n = 6t+ 5§t ∈ [3, 10] ∪ {12, 14}§A3(n, 6, 4) = U(n, 3)"

y². é?¿n = 6t + 5§lÚn5.35–5.38��.�23t51�(4, 2)-CRSSGDD"

�{B§·�r��5�|¥�:P�6t§6t + 1§6t + 2§6t + 3§6t + 4"

=µ�t = 3�§·�©Ora0§b0§c0§a1§a2^6t§6t + 1§6t + 2§6t + 3§

6t+ 4O�¶�t ∈ [4, 9] ∪ {12}�§·�©Ora0§b0§c0§d0§e0^6t§6t+ 1§

6t+ 2§6t+ 3§6t+ 4O�¶�t ∈ {10, 14}�§·�©Or∞1§∞2§∞3§∞4§

∞5^6t§6t+ 1§6t+ 2§6t+ 3§6t+ 4O�"ùp§ét = 9§·�kI�^N

�(a, b) 7→ 18 · a+ b�^31��GDD�:8þ"3��5�|þW\��#

�«|{6t, 6t + 1, 6t + 2, 6t + 3}§l ��3:8{0, 1, 2, . . . , 6t + 4}þ��ê�1��`W¿"^L5.2¥��^3:8þ§Ò��1��`W

¿§¿�ùü��`W¿÷vCRSS5�"dÚn5.5�y²§·�Ò��

A��`(6t+ 5, 6, 4)3è"

Ún 5.41. é?¿u ∈ {24, 30, 42}§�3��.�2u231�(4, 2)-CRSSGDD"


y². -:8�Z2u ∪ {∞1,∞2, . . . ,∞5} ∪ M§|8�{{0, u} + i : 0 ≤ i ≤u− 1} ∪ {{∞1,∞2, . . . ,∞5} ∪M}"-2u = 3x"¤I�O�Ä«|´düÜ©

|¤"1�Ü©´dÚn5.37¥�1�Ü©Ä«|+3 (mod 2u)Ðm��"1

�Ü©´dXeÄ«|+1½+2 (mod 2u)Ðm��"

224231µ+2 (mod 48)¶M=({a} × Z12) ∪ ({b} × Z6)


(0, 41, 3, a0) (7, 18, 13, a0) (9, 4, 29, a0) (14, 6, 36, a0) (15, 2, 8, a0) (11, 23, 44, a0)

(22, 43, 34, a0) (5, 35, 27, b0) (4, 14, 1, b0) (8, 12, 31, b0) (1, 16, 45, a0) (6, 34, 45, b0)


(15, 40, 22, a0) (20, 17, 8, a0) (6, 44, 2, b0) (11, 41, 22, b0) (21, 14, 34, a0) (18, 43, 47, a0)

(0, 11, 26, a0) (12, 4, 7, a0) (1, 29, 37, a0) (7, 45, 40, b0) (3, 9, 30, a0) (0, 13, 39, b0)

230231µ+1 (mod 60)¶M=({a} × Z15) ∪ ({b} × Z3)


(7, 35, 23, a0) (3, 45, 58, a0) (1, 5, 12, b0) (1, 4, 40, a0)

(9, 26, 59, a0) (2, 27, 36, a0) (0, 6, 14, 29)


(1, 11, 45, b0) (5, 19, 48, a0) (8, 55, 47, a0) (0, 11, 7, a0)

(9, 42, 14, a0) (1, 13, 36, a0) (0, 3, 9, 45)

242231µ+1 (mod 84)¶M=({a, b, c} × Z6)


(1, 45, 48, a0) (2, 10, 23, a0) (0, 5, 14, b0) (1, 46, 81, b0) (0, 10, 33, c0)

(1, 17, 44, c0) (0, 6, 73, 25) (0, 24, 31, 46) (0, 12, 30, 64)


(0, 3, 7, a0) (2, 11, 16, a0) (0, 8, 19, b0) (3, 41, 16, b0) (0, 15, 31, c0)

(5, 40, 50, c0) (0, 6, 54, 33) (0, 12, 34, 52) (0, 17, 37, 60)

Ún 5.42. é?¿t ≥ 3§t 6∈ {16, 20, 22, 29, 32, 40}§A3(6t+5, 6, 4) = U(6t+5)"

y². ét ∈ [3, 10]∪{12, 14}§¤I�(6t+ 5, 6, 4)3è3Ún5.40¥�E��"é

Ù{�t§·��E.�23s231§s ≥ 8§s 6∈ {9, 11, 13, 17, 19, 26, 29, 37}�(4, 2)-

CRSSGDD"

72 Aa|Ü?è¯KïÄ

és ∈ {8, 10, 14}§¤I�O3Ún5.41¥�E��"és ∈ {12, 15, 18, 21,

24, 27, 30, 33, 36}§�.�6u§5 ≤ u ≤ 13�(4, 2)-CRSSGDD§¿^3)ä"3|

þW\.�2951�(4, 2)-CRSSGDDÒ��.�29(u−1)231�(4, 2)-CRSSGDD"

és = 16§��TD(5, 4)§^WFCéc4�|�¤k:§��|�3�:

\�6§Ù{:\�3"O\2�Ã¡:¿3|þëÓÃ¡:W\.�213�(4, 2)-

CRSSGDDÒ��¤I�O"és = 28§�.�6431�{4}-GDD§^WFC\

�7§O\2�Ã¡:¿W\.�222�(4, 2)-CRSSGDDÒ��¤I�O"és =

32§��TD(4, 4)§^WFC \�12§O\23�Ã¡:§¿ëÓÃ¡:W\.

�224231�(4, 2)-CRSSGDDÒ��¤I�O"

és ∈ {20, 22, 23, 25}§��TD(6, 5)§^WFCéc4�|�:§15�|

�x�:§16�|�2�:\�6§16�|�Ù{:\�3"Ù{:\�0"ùp

�Ñ\�O´.�65§66§6431Ú6531�(4, 2)-CRSSGDD"O\2�Ã¡:§¿

W\.�27§210½216�(4, 2)-CRSSGDDÒ��¤I�O"-x = 5§Ò��.

�275231�(4, 2)-CRSSGDD¶-x = 3§Ò��.�269231�(4, 2)-CRSSGDD¶

-x = 2§Ò��.�266231�(4, 2)-CRSSGDD¶-x = 0§Ò��.�260231�

(4, 2)-CRSSGDD"

és ∈ {31, 34}§��TD(6, 7)§^WFCéco�|�¤k:§15�

|�x�:\�6§��|�:\�3§Ù{:\�0"ùp�Ñ\�O

´.�6431Ú6531�(4, 2)-CRSSGDD"O\2�Ã¡:§3|þëÓÃ¡:W

\.�222½23x+1�(4, 2)-CRSSGDD§Ò��¤I�O"-x = 6§Ò��.

�2102231�(4, 2)-CRSSGDD¶-x = 3§Ò��.�293231�(4, 2)-CRSSGDD"

��TD(12, 11)§^WFCéc5�|�¤k:§��|�3�:§�

{�1i�|�xi�:§1 ≤ i ≤ 6§\�6"Ù{:\�0"�xi = 0½3 ≤ xi ≤11"O\5�Ã¡:§3|þëÓÃ¡:W\.�23u51§u ∈ [3, 11]�(4, 2)-

CRSSGDD§Ò��.�25·33+3∑xi231 ≡ 23s231�(4, 2)-CRSSGDD§Ù¥s =

5· 11 +∑xi ∈ {55} ∪ [58, 121]"aq/§XJ�TD(9, 8)§·�ò��s ∈

{40} ∪ [43, 64]¶XJ�TD(8, 7)§·�ò��s ∈ {35} ∪ [38, 49]"

és ≥ 83§lÚn2.9��TD(7, n)§^WFCéc5�|�¤k:§16�

|�x�:§��|�3�:\�6"Ù{:Ñ\�0"·��.

�(6n)5(6x)1181§x = 0½3 ≤ x ≤ n�(4, 2)-CRSSGDD"O\5�Ã¡:§

3c6�|ëÓÃ¡:W\.�23n51½23x51�(4, 2)-CRSSGDD§Ò��.


�23(5n+x)231�(4, 2)-CRSSGDD§Ù¥s = 5n + x�±��Ø�u83�?Û

��ê"

dÚn5.5§·�Ò��.�23s231§s ≥ 8§s 6∈ {9, 11, 13, 17, 19, 26, 29,

37}�GDC"3��23�|þW\�`(23, 6, 4)3èÒ��¤I�è"

Ún 5.43. é?¿t ∈ {16, 20, 22, 29, 32, 40}§A3(6t+ 5, 6, 4) = U(6t+ 5, 3)"

y². Äk§·��E.�23s291§s ∈ {12, 16, 18, 25, 28, 36}�(4, 2)-CRSSGDD"

és ∈ {12, 18}§©O�.�6491½6691�(4, 2)-CRSSGDD§^3)ä§��

.�184271½186271�(4, 2)-CRSSGDD"O\2�Ã¡:§3��18�|þ

ëÓÃ¡:W\.�210�(4, 2)-CRSSGDD��¤I�O"és = 16§��

�TD(5, 4)§^WFC\�6��.�245�(4, 2)-CRSSGDD"O\5�Ã¡:§

3c4��24�|þëÓÃ¡:W\.�21251�(4, 2)-CRSSGDD��¤

I�O"

és = 25§��TD(6, 5)§^WFCéc5�|�:§��|�4�

:\�6"Ù{:\�0"��.�305241�(4, 2)-CRSSGDD"O\5�Ã¡

:§3��30�|ëÓÃ¡:W\.�21551�(4, 2)-CRSSGDD��¤I

�O"és = 28§�.�68�{4}-GDD§^WFC\�4��.�248�(4, 2)-

CRSSGDD"O\5�Ã¡:§3c7�|ëÓÃ¡:W\.�21251�(4, 2)-

CRSSGDD��¤I�O"és = 36§��TD(7, 7)§�K��:��

.�6671�{6, 7}-GDD"^WFCéc6�|�:§��|�4�:\�6§

Ù{:\�0§��.�366241�(4, 2)-CRSSGDD"O\5�Ã¡:§3��

�36�|þëÓÃ¡:W\.�21851�(4, 2)-CRSSGDD��¤I�O"

dÚn5.5§·��.�23s291§s ∈ {12, 16, 18, 25, 28, 36}�GDC"3ù

GDC��29�|þW\�`(29, 6, 4)3èÒ��¤I�è"


½n 5.44. é?¿t ≥ 3§A3(6t+5, 6, 4) = U(6t+5, 3)¶A3(11, 6, 4) ≥ U(11, 3)−1¶A3(17, 6, 4) ≥ U(17, 3)− 2"

5.4 (((ØØØ

3�Ù¥§·�A��(½�`(n, 6, 4)3è�èi�ê"·�r(J

o(Xeµ

74 Aa|Ü?è¯KïÄ

½n 5.45. é?¿�ên ≥ 4§

A3(n, 6, 4) =

1§ �n ≤ 5�

3§ �n = 7�

5§ �n = 8�⌊n2

⌊n−13

⌋⌋§ �n ≥ 6§n 6∈ {7, 8, 11, 17}�

A3(11, 6, 4) ∈ [15, 16]§A3(17, 6, 4) ∈ [40, 42]"

5µ©[159]¥��"v<^O�Å|¢(½A3(11, 6, 4) = 15"

Chapter 6

|||��ooo��üüü��OOOÚÚÚ��'''��üüüWWW¿¿¿


��ü£CRSS¤�O�q�~è£CWC¤��'"��(v, 4, q)-

CRSS�OÒ´��`(v, 6, 4)q+1è"CRSS�©|�O£CRSSGDD¤�±^

5�Eq��©|è£GDC¤§ �©|è3q�CWC��E¥å��^"

315Ù¥§·�^þãCRSS�OÚCWC�éXA��(½�`(v, 6, 4)3

è�èi�ê", §�,éCWC®²kéõ<ïÄL§8céCRSS�O�

(J%Øõ§X[5]"

3�Ù¥§·�òUYïÄ«|��4�CRSS�O��35"Äk§·

��(½(v, 4, 3)-CRSS�O��35"Ïd§·��#��

`(v, 6, 4)4è"Ùg§·��Ñ��^�Room frame�E(4, 4)-CRSSGDD�

�{§¿y²��.�gu�(4, 2)-CRSSGDD�3�7�^�§ØA|

�(g, u) ∈ {(2, 4), (3, 4), (6, 4)}§�´¿©�"�ü�O´dGronau�u1992cJÑ�[83]"�ü�OØ�Ù�35��

´|Ü�OnØ¥��k��ïÄ¯K§ �3¢S¥�k�«��A^"

~Xù«�O�±^5�E�{MF¼ê[130]ÚCX�O[13]§�#��

O[12]§ÚU\è[100]�"

8c§'u�ü�O��õêó�Ñ´�Ä«|��4§�êλ ∈{2, 3, 4, 5, 6, 9}£�©[5, 21, 29–31, 81, 83, 99]¤§½ö«|��5§�êλ ∈{2, 4, 5}£�©[1, 2, 34, 35, 82]¤��üBIBD��35", §é�üW¿

��35§%A�vk?Û(J", §l©[159]¥�±wÑù«�O%

�CWCkX��éX"¤±§3�Ù¥§·�òïÄ�`�üW¿��35§

¿�y²é?¿v ≥ 4§Ø(½��v ∈ {4, 5, 6, 9}§�`�ü(v, 4, 2)-W¿

Ñ´�3�"

ù�Ù�(�Xeµ316.2!¥§·�ò0��Ä�VgÚ(J¶3

16.3!¥§·�ò�E(v, 4, 3)-CRSS�O§¿��a�`(v, 6, 4)3è¶3

76 Aa|Ü?è¯KïÄ

16.4!¥§·�ò�Ñ��^�Room frame�E(4, 4)-CRSSGDD��{§¿

316.5!¥§y².�gu�(4, 2)-CRSSGDD��35¶316.6!¥§·�ò

)û�`�ü(v, 4, 2)-W¿��35"


�©[5]¥y²aq§·�kXeCRSSGDD�3�7�^�"

Ún 6.1. ��.�gu�(4, λ)-CRSSGDD�3�7�^�´µ

(i) u ≥ 4§

(ii) (u− 1)g ≡ 0 (mod 3)§

(iii) u(u− 1)g2 ≡ 0 (mod 12)§

(iv) ug ≥ 2λ+ 2g§eg 6= 1"

1o�^�´��y¤k�U�n�|ê�uÑy3�O�«|¥�n

�|ê"AO�§�g = 1�§·�Ò��(v, 4, λ)-CRSS�O�3�7�^

�´v ≡ 1½4 (mod 12)§v ≥ 2λ+ 2½v = 1"

éGDDÚPBD�Ì��E§Ï~^�´“\�”�EÚWilson’sÄ��E{

£�[147]¤§Ò´Ï~l��“Ì”GDDÚ��Ñ\�O5��#�GDD"

·�éCRSSGDD��E�òæ^ù��{"·�l��CRSSGDDÑu§

^TD��Ñ\�O§½öl��GDDÑu§r�CRSSGDD��Ñ\�O"

äN5`§·�ò^�Xeü��E§=µ)ä{ÚGDD�Wilson’sÄ��E

{£WFC¤£�©[41]¤"

�E 6.2. £)ä{¤ b��3��.�{h1, h2, . . . , hn}�(K,λ)-CRSSGDD§

�é?¿k ∈ K§�3��TD(k,m)"@o�3��.�{mh1,mh2, . . . ,mhn}�(K,λ)-CRSSGDD"

�E 6.3.£WFC¤-(X,G,B)´��GDD§-w : X → Z+∪{0}´Xþ�\�¼ê"b�é?¿B ∈ B§Ñ�3��.�{w(x) : x ∈ B}�(K,λ)-CRSSGDD"

@o�3��.�{∑

x∈Gw(x) : G ∈ G}�(K,λ)-CRSSGDD"

3GDDÚPBD��E¥§“ÖÉ”��{åX��^"

CHAPTER 6 |��o��ü�OÚ�'�üW¿ 77

�E 6.4. (ÖÉ)

(i) b��3��.�{si : 1 ≤ i ≤ n}�(K,λ)-CRSSGDD"-a ≥ 0��

��ê"XJ§é?¿1 ≤ i ≤ n − 1§Ñ�3��.�{sij : 1 ≤ j ≤ki} ∪ {a}�(K,λ)-CRSSGDD§Ù¥si =

∑1≤j≤ki sij"@o�3��.

�{sij : 1 ≤ j ≤ ki, 1 ≤ i ≤ n− 1} ∪ {a+ sn}�(K,λ)-CRSSGDD"

(ii) b��3��.�{si : 1 ≤ i ≤ n}�(K,λ)-CRSSGDD"?�Úb��3

��.�{tj : 1 ≤ j ≤ t}�(K,λ)-CRSSGDD§Ù¥sn =∑

1≤j≤t tj"@o

�3��.�{si : 1 ≤ i ≤ n− 1} ∪ {tj : 1 ≤ j ≤ t}�(K,λ)-CRSSGDD"

��É�H�£|H| = w¤�(v, k, λ)-W¿§P�(v, w; k, λ)-W¿§´��

n�|(X,H,B)§Ù¥X´��v�:8§H�§��w�f

8§¡�É§«|8B´X¥�k�f8�8Ü§¦�Ø3Ép�?¿�É:é�õÑy3λ�«|¥§�vk«|�¹É¥�:é"

��É��w�(v, k, λ)-��Ø��W¿�O£maximum incomplete

packing design§MIPD¤§P�(v, w; k, λ)-MIPD§Ò´��n�|(X, Y,A)§

Ù¥X´��v�8Ü£¡�:¤§Y ⊆ X´��w�8Ü£¡�É¤§A´X¥��k�f8�8Ü£¡�«|¤§�÷vXe^�µ

1. éX¥�?¿�É:éx!y§XJxÚy��k��ØÑy3Y¥§@où

�:é�õÑy3A�λ�«|¥¶

2. Y¥�:éØÑy3?Û«|¥¶

3. λ(v − 1) ≡ λ(w − 1) ≡ d (mod k − 1)§Ù¥d´÷v0 ≤ d ≤ k − 2��

�ê¶

4. (X×X)\(Y ×Y )¥ØÑy3A�?Û«|¥�:éTÐkd(v−w)/2�"

MIPD�Vg´dYinÚAssaf3©[155]¥JÑ�§¦��Ñ�Ì�

�E�{"

�E 6.5 (Yin§Assaf [155]). b��3��.�{t1, t2, . . . , tn}�(k, λ)-GDD§�

é?¿1 ≤ i ≤ n−1§�3��(ti+w,w; k, λ)-MIPD"@o�3��(t+w, tn+

w; k, λ)-MIPD§Ù¥t =∑

1≤i≤n ti"

78 Aa|Ü?è¯KïÄ

�E 6.6 (Yin§Assaf [155]). b��3��(v, w; k, λ)-MIPD"XJw ≤ k − 1§

½ö�3��`(w, k, λ)-W¿§@o�3��(v, k, λ)-W¿"

·�r��ü(v, k, λ)-W¿�W¿êP�D′λ(v, k, 2)§w,D′λ(v, k, 2) ≤Dλ(v, k, 2)§Ù¥Dλ(v, k, 2)��(v, k, λ)-W¿�W¿ê"

é��(v, 4, 2)-W¿§kXe(Jµ

Ún 6.7 (Assaf [7]§Billington�[11]). é?¿v 6= 9§D2(v, 4, 2) = U2(v, 4, 2)¶

D2(9, 4) = U2(9, 4, 2) −1"

6.3 (v, 4, 3)-CRSS��OOO��333555

3�!¥§·�ò��(½(v, 4, 3)-CRSS�O��35"Ó�·��

�a#�(v, 6, 4)4è"

Äk§·�ò^�©[159]¥�Ó��{��E��O"�é�

��ê�λ�CRSS�O�§·�ò©O�E:8Ú|8�Ó�´�ê�1��

O§¦�?¿ü�«|�õ�uü�:§,�2r§�Ü¿å5�¤��ê

�λ�CRSS�O"e©¥§·�²~^¦f¦±1��O�Ä«|��1�

�½1n��O�Ä«|"

Ún 6.8. ©O�3.�47Ú126�(4, 3)-CRSSGDD"

y². é.�47�(4, 3)-CRSSGDD§-:8�Z28§|8�{{0, 7, 14, 21} + i :

0 ≤ i ≤ 6}"¤I�OdXeÄ«|3Z28¥+4 (mod 28)Ðm��"1�

�GDD�Ä«|µ{2, 5, 10, 21}§{0, 2, 18, 20}§{0, 11, 24, 27}§{3, 18, 22, 23}§{1, 3, 7, 9}§{0, 9, 19, 22}§{0, 1, 12, 25}§{1, 2, 19, 24}"1��GDD�Ä«|d

1��GDD�Ä«|¦±¦f5��"1n�GDD�Ä«|d1��GDD�

Ä«|¦±¦f17��"

é.�126�(4, 3)-CRSSGDD§-:8�Z72§|8�{{0, 6, . . . , 66} + i :

0 ≤ i ≤ 5}"¤I�OdXeÄ«|3Z72¥+1 (mod 72)Ðm��"1�

�GDD�Ä«|µ{0, 39, 55, 64}§{0, 15, 29, 52}§{0, 28, 62, 69}§{0, 22, 26, 27}§{0, 2, 13, 53}"1��GDD�Ä«|d1��GDD�Ä«|¦±¦f11��"

1n�GDD�Ä«|d1��GDD�Ä«|¦±¦f13��"

Ún 6.9. é?¿t ≥ 4§�3��.�12t�(4, 3)-CRSSGDD"


y². �t ≡ 0½1 (mod 4)§t ≥ 4�§dÚn2.7§�3��(3t + 1, {4}, 1)-

PBD"lù�PBD�:8�K��:§��.�3t�{4}-GDD"�t ≡2½3 (mod 4)§t ≥ 7�§dÚn2.7§�3��(3t + 1, {4, 7?}, 1)-PBD"lù

�PBD�:8�K��Ø3��7�«|¥�:§��.�3t�{4, 7?}-GDD"

Ïd§é?¿t ≥ 4§t 6= 6§·�Ñk.�3t�{4, 7}-GDD"

éù�GDD^WFC\�4��.�12t�(4, 3)-CRSSGDD§Ù¥t ≥ 4§t 6=6"ùp�Ñ\�O´.�44Ú47�(4, 3)-CRSSGDD£Ún5.4Ú6.8¤"ét = 6§

¤I�OdÚn6.8��E��"

Ún 6.10. é?¿v ∈ {25, 28, 37, 40}§�3��(v, 4, 3)-CRSS�O"

y². év = 25§-:8�GF (25)"-��õ�ª�f(x) = x2 + x + 2"1�

�BIBD�«|´3dÄ«|{0, 1, x8, x16}§{0, x2, x10, x18}3GF (25)¥�\{

+Ðm��"1��BIBD�Ä«|d1��BIBD�Ä«|¦±¦fxÐm�

�"1n�BIBD�Ä«|d1��BIBD�Ä«|¦±¦fx4Ðm��"

év = 28§-:8�Z28"¤I�OdXeÄ«|3Z28¥+4 (mod 28)Ð

m��"1��BIBD�Ä«|µ{2, 3, 11, 14}§{3, 13, 25, 27}§{1, 11, 24, 26}§{0, 3, 9, 10}§{2, 7, 8, 9}§{0, 6, 14, 16}§{1, 9, 14, 18}§{3, 8, 12, 19}§{1, 5, 8, 16}"1��BIBD�Ä«|d1��BIBD�Ä«|¦±¦f5Ðm��"1n

�BIBD�Ä«|d1��BIBD�Ä«|¦±¦f17Ðm��"

év = 37§-:8�Z37"¤I�OdXeÄ«|3Z37¥+1 (mod 37)Ð

m��"1��BIBD�Ä«|µ{0, 1, 3, 24}§{0, 4, 9, 15}§{0, 7, 17, 25}"1��BIBD�Ä«|d1��BIBD�Ä«|¦±¦f6Ðm��"1n�BIBD�

Ä«|d1��BIBD�Ä«|¦±¦f31Ðm��"

év = 40§-:8�Z39 ∪{∞}"¤I�OdXeÄ«|+3 (mod 39)��§

Ù¥§∞3gÓ��^e�±ØÄ"1��BIBD�Ä«|µ

{1, 5, 24, 34} {1, 14, 18, 31} {1, 4, 20, 25} {1, 9, 28, 33} {0, 1, 3, 12}{0, 4, 29,∞} {0, 11, 33, 38} {2, 9, 23, 34} {0, 2, 18, 26} {2, 4, 5, 35}

1��BIBD�Ä«|µ

{0, 7, 8, 11} {0, 34, 2,∞} {1, 21, 7, 20} {2, 27, 32, 12} {1, 37, 10, 35}{0, 22, 4, 33} {2, 10, 9, 26} {1, 27, 25, 9} {1, 11, 23, 29} {0, 27, 23, 36}

1n�BIBD�Ä«|d1��BIBD�Ä«|¦±¦f5Ðm��"

80 Aa|Ü?è¯KïÄ

½n 6.11. é?¿v ≡ 1, 4 (mod 12)§v ≥ 13§�3��(v, 4, 3)-CRSS�O"

y². év ∈ {13, 16}§¤I�O3©[5]¥�Ñ"év ∈ {25, 28, 37, 40}§¤I�O3Ún6.10¥�E��"

é?¿v ≥ 49§lÚn6.9��.�12t�(4, 3)-CRSSGDD"XJO\�

�:§¿3|þëÓÃ¡:W\(13, 4, 3)-CRSS�O§·�Òé?¿t ≥ 4��

(12t + 1, 4, 3)-CRSS�O"XJO\4�Ã¡:§¿3��|þëÓÃ¡:W

\(16, 4, 3)-CRSS�O§Ù§|þëÓÃ¡:W\.�11241�(4, 3)-CRSSGDD

£�©[5]¤§·�Òé?¿t ≥ 4��(12t+ 4, 4, 3)-CRSS�O"

dÚn5.5§·��a#��`(v, 6, 4)4è§=µ

íØ 6.12. é?¿v ≡ 1, 4 (mod 12)§v ≥ 13§A4(v, 6, 4) = v(v−1)4"

6.4 ^��Room frame��EEE(4, 4)-CRSSGDD

3�!¥§·�ò�Ñ��^�Room frame�E(4, 4)-CRSSGDD��

{"

-{S1, . . . , Sn}´8ÜS�y©§��{S1, . . . , Sn}-Room frameÒ´��

^S¥��IP�|S| × |S|�F§÷vµ

1. F¥�z�ü�½ö��§½ö�¹S¥��ÃS:é§

2. é?¿1 ≤ i ≤ n§f�Si × Si´��§£ùf�¡�É¤§

3. ?¿x 6∈ SiTÐ�¹31s1£�¤§s ∈ SiTÐ�g§

4. F¥�:é´{s, t}§Ù¥(s, t) ∈ (S × S)\⋃ni=1(Si × Si)"

��{S1, . . . , Sn}-Room frame F�.´��õ8{|S1|, . . . , |Sn|}"XJé1 ≤ j ≤ k§kuj�Si��tj§·�`F�.�t

u11 . . . tukk "��Room

frame´��XJü�(i, j)��§@oü�(j, i)�½´��".�1n�Room

frame�¡�Room�"

�Room frameé�E«|��4�BIBDÚGDD§)ûf3-XÚBIBD�

�35¯KÑå��^£�©[119]Ú[120]¤"


l��.�tu��Room frame F§·��±��.�(6t)u�{4}-GDD

£�©[119]¤"Ù¥§ù�{4}-GDD�:8�{Si × Z6 : 1 ≤ i ≤ n}§«|8B�¹¤k�«|{(a, j), (b, j), (c, 1 + j), (r, 4 + j)}§Ù¥j ∈ Z6§{a, b}´3F�1c�1r1��"

Ún 6.13 (Chen§Zhu[36]§Zhang§Ge[163]). ��.�tu��Room frame�3

�7�^�§=µu ≥ 4§t(u−1)´óê§Ø(½��(t, u) ∈ {(1, 5), (2, 4)}ÚXeØ(½��±§�´¿©�µ

(i) u = 4§t ≡ 2 (mod 4)§

(ii) u = 5§t ∈ {17, 19, 23, 29, 31}"

�E 6.14. e�3��.�tu��Room frame§K�3��.�(6t)u�(4, 4)-

CRSSGDD"

y². -F´�½�.�tu��Room frame"·��Eo�.�(6t)u�(4, 1)-

GDD§Ù¥:8�{{(i+k, j) : 0 ≤ i ≤ t−1, j ∈ Z6} : k = 0, t, . . . , t(u−1)}§o�ØÓ��O©O�¹«|{(x, j), (y, j), (c, 1+j), (r, 4+j)}§{(x, j), (y, j), (c, 4+

j), (r, 1+j)}§{(x, j), (y, j), (c, 2+j), (r, 5+j)}§{(x, j), (y, j), (c, 5+j), (r, 2+j)}Ù¥j ∈ Z6§{x, y} ∈ F§{x, y}´F�1c�1r1��"éN´�yù�Ò��o�.�(6t)u�{4}-GDD§¿�?¿ü�«|�õ�uü�:"

(ÜÚn6.13Ú�E6.14§·�kXe(Jµ

½n 6.15. eu ≥ 4§t(u−1)´óê§Ø(t, u) ∈ {(1, 5), (2, 4)}ÚXe�(t, u)§

�3��.�(6t)u�(4, 4)-CRSSGDDµ

(i) u = 4§t ≡ 2 (mod 4)§

(ii) u = 5§t ∈ {17, 19, 23, 29, 31}"

-t = 1§·�Ò��Xe(Jµ

íØ 6.16. eu´Ûê§�u ≥ 7§K�3��.�6u�(4, 4)-CRSSGDD"

82 Aa|Ü?è¯KïÄ

6.5 ...��gu��(4, 2)-CRSSGDD��333555

3�!¥§·�òy².�gu�(4, 2)-CRSSGDD�3�7�^�§Ø(

½��(g, u) ∈ {(2, 4), (3, 4), (6, 4)}±�´¿©�"·�3L6.1¥©a�ÑÚn6.1¥.�gu�(4, 2)-CRSSGDD�3�7�^

�"

L 6.1: .�gu�(4, 2)-CRSSGDD�3�7�^�

g u

g ≡ 0 (mod 12) u ≥ 4§u ∈ Ng ≡ 1, 5, 7, 11 (mod 12) u ≥ 4§u ≡ 1, 4 (mod 12)§(g, u) 6= (1, 4)

g ≡ 2, 10 (mod 12) u ≥ 4§u ≡ 1 (mod 3)

g ≡ 3, 9 (mod 12) u ≥ 4§u ≡ 0, 1 (mod 4)

g ≡ 4, 8 (mod 12) u ≥ 4§u ≡ 1 (mod 3)

g ≡ 6 (mod 12) u ≥ 4§u ∈ N

d)ä{ÚÚn5.2§Ún6.9¥�®�(J§·��I��Ä�g ∈{2, 3, 4, 6}�.�gu�(4, 2)-CRSSGDD��35"

a. ...��6u��(4, 2)-CRSSGDD

Ún 6.17. é?¿u ∈ {14, 18}§�3��.�6u�(4, 2)-CRSSGDD"

y². -:8�Z6u§|8�{{0, u, 2u, . . . , 5u} + i : 0 ≤ i ≤ u − 1}"¤I�OdXeÄ«|3Z6u¥+2 (mod 6u)Ðm��"

u = 14µ


{1, 10, 11, 18} {1, 21, 32, 69} {0, 9, 12, 27} {0, 18, 38, 59} {0, 22, 45, 47} {0, 2, 7, 51}{0, 26, 55, 61} {0, 48, 79, 80} {0, 6, 50, 74} {1, 14, 53, 77} {0, 13, 17, 43} {0, 3, 54, 65}{0, 19, 57, 69}

1��GDD�Ä«|d1��GDD�Ä«|¦±¦f25��"

u = 18µ


{1, 2, 64, 75} {1, 22, 48, 51} {1, 5, 33, 100} {1, 18, 32, 38} {0, 15, 22, 38} {0, 23, 75, 105}{1, 3, 23, 52} {0, 24, 51, 63} {0, 28, 60, 93} {0, 34, 69, 83} {0, 37, 53, 64} {0, 42, 98, 100}{0, 7, 55, 96} {1, 41, 47, 85} {0, 21, 31, 40} {0, 1, 43, 104} {0, 17, 25, 30}

1��GDD�Ä«|d1��GDD�Ä«|¦±¦f5��"


½n 6.18. é?¿u ≥ 5§�3��.�6u�(4, 2)-CRSSGDD"

y². éu ∈ [5, 13]§¤I�O3©[159, Ún3.3]¥®�Ñ"éu ∈ {14, 18}§¤I�OdÚn6.17��"éu ∈ {15, 17, 19, 23, 27, 29, 33}§¤I�OdíØ6.16��"

éu ∈ {16, 22, 28, 34}§¤I�Odé.�2u§u ∈ {16, 22, 28, 34}�(4, 2)-

CRSSGDD^3)ä��"éu ∈ {20, 24, 32}§©O�.�304£½n6.30¤§

364½484£Ún5.4¤�(4, 2)-CRSSGDD§¿3|þ©OW\.�65§66½68�

(4, 2)-CRSSGDD§Ò��¤I�O"é?¿u ∈ {21, 25, 26, 30, 31}½u ≥ 35§

lÚn2.4��(u, {5, 6, 7, 8, 9}, 1)-PBD"^WFC\�6¿W\.�6s§s ∈{5, 6, 7, 8, 9} �(4, 2)-CRSSGDDÒ��¤I�O"

b. ...��4u��(4, 2)-CRSSGDD

Ún 6.19. �3��.�410�(4, 2)-CRSSGDD"

y². -:8�Z40§|8�{{0, 10, 20, 30}+ i : 0 ≤ i ≤ 9}"1��GDD´dÄ

«|{0, 1, 4, 13}§{0, 2, 7, 24}§{0, 6, 14, 25}3Z40¥+1 (mod 40)Ðm��"1

��GDD�Ä«|d1��GDD�Ä«|¦±¦f7��"

½n 6.20. é¤ku ≡ 1 (mod 3)§u ≥ 4§�3.�4u�(4, 2)-CRSSGDD"

y². éu ∈ {4, 7, 10}§¤I�OdÚn5.4§6.8Ú6.19��"é?¿u ≥ 13§

3Ún6.9¥�.�12t§t ≥ 4�(4, 2)-CRSSGDD§O\��Ã¡:§¿3|

þëÓÃ¡:W\.�44�(4, 2)-CRSSGDDÒ��.�43t+1§t ≥ 4�(4, 2)-

CRSSGDD"

c. ...��3u��(4, 2)-CRSSGDD

Ún 6.21. é?¿u ∈ {9, 13}§�3��.�3u�(4, 2)-CRSSGDD"

y². éu = 9§-:8�Z27§|8�{{0, 9, 18} + i : 0 ≤ i ≤ 8}"¤I�OdXeÄ«|3Z27¥+9 (mod 27)Ðm��"


{25, 14, 13, 2} {6, 10, 25, 20} {2, 6, 22, 26} {3, 10, 7, 18} {2, 12, 4, 10} {24, 23, 21, 2}{15, 3, 23, 25} {20, 23, 9, 19} {0, 21, 5, 22} {6, 4, 18, 19} {16, 2, 8, 9} {22, 14, 19, 17}{25, 5, 26, 19} {12, 26, 11, 9} {8, 3, 19, 24} {0, 4, 24, 25} {0, 6, 8, 23} {26, 16, 3, 13}

84 Aa|Ü?è¯KïÄ

1��GDD�Ä«|d1��GDD�Ä«|¦±¦f10��"

éu = 13§-:8�Z39§|8�{{0, 13, 26}+ i : 0 ≤ i ≤ 12}"¤I�OdXeÄ«|3Z39¥+1 (mod 39)Ðm��"1��GDD�Ä«|{0, 1, 6, 31}§{0, 2, 12, 23}§{0, 3, 7, 22}"1��GDD�Ä«|d1��GDD�Ä«|¦±

¦f14��"

Ún 6.22. é¤ku ≡ 1 (mod 4)§u ≥ 5§�3.�3u�(4, 2)-CRSSGDD"

y². éu = 5§¤I�O3©[5]¥�Ñ"éu ∈ {9, 13}§¤I�O3Ún6.21¥�E��"é?¿u ≥ 17§lÚn6.9¥�.�12t§t ≥ 4�(4, 2)-

CRSSGDD§O\3�:§¿3|þëÓÃ¡:W\.�35�(4, 2)-CRSSGDD§

Ò��.�34t+1§t ≥ 4�(4, 2)-CRSSGDD"

Ún 6.23. é?¿u ∈ {8, 12, 16, 24, 28}§�3��.�3u�(4, 2)-CRSSGDD"

y². é?¿�½�u§-:8�Z3(u−1)∪({a}×Z3)§|8�{{0, u−1, 2u−2}+i : 0 ≤ i ≤ u−2}∪{{a}×Z3}"¤I�Od[160,L 2]¥�Ä«|3Z3(u−1)¥+1

(mod 3(u − 1))Ðm��"Ù¥§��x0 ∈ {x} × Zn�eI´3Z3(u−1)��

�n�f+Ðm"

Ún 6.24. Ø�3.�34�(4, 2)-CRSSGDD"

y². b��3ù��O"-|�{{0, 4, 8}, {1, 5, 9}, {2, 6, 10}, {3, 7, 11}}§B1�1��GDD�9�«|�¤�8Ü§B2�1��GDD�9�«|�¤�

8Ü"P1��GDD¥�¹x�«|/¤n�|8Fx = {{a, b, c}|{a, b, c, x} ∈B1}§1��GDD¥�¹x�«|/¤n�|8Sx = {{a, b, c}|{a, b, c, x} ∈ B2}"Ø��5§·�b�F0 = {{1, 2, 3}, {5, 6, 7}, {9, 10, 11}}"Ï�?¿ü�«|�õ�uü�:§S0�½´{{1, 6, 11}, {2, 7, 9}, {3, 5, 10}}½ö{{1, 7, 10},{2, 5, 11}, {3, 6, 9}}"?Û�«�¹¥§ÑN´wÑvk�{��ES1��{«

|"

Ún 6.25. é¤ku ≡ 0 (mod 4)§u ≥ 8§�3.�3u�(4, 2)-CRSSGDD"

y². éu ∈ {8, 12, 16, 24, 28}§¤I�O3Ún6.23¥�Ñ"éu ∈ {20, 32, 36}§é.�s4§s ∈ {5, 8, 9}£Ún5.4¤�(4, 2)-CRSSGDD^3)ä§¿3|þW\

.�3s�(4, 2)-CRSSGDD��¤I�O"


é?¿u ≡ 0 (mod 8)§u ≥ 40§lÚn2.8�.�6t§t ≥ 5�{4}-GDD"

rù�GDD^WFC\�4��.�24t�(4, 2)-CRSSGDD§23|þW\.

�38�(4, 2)-CRSSGDDÒ��.�38t§t ≥ 5�(4, 2)-CRSSGDD"é?¿u ≡4 (mod 8)§u ≥ 44§�.�6t91§ t ≥ 4�{4}-GDD£�[77, ½n1.6]¤"r

ù�GDD^WFC\�4§��.�24t361�(4, 2)-CRSSGDD§23��24�

|þW\.�38�(4, 2)-CRSSGDD§3��36�|þW\.�312�(4, 2)-

CRSSGDD§Ò��.�38t+12§t ≥ 4�(4, 2)-CRSSGDD"

d. ...��g4§§§g ≡ 2 (mod 4)��(4, 2)-CRSSGDD

��|��n§«|��k§�ê�λ§É��h1, . . . , hs�Ø��î

��O§P�ITDλ(k, n;h1, . . . , hs)£½TDλ(k, n) −∑

1≤i≤s TDλ(k, hi)¤§´�

�o�|(X,G,H,B)§Ù¥

1. X´kn��8Ü¶

2. GrXy©k�Ü©£|¤§z��n¶

3. H = {H1, . . . , Hs}´X��Ø��f8§¡�É§�é?¿1 ≤ i ≤sÚ?¿G ∈ G§|G ∩Hi| = hi¶

4. B´X¥��k�f8�8Ü£«|¤§¦�Ø3Ó��|ÚÓ��Ép�?¿:éÑTÐÑy3λ�«|¥¶

5. �¹3Ó��|½öÓ��Ép�?¿:éÑØ�¹3?Û«|¥"

�λ = 1�§·��±òeI�Ñ"

Xe�E´Wilson'uMOLS��E{�Ä�/ª£�[40]¤"éN´�y

XJÑ\�OäkCRSS5�§@od�E6.26��O�äkCRSS5�"

�E 6.26. b��3��TDµ(k + l, t) (X,G,B)§�é?¿B ∈ B§Ñ�3��TDλ(k,m +

∑1≤i≤l w

Bi ) −

∑1≤i≤l TDλ(k, w

Bi )"@o�3��TDλµ(k,mt +∑

1≤i≤l∑

1≤j≤twij)−∑

1≤i≤l TDλµ(k,∑

1≤j≤twij)"

Äk§·��E��O"·�ò^ØÓ��{5�EMOLS

£�[41]¤§,�^MOLSÚTD�m�éX5�ETD£½ITD¤"�·�k1

86 Aa|Ü?è¯KïÄ

��TD £½ITD¤§1��±^·��^31��:8þ��"é

��ÝM§-M(i, j)L«M�1i11j��"

Ún 6.27. é?¿g ∈ {10, 18, 22, 26}§�3��.�g4�(4, 2)-CRSSGDD"

y². ég = 10§�[109, 5 35.19]¥�ü�MOLS(10)Xeµ

0 4 7 6 2 1 9 8 3 5 0 8 6 1 7 9 2 5 4 3

2 1 5 0 8 9 4 3 6 7 3 1 4 6 9 0 5 2 7 8

3 5 2 8 7 6 0 1 9 4 1 7 2 4 0 8 9 3 5 6

1 6 9 3 5 7 8 2 4 0 2 9 8 3 6 4 7 1 0 5

7 8 1 2 4 3 5 9 0 6 9 0 5 8 4 7 1 6 3 2

8 7 4 9 6 5 1 0 2 3 6 2 1 7 3 5 4 8 9 0

5 9 0 1 3 8 6 4 7 2 8 3 7 0 5 2 6 9 1 4

9 2 6 5 0 4 3 7 1 8 4 5 0 9 2 3 8 7 6 1

6 0 3 4 9 2 7 5 8 1 5 4 9 2 1 6 3 0 8 7

4 3 8 7 1 0 2 6 5 9 7 6 3 5 8 1 0 4 2 9

·�^L1ÚL25©OL«þãü�MOLS(10)§¿©O^��p1 = (0 1 2 6)

(3 9 8 4)(5 7)Úp2 = (0 1 7 8 5 6 2 4 3 9)�^3:8þ��ü�#

�MOLS(10)§L3ÚL4"·�^ùü�MOLS(10)5�Eü�äk�ü5�

�TD(4, 10)§Ù¥µ:8�{{(0, i), (1, i), . . . , (9, i)} : 0 ≤ i ≤ 3}§«|8©O�{{(i, 0), (j, 1), (L1(i, j), 2), (L2(i, j), 3)} : 0 ≤ i, j ≤ 9}Ú{{(i, 0), (j, 1), (L3(i, j),

2), (L4(i, j), 3)} : 0 ≤ i, j ≤ 9}"ég = 18§�[3, ½n 3.48]¥�ÝXeµ

0 0 10 1 8 5 7 0 4 6 13

7 1 2 3 11 9 12 − − − −1 7 12 0 6 2 3 8 9 10 5

8 4 11 − − − − 13 7 4 1

éz��(a, b, c, d)T§Ø1��±§©O^�(a, b, c, d)TÚ(b, a, d, c)TO

��ü�§¿O\�(0, 0, 0, 0)T��#�ÝM"rM¥z1�Î

Ò“−”©O^“∞0,∞1,∞2,∞3”O�"2rM��(+1 (mod 14),−)ÐmÒ��

��ITD(4, 18; 4)§Ù¥|8�{{(0, i), (1, i), . . . , (13, i), (∞0, i), . . . , (∞3, i)} :

0 ≤ i ≤ 3}§É�{{(∞0, i), (∞1, i), (∞2, i), (∞3, i)} : 0 ≤ i ≤ 3}§Ä«|�{{(M(0, j), 0), (M(1, j), 1), (M(2, j), 2), (M(3, j), 3)} : 0 ≤ j ≤ 21}"3Ó��|8ÚÉe�E1��ITD(4, 18; 4)§Ä«|�{{(M(0, j) + 1, 0), (M(1, j) +

1, 1), (M(2, j) + 2, 2), (M(3, j) + 2, 3)} : 0 ≤ j ≤ 21}§Ù¥eM(i, j) ∈ Z14§


M(i, j)+a = M(i, j)+a (mod 14)£i ∈ {2, 3}¤¶ÄKeM(i, j) =∞k§k ∈ Z4§

M(i, j) + a = ∞k+a£i ∈ {2, 3}¤"��§é��ü�ITD©OW\ü�

äk�ü5��TD(4, 4)£�Ún5.4¤Ò��.�184��ü(4, 2)-

CRSSGDD"

ég = 22§�[4, íØ 3.9]¥�ÝXeµ

− 0 0 0 0 − 0 0 0 0 0

0 − 2 3 4 1 − 5 6 7 8

2 2 − 6 8 9 7 − 12 14 13

5 6 9 − 10 16 2 13 − 3 5

13 12 13 11 − 13 9 15 3 − 10

rz��(a, b, c, d, e)T©O^�(a, b, c, d, e)TÚ(−a,−b,−c,−d,−e)T3Z17O��ü�§ù��#�ÝM"rM�z�1�ÎÒ“−”©O^“∞0,∞1,∞2,∞3”O�"2rXeÝB�ÝMÜ¿§Ò��#�k27��ÝM ′"

B =

∞4 0 1 1 0

0 ∞4 0 1 1

1 0 ∞4 0 1

1 1 0 ∞4 0

0 1 1 0 ∞4

�M ′�co1§rM ′�z��(+1 (mod 17),−)Ðm��ITD(4, 22; 5)§

Ù¥|8�{{(0, i), (1, i), . . . , (16, i), (∞0, i), (∞1, i), (∞2, i), . . . , (∞4, i)} : 0 ≤i ≤ 3}§É�{{(∞0, i), (∞1, i), (∞2, i), (∞3, i), (∞4, i)} : 0 ≤ i ≤ 3}§Ä«|�{{(M ′(0, j), 0), (M ′(1, j), 1), (M ′(2, j), 2), (M ′(3, j), 3)} : 0 ≤ j ≤ 26}"3Ó��|8ÚÉþ�E1��ITD(4, 22; 5)§Ä«|�{{(M ′(0, j), 0), (M ′(1, j), 1),

(M ′(2, j) + 1, 2), (M ′(3, j) + 3, 3)} : 0 ≤ j ≤ 26}§Ù¥eM ′(i, j) ∈ Z17§

M ′(i, j)+a = M ′(i, j)+a (mod 17)£i ∈ {2, 3}¤¶ÄKeM(i, j) =∞k§k ∈ Z5§

M ′(i, j) + a = ∞k+a£i ∈ {2, 3}¤"��3��ü�ITD¥©OW\ü�äk

�ü5��TD(4, 5)£�5.4¤Ò��.�224�(4, 2)-CRSSGDD"

ég = 26§�[3, ½n 3.53]¥�ÝXeµ

− − − − −0 0 0 0 0

1 6 7 8 14

3 11 20 18 10

6 10 14 1 5

4 19 5 12 2

88 Aa|Ü?è¯KïÄ

rz��^rd��Ì��8�O�§¿O\�(0, 0, 0, 0, 0, 0)T�

��#�ÝM"rM�z1¥�ÎÒ“−”©O^“∞0,∞1,∞2,∞3,∞4”O

�"�co1§rM�z�(+1 (mod 21),−)Ðm��ITD(4, 26; 5)§Ù

¥|8�{{(0, i), (1, i), . . . , (20, i), (∞0, i), (∞1, i), (∞2, i), (∞3, i), (∞4, i)} : 0 ≤i ≤ 3}§É�{{(∞0, i), (∞1, i), (∞2, i), (∞3, i), (∞4, i)} : 0 ≤ i ≤ 3}§Ä«|�{{(M(0, j), 0), (M(1, j), 1), (M(2, j), 2), (M(3, j), 3)} : 0 ≤ j ≤ 30}"3Ó��|8ÚÉþ�E1��ITD(4, 26; 5)§Ä«|�{{(M(0, j), 0), (M(1, j), 1),

(M(2, j)+1, 2), (M(3, j)+8, 3)} : 0 ≤ j ≤ 30}§Ù¥eM(i, j) ∈ Z21§M(i, j)+

a = M(i, j)+a (mod 21)£i ∈ {2, 3}¤¶ÄKeM(i, j) =∞k§k ∈ Z5§M(i, j)+

a = ∞k+a£i ∈ {2, 3}¤"��3��ü�ITD¥©OW\äk�ü5��ü

�TD(4, 5)£Ún5.4¤Ò��.�264�(4, 2)-CRSSGDD"

Ún 6.28. ©O�3��CRSSITD2(4, 6; 2)Ú��CRSSITD2(4, 14; 4)"

y². éCRSSITD2(4, 6; 2)§-:8�{0, 1, . . . , 23}§|8�{{0, 4, . . . , 20}+ i :

0 ≤ i ≤ 3}§É�{16, 17, . . . , 23}"¤I�«|Xeµ1��ITD�«|µ

{0, 11, 22, 5} {10, 9, 19, 12} {17, 7, 2, 12} {1, 2, 11, 16} {16, 9, 6, 3} {23, 0, 9, 2}{19, 8, 13, 2} {13, 22, 12, 3} {6, 17, 0, 15} {16, 14, 5, 7} {7, 22, 4, 9} {8, 7, 6, 21}{13, 7, 18, 0} {13, 23, 14, 4} {1, 15, 22, 8} {8, 23, 5, 10} {5, 2, 3, 20} {13, 6, 11, 20}{14, 17, 8, 3} {12, 5, 18, 15} {9, 11, 8, 18} {1, 23, 12, 6} {5, 19, 6, 4} {10, 16, 13, 15}{0, 14, 1, 19} {15, 9, 20, 14} {20, 1, 7, 10} {21, 0, 3, 10} {1, 4, 3, 18} {11, 14, 12, 21}{2, 4, 15, 21} {4, 10, 11, 17}

1��ITD�«|µ

{19, 9, 8, 6} {12, 1, 7, 22} {4, 17, 14, 15} {6, 3, 21, 4} {10, 19, 13, 0} {10, 16, 1, 3}{17, 3, 0, 2} {6, 1, 15, 20} {11, 6, 17, 12} {3, 8, 5, 22} {14, 3, 13, 20} {12, 9, 3, 18}{2, 19, 1, 4} {0, 9, 15, 22} {4, 11, 13, 22} {5, 23, 0, 6} {9, 14, 11, 16} {1, 14, 8, 23}{23, 4, 10, 9} {15, 5, 16, 2} {10, 20, 5, 11} {9, 2, 7, 20} {23, 2, 13, 12} {13, 7, 16, 6}{11, 2, 21, 8} {8, 7, 17, 10} {19, 14, 5, 12} {4, 5, 7, 18} {8, 13, 15, 18} {21, 12, 10, 15}{11, 18, 0, 1} {7, 21, 14, 0}

éCRSSITD2(4, 14; 4)§-:8�{0, 1, . . . , 55}§|8�{{0, 4, . . . , 52} + i :

0 ≤ i ≤ 3}§É�{40, 41, . . . , 55}"¤I«|dXeÄ«|ÏLgÓ�+G =

〈(0 4 . . . 36)(1 5 . . . 37)(2 6 . . . 38)(3 7 . . . 39)(40)(41)(42)(43)(44)(45)(46)(47)

(48)(49)(50)(51)(52)(53)(54)(55)〉Ðm��"


1��ITD�Ä«|µ

{1, 34, 47, 12} {2, 41, 4, 31} {2, 19, 13, 48} {1, 50, 15, 24} {1, 35, 28, 46} {1, 2, 3, 0}{1, 32, 22, 55} {2, 5, 24, 23} {0, 45, 14, 11} {2, 29, 40, 15} {2, 35, 49, 16} {1, 39, 42, 4}{1, 16, 51, 10} {1, 36, 43, 6} {2, 17, 27, 52} {2, 11, 36, 53} {1, 18, 23, 44} {1, 8, 54, 31}

1��ITD�Ä«|µ

{0, 15, 29, 54} {47, 8, 29, 18} {28, 50, 37, 7} {37, 55, 36, 30} {31, 1, 6, 40} {52, 7, 5, 2}{38, 7, 45, 20} {0, 11, 13, 22} {36, 19, 49, 2} {31, 13, 28, 46} {27, 48, 6, 5} {39, 41, 8, 38}{29, 43, 32, 6} {29, 2, 44, 35} {29, 3, 42, 36} {29, 14, 51, 12} {0, 5, 26, 39} {27, 30, 53, 32}

Ún 6.29. �3��.�46101�(4, 2)-CRSSGDD"

y². -:8�Z24 ∪ ({a}×Z8)∪ ({b}×Z2)§|8�{{0, 6, 12, 18}+ i : 0 ≤ i ≤5} ∪ {({a} × Z8) ∪ ({b} × Z2)}"¤I�OdXeÄ«|3Z24¥+3 (mod 24)Ð

m��§Ù¥��x0 ∈ {x} × Zn�eI3Z24¥��n�f+Ðm"


{1, 6, 3, a0} {2, 9, 23, a0} {10, 20, 11, a0} {4, 8, 21, a0} {7, 14, 18, a0}{0, 23, 15, b0} {0, 5, 16, a0} {22, 12, 13, a0} {1, 4, 20, b0} {15, 17, 19, a0}


{3, 2, 13, a0} {4, 21, 13, b0} {8, 23, 16, a0} {4, 5, 15, a0} {11, 9, 0, a0}{12, 17, 7, a0} {21, 19, 18, a0} {1, 20, 22, a0} {6, 10, 14, a0} {0, 17, 20, b0}

½n 6.30. é?¿g ≡ 2 (mod 4)§g ≥ 10§�3��.�g4�(4, 2)-CRSSGDD"

y². ég ∈ {10, 18, 22, 26}§¤I�O3Ún6.27¥�E��"ég = 14§�Ú

n6.28¥�CRSSITD2(4, 14; 4)§3É¥W\��.�44�(4, 2)-CRSSGDD£Ú

n5.4¤Ò�¤I�O"ég = 34§é.�46101�(4, 2)-CRSSGDD^4)ä§^

.�44�{4}-MGDD��Ñ\�O§Ò��.�(16, 44)6(40, 104)1�(4, 2)-

CRSSDGDD"3É¥©OW\CRSSTD2(4, 4)ÚCRSSTD2(4, 10)Ò��¤I

�O"

ég = 30½g ≥ 38§A^�E6.26§-µ = l = 1§k = m = 4§λ = 2§

wij ∈ {0, 2}Ò��CRSSITD2(4, 50; 14)§¿é?¿t ≥ 5§t 6∈ {6, 10}�

90 Aa|Ü?è¯KïÄ

�CRSSITD2(4, 4t + 10; 10)"ùp§Ñ\�O�CRSSTD2(4, 4)£Ún5.4¤

ÚCRSSITD2(4, 6; 2)£Ún6.28¤"2W\CRSSTD2(4, 14)ÚCRSSTD2(4, 10)�

�¤I�O"

(ÜÚn2.8§5.4§6.24Ú½n6.30§·��Xe(J"

½n 6.31. ��.�g4�(4, 2)-CRSSGDD�3��=�g ≥ 4§g 6= 6"

e. ...��gu��(4, 2)-CRSSGDD��ÌÌÌ��(((JJJ

½n 6.32. .�gu�(4, 2)-CRSSGDD�3�7�^�§Ø(½��(g, u) ∈{(2, 4), (3, 4), (6, 4)}±§�´¿©�"

y². éu = 4§d�O��35d½n6.31�Ñ§Ïdùp·��Äu ≥ 5�

�¹"

£i¤�g ≡ 0 (mod 6)�

ég ∈ {6, 12}§u ≥ 5§¤I�OdÚn6.18Ú6.9��"ég = 36§u ≥ 5§

é.�12u§u ≥ 5�(4, 2)-CRSSGDD^3)ä��¤I�O"é?¿g > 12§

g 6= 36§é.�6u�(4, 2)-CRSSGDD^g/6)äÒ��.�gu§u ≥ 5�(4, 2)-

CRSSGDD"

£ii¤�g ≡ 1, 5, 7, 11 (mod 12)�

ég = 1§u ≡ 1, 4 (mod 12)§u ≥ 13§¤I�OdÚn5.2��"ég ≥5§u ≡ 1, 4 (mod 12)§u ≥ 13§é(u, 4, 2)-CRSS�O^g)ä��¤I�.

�gu�(4, 2)-CRSSGDD"

£iii¤�g ≡ 2, 10 (mod 12)�

ég = 2§u ≡ 1 (mod 3)§u ≥ 7§¤I�OdÚn5.19Ú5.20��"ég ≥10§u ≡ 1 (mod 3)§u ≥ 7§é.�2u�(4, 2)-CRSSGDD^g/2)ä��.

�gu�(4, 2)-CRSSGDD"

£iv¤�g ≡ 3, 9 (mod 12)�

ég = 3§u ≡ 0, 1 (mod 4)§u ≥ 5§¤I�OdÚn6.22Ú6.25��"

ég ≥ 9§u ≡ 0, 1 (mod 4)§u ≥ 5§é.�3u�(4, 2)-CRSSGDD^g/3)ä�

�.�gu�(4, 2)-CRSSGDD"

£v¤�g ≡ 4, 8 (mod 12)�


ég = 4§u ≡ 1 (mod 3)§u ≥ 7§¤I�OdÚn6.20��"ég = 8§

u ≡ 1 (mod 3)§u ≥ 7§é.�2u�(4, 2)-CRSSGDD^4)ä��¤I�O"

ég ≥ 16§u ≡ 1 (mod 3)§u ≥ 7§é.�4u�(4, 2)-CRSSGDD^g/4)ä��

.�gu�(4, 2)-CRSSGDD"

6.6 ��`��üüü(v, 4, 2)-WWW¿¿¿��333555

3�!¥§·�òïÄ�ü(v, 4, 2)-W¿��35"Äk§·��Ñ

��`(v, 6, 4)3èÚ�`�ü(v, 4, 2)-W¿�éX",�§·�y²é?

¿v ≥ 4§Ø(½��v ∈ {4, 5, 6, 9}§�`�ü(v, 4, 2)-W¿Ñ´�3�"

a. ��v ≡ 1, 2 (mod 3)��

3©[159]¥§�ö(½�`(v, 6, 4)3è��35§=µXJv ≥ 4§v 6∈{4, 5, 7, 8, 11, 17}§A3(v, 6, 4) =

⌊v2

⌊v−13

⌋⌋"�±y²�v ≡ 1, 2 (mod 3)�§ä

k⌊v2

⌊v−13

⌋⌋�èi��`(v, 6, 4)3èÒ´��`�ü(v, 4, 2)-W¿", §�

�%Ø¤á"

Ún 6.33. �v ≡ 1, 2 (mod 3)�§XJ�3��äk⌊v2

⌊v−13

⌋⌋�èi��

`(v, 6, 4)3è§@o�3��`�ü(v, 4, 2)-W¿"

y². -X��v�:8§D��`(v, 6, 4)3è�¤kèi�| 8�

¤�8Ü"@oD/¤��«|��4��O"

Äk§·��ÑX¥�?¿:é3D¥�õÑy3ü�«|¥"XJ��:éÑy3ü�«|¥§@o3ùü�èi¥ùü� �þ��〈1, 1〉§〈2, 2〉½〈1, 2〉§〈2, 1〉"Ïd§D´��(v, 4, 2)-W¿"Ùg§·��ÑDäk�ü5�"XJkü�«|�uõuü�:§@oùü�èi��ål�õ

´5§ùÒ�ål^�gñ"¤±D´�ü�"��§éN´u��v ≡ 1, 2

(mod 3)�§��`(v, 6, 4)3è�èi�ê§=⌊v2

⌊v−13

⌋⌋§TÐ��`�

ü(v, 4, 2)-W¿�«|�ê"Ïd§·�Òy²dè��ü(v, 4, 2)-W¿

�´�`�"

Ún 6.34. é?¿v ∈ {7, 8, 11, 17}§�3��`�ü(v, 4, 2)-W¿"

92 Aa|Ü?è¯KïÄ

y². év = 7§¤I�OÒ´�ü(7, 4, 2)-BIBD£�©[83]¤"év ∈ {8, 11}§¤I�O3:8{0, 1, . . . , v − 1}þ�E§«|Xeµv = 8µ{2, 1, 6, 5} {1, 2, 4, 7} {3, 6, 0, 7} {5, 7, 4, 0} {7, 5, 1, 3} {0, 6, 4, 1} {0, 3, 5, 2} {4, 6, 2, 3}

v = 11µ

{0, 1, 2, 3} {0, 4, 5, 6} {0, 7, 8, 9} {1, 4, 0, 10} {1, 5, 8, 7} {2, 1, 6, 9} {2, 5, 10, 0} {3, 2, 4, 7}{3, 5, 9, 1} {3, 6, 8, 0} {4, 9, 2, 8} {6, 7, 3, 10} {7, 6, 4, 1} {8, 5, 4, 3} {8, 10, 6, 2} {9, 10, 7, 5}

év = 17§¤I�O3:8Z15 ∪ {∞0,∞1}þ�E"¤I«|dXeÄ«|3Z15¥+5 (mod 15)Ðm��§∞0Ú∞13gÓ��^e�±ØÄ"

{0, 2, 4, 6} {0, 1, 2, 3} {0, 9, 12,∞1} {0, 3, 6, 11} {12, 5,∞0, 3} {5, 3, 4,∞1} {13, 7,∞1, 6}{0, 1, 4, 5} {3, 7, 9, 14} {5, 12, 2, 13} {3, 9, 1, 13} {13, 14,∞0, 5} {1, 4, 7, 11} {∞0, 11, 9, 2}

é?¿v ≡ 1, 2 (mod 3)§v ≥ 7§v 6∈ {7, 8, 11, 17}§·�dÚn6.33��

�`�ü(v, 4, 2)-W¿"év ∈ {4, 5}§w,¤I�O´²��§�k��«|"Ïd§·��µ

Ún 6.35. é?¿v ≡ 1, 2 (mod 3)§v ≥ 7§D′2(v, 4, 2) = U2(v, 4, 2)¶é?

¿v ∈ {4, 5}§D′2(v, 4, 2) = U2(v, 4, 2)− 1"

b. ��v ≡ 0 (mod 3)��

Ún 6.36. é?¿v ∈ {6, 9}§D′2(v, 4, 2) = U2(v, 4, 2)− 1"

y². év = 6§w,µ3��`�ü(6, 4, 2)-W¿¥§?¿��:�õ�¹

3ü�«|¥"¤±d�O�õk2×64

= 3�«|"·�3:8{0, 1, 2, 3, 4, 5}þ�E¤I�O�3�«|{0, 1, 2, 3}§{2, 3, 4, 5}§{0, 1, 4, 5}"

év = 9§dÚn6.7§D2(9, 4, 2) = U2(9, 4, 2)−1"Ïd§é��ü(9, 4, 2)-W¿§D′2(9, 4, 2) ≤ U2(9, 4, 2)− 1"·�3:8{0, 1, 2, 3, 4, 5, 6, 7, 8}þ�E¤I�10�«|Xeµ

{0, 1, 2, 3} {0, 4, 5, 6} {1, 0, 7, 8} {2, 1, 5, 4} {2, 6, 7, 0}{3, 5, 7, 1} {5, 8, 3, 0} {6, 8, 4, 1} {8, 6, 5, 2} {3, 4, 2, 8}

Ún 6.37. é?¿v ∈ {12, 24, 36}§�3��`�ü(v, 4, 2)-W¿"


y². ¤I�O3Zvþ�E§¿dXeÄ«|3Zv¥+4 (mod v)Ðm��"v = 12µ{0, 1, 2, 3} {0, 4, 10, 9} {0, 7, 11, 8} {3, 9, 6, 2} {3, 5, 1, 11} {6, 10, 3, 8} {2, 5, 9, 8}

v = 24µ

{0, 1, 2, 3} {19, 7, 17, 15} {0, 6, 10, 13} {1, 5, 10, 16} {12, 10, 5, 21} {21, 17, 20, 11}{0, 7, 11, 14} {12, 16, 11, 8} {1, 7, 13, 20} {6, 11, 16, 4} {19, 18, 10, 9} {20, 17, 1, 14}{6, 18, 2, 15} {3, 19, 21, 14} {2, 19, 20, 4}

v = 36µ

{10, 23, 30, 35} {0, 19, 16, 20} {35, 34, 28, 26} {20, 5, 6, 32} {23, 31, 18, 9} {14, 29, 35, 18}{16, 10, 29, 19} {3, 16, 24, 33} {32, 21, 31, 19} {8, 9, 22, 18} {5, 34, 16, 22} {19, 22, 23, 15}{15, 30, 25, 4} {34, 14, 0, 31} {20, 25, 10, 22} {20, 2, 16, 8} {5, 28, 19, 35} {33, 15, 13, 32}{7, 13, 18, 16} {6, 14, 13, 17} {13, 15, 20, 35} {27, 9, 17, 29} {1, 32, 29, 25}

Ún 6.38. é?¿v ∈ {15, 27, 39}§�3��`�ü(v, 4, 2)-W¿"

y². ¤I�O3Zv−3 ∪ ({a} × Z3)þ�E§Ù¥{{a} × Z3}��É"¤I�OdXeÄ«|3Zv−3¥+4 (mod v − 3)Ðm��§Ù¥x0 ∈ {x} × Zn��eI´3Zv−3¥��n�f+¥Ðm"v = 15µ

{9, 3, 7, 0} {0, 4, 9, a0} {2, 7, a1, 5} {a1, 4, 3, 11} {a2, 6, 1, 5} {0, a1, 10, 8}{0, 1, 3, 2} {5, 2, 4, 10} {2, 6, a0, 3} {3, a2, 10, 4} {3, 9, 5, a1}

v = 27µ

{2, 23, 4, 10} {9, 2, 16, 5} {23, 13, 8, a1} {11, 9, 7, 19} {20, 14, 10, a0} {10, 19, a1, 20}{0, 16, 23, 2} {21, 5, 8, 9} {14, 21, 13, 2} {14, 3, 18, 9} {5, 23, a0, 20} {12, a0, 15, 17}{0, 8, 20, 17} {10, 2, 8, 3} {a2, 5, 19, 14} {19, 8, 12, 23} {17, a1, 18, 23} {a0, 13, 12, 10}{3, 19, 2, 13}

v = 39µ

{2, 5, 25, 7} {10, 11, 19, 1} {6, 31, 28, 27} {30, 8, 0, 10} {22, 24, 29, 5} {8, 30, 22, 20}{1, 0, 30, 29} {28, 3, 12, 25} {19, 29, 17, 7} {0, 34, 6, 18} {1, 11, 18, 31} {16, 12, a1, 19}{a2, 9, 17, 6} {17, a2, 0, 26} {6, 35, 10, a2} {1, 33, 2, a2} {13, 29, 35, 0} {2, 19, 35, 34}{3, 6, 17, 30} {16, 7, 11, a0} {0, 19, 21, 25} {a0, 26, 5, 0} {21, 19, 6, 24} {a1, 12, 35, 27}{29, 4, 20, 8} {3, 24, 32, 33} {3, 15, 34, 28}

5¿�Ún6.38¥�E��`�ü(v, 4, 2)-W¿§v ∈ {15, 27, 39}�´�ü(v, 3; 4, 2)-MIPD"

Ún 6.39. é?¿v ≡ 0, 3 (mod 12)§v ≥ 12§�3�`�ü(v, 4, 2)-W¿"

94 Aa|Ü?è¯KïÄ

y². é12 ≤ v ≤ 39§¤I�O3Ún6.37Ú6.38¥�E��"é?¿v ≥ 48§

lÚn6.9¥��.�12t§t ≥ 4�(4, 2)-CRSSGDD"d�E6.5Ú6.6§XJ

·�3|þW\�`�ü(12, 4, 2)-W¿§·�Òé?¿t ≥ 4��`�

ü(12t, 4, 2)-W¿"XJO\3�:§¿3|þëÓÃ¡:W\�ü(15, 3; 4, 2)-

MIPD§·�Òé?¿t ≥ 4��`�ü(12t+ 3, 4, 2)-W¿"

Ún 6.40. é?¿v ∈ {18, 21, 30}§�3��`�ü(v, 4, 2)-W¿"

y². év = 18§¤I�O3{0, 1, . . . , 17}þ��E§«|Xeµ

{1, 6, 5, 9} {6, 8, 7, 15} {9, 8, 14, 12} {15, 2, 13, 0} {8, 16, 10, 17} {10, 7, 1, 11}{5, 2, 7, 3} {7, 0, 5, 10} {14, 17, 7, 9} {12, 7, 6, 17} {17, 3, 12, 10} {15, 16, 9, 3}{3, 8, 4, 1} {0, 14, 6, 3} {14, 1, 2, 10} {17, 8, 0, 11} {13, 1, 15, 12} {16, 7, 12, 4}{0, 2, 8, 9} {2, 17, 1, 4} {13, 3, 11, 7} {11, 12, 8, 5} {10, 13, 4, 14} {16, 11, 6, 1}{4, 3, 6, 16} {0, 4, 15, 7} {12, 0, 1, 16} {15, 4, 10, 8} {2, 12, 15, 11} {5, 15, 1, 14}{4, 5, 0, 12} {9, 5, 4, 11} {10, 3, 15, 5} {5, 17, 6, 13} {12, 3, 14, 13} {14, 5, 16, 8}{1, 7, 13, 8} {11, 3, 9, 2} {6, 14, 11, 4} {17, 16, 2, 5} {13, 9, 10, 16} {4, 9, 13, 17}{1, 3, 0, 17} {10, 9, 6, 0} {6, 10, 12, 2} {7, 2, 16, 14} {11, 16, 0, 13} {11, 17, 14, 15}{8, 13, 2, 6}

év = 21§¤I�O3Z20∪{∞}þ�E"¤I«|dXeÄ«|3Z20¥+5(mod 20)Ðm��§Ù¥∞3gÓ��^e�±ØÄ"

{0, 2, 5, 7} {4, 3, 14, 5} {1, 17, 9, 13} {11, 16, 0, 19} {10, 0, 11, 4} {3, 18, 7, 17}{6, 3,∞, 9} {6, 18, 16, 7} {4, 7, 10, 19} {5, 18, 12,∞} {0, 12, 13, 6} {18, 10, 15, 13}{3, 13, 6, 4} {17, 4, 19, 2} {13, 11, 5, 9} {11, 16, 7, 15} {1, 7, 14,∞}

év = 30§¤I�O3Z30þ�E"¤I«|dXeÄ«|3Z30¥+15(mod 30)Ðm��"

{19, 8, 0, 11} {19, 20, 0, 21} {15, 12, 0, 11} {23, 6, 19, 24} {17, 26, 12, 9} {16, 29, 23, 28}{20, 1, 24, 5} {16, 24, 1, 27} {21, 22, 27, 6} {20, 11, 4, 13} {21, 5, 29, 13} {18, 20, 15, 21}{17, 8, 11, 6} {22, 10, 8, 12} {16, 13, 0, 17} {21, 3, 26, 10} {22, 21, 29, 7} {14, 24, 26, 11}{18, 8, 2, 13} {21, 5, 27, 14} {21, 15, 9, 24} {18, 17, 29, 0} {16, 12, 0, 18} {16, 15, 17, 18}{23, 9, 3, 13} {18, 27, 6, 26} {18, 5, 25, 28} {24, 28, 12, 2} {22, 26, 17, 28} {20, 18, 23, 22}{22, 4, 24, 8} {17, 1, 23, 20} {23, 11, 5, 29} {10, 21, 18, 4} {19, 17, 22, 14} {20, 22, 26, 16}{20, 8, 3, 25} {22, 11, 28, 3} {17, 10, 27, 5} {19, 29, 27, 8} {22, 15, 13, 25} {25, 12, 13, 27}{18, 12, 7, 1} {25, 14, 16, 7} {20, 27, 2, 29} {24, 19, 3, 12} {14, 16, 13, 10} {21, 13, 28, 19}{17, 10, 6, 2} {18, 19, 22, 1} {17, 7, 19, 15} {21, 0, 14, 28} {20, 15, 11, 10} {22, 12, 23, 15}{18, 14, 3, 9} {24, 7, 10, 26} {22, 13, 5, 24} {23, 0, 25, 10} {10, 14, 29, 24} {21, 24, 25, 16}{16, 8, 23, 6} {21, 1, 11, 17} {20, 17, 7, 24} {26, 19, 1, 13} {16, 15, 29, 19} {14, 27, 11, 23}{2, 9, 10, 19} {15, 8, 9, 28} {18, 14, 2, 19} {19, 4, 20, 27} {19, 10, 16, 11}


Ún 6.41. é?¿v ∈ {54, 57}§�3��`�ü(v, 4, 2)-W¿"

y². ·�^©[7]¥��{��E¤I(v, 4, 2)-W¿"Äk§·�3:8Z56þ�E��.�228�{4}-GDD§Ù¥|8�{{0, 28} + i : 0 ≤ i ≤ 27}§«|8D1dXeÄ«|+2 (mod 56)Ðm��"

{0, 3, 33, 52} {0, 35, 43, 55} {0, 8, 19, 22} {0, 9, 15, 20} {1, 15, 19, 44}{0, 2, 12, 18} {0, 26, 47, 49} {0, 1, 17, 32} {0, 5, 29, 39}

év = 54§·�3Z52þ�E��(52, 4, 1)-BIBD§Ù¥«|8dXeÄ«|+4(mod 52)Ðm��"PÙ«|8�D2"

{7, 29, 50, 30} {0, 12, 16, 6} {0, 50, 41, 24} {7, 13, 10, 38} {7, 11, 1, 24} {14, 49, 25, 39}{0, 15, 43, 44} {21, 8, 26, 40} {21, 28, 6, 13} {7, 28, 39, 47} {7, 4, 18, 26} {21, 12, 33, 17}{0, 49, 47, 10} {7, 14, 32, 33} {14, 15, 50, 2} {7, 21, 43, 41} {0, 7, 25, 2}

·�Uw�3D1¥k��«|{15, 30, 54, 55}§§´d{0, 1, 17, 32}Ðm��"3D1¥©O^:52, 53, 15, 30O�15, 30, 52, 53Ò��#�«|

8D′1§Ù¥{52, 53, 54, 55}´��«|"�Kù�«|§¿3�{�«|D′1¥^52�O55§^53�O54Ò��«|8D′′1"^��(0 48 31 6 21 7 28

27 12 5 44 45 13 23 11 34 32 15)(1 8)(2)(3 43 20 4 30)(9)(10)(14)(16)(17)

(18)(19)(22)(24)(25)(26)(29)(33)(35)(36)(37)(38)(39)(40)(41)(42)(46 50)(47)(49)

(51)(52)(53)�^3D2�z�:þÒ��D′2"��§rD′′1ÚD′2Ü¿Ò��I��O"

év = 57§·�3:8{0, 1, . . . , 57}þ�E��.15171�{4}-GDD§Ù¥|8�{{{i} : 0 ≤ i ≤ 50} ∪ {51, 52, . . . , 57}}"^gÓ�+G = 〈(0 3 · · · 48)(1 4 · · · 49)(2 5 · · · 50)(51)(52)(53)(54)(55)(56)(57)〉éXeÄ«|Ðm��«|8D3"

{1, 4, 34, 46} {2, 3, 23, 26} {1, 20, 26, 35} {0, 6, 10, 18} {0, 26, 36, 38} {0, 21, 34, 48}{0, 8, 19, 57} {2, 22, 35, 37} {2, 16, 21, 54} {2, 7, 42, 51} {1, 30, 44, 56} {0, 25, 47, 55}{0, 9, 40, 44} {2, 36, 43, 53} {2, 46, 48, 52} {0, 1, 28, 29}

éD1¥�z�:x§XJx < 28§@o^N�x 7→ 2x§ÄK§^N�x 7→2(x − 28) + 1Ò��D′′′1"3D3¥§^56�O57§¿O\«|{51, 52, 53, 56}§{51, 54, 55, 56}§{52, 53, 54, 55}��D′3"^��(0 10 29)(1 11)(2 31 48 12)(3)

(4 26 22 27 50 35 14 37 51 44 47 33 6)(5 21)(7 8)(13)(23)(28)(9 43)(15 49 20 19 18

17 16)(24 25)(30 36)(32)(34)(38)(39)(40)(41)(42)(45)(46)(52)(53)(54)(55)(56)�

^3D′3¥�z�:þÒ��D′′3"��§rD′′′1ÚD′′3 Ü¿Ò��¤I�O"

96 Aa|Ü?è¯KïÄ

Ún 6.42. é?¿(v, w) ∈ {(33, 7), (42, 10), (45, 13)}§©O�3«|�ê�166§

268§300��ü(v, w; 4, 2)-W¿"

y². 3:8{0, 1, . . . , 32}þ�Eäk166�«|��ü(33, 7; 4, 2)-W¿§Ù¥É�{26, 27, . . . , 32}"¤I�OdXeÄ«|ÏLgÓ�+G = 〈(0 13)(1 14)(2 15) · · · (12 25)(26 27)(28 29)(30 31)(32)〉Ðm��"

{1, 25, 9, 5} {18, 20, 8, 1} {32, 5, 24, 2} {18, 27, 14, 9} {28, 19, 3, 21} {22, 15, 24, 23}{2, 28, 8, 7} {19, 17, 2, 7} {5, 17, 27, 4} {2, 15, 26, 23} {28, 25, 4, 11} {22, 17, 32, 16}{3, 8, 30, 5} {2, 11, 3, 20} {5, 26, 7, 18} {2, 20, 25, 21} {29, 10, 14, 1} {23, 30, 13, 20}{7, 1, 17, 6} {2, 31, 24, 4} {6, 5, 32, 21} {21, 10, 17, 0} {3, 32, 12, 23} {24, 13, 18, 25}{0, 13, 26, 6} {20, 8, 0, 31} {8, 2, 31, 22} {22, 29, 13, 2} {5, 10, 19, 23} {25, 10, 14, 12}{0, 22, 7, 32} {21, 4, 1, 32} {9, 11, 7, 27} {23, 14, 31, 6} {5, 14, 22, 13} {27, 14, 19, 24}{0, 6, 25, 27} {22, 6, 8, 19} {1, 19, 22, 11} {23, 17, 2, 18} {10, 20, 31, 16} {28, 13, 10, 21}{11, 31, 3, 5} {24, 21, 1, 8} {10, 30, 6, 17} {24, 20, 28, 9} {10, 22, 27, 20} {29, 13, 18, 10}{11, 4, 0, 24} {27, 2, 13, 1} {12, 18, 28, 3} {25, 19, 3, 24} {12, 26, 10, 24} {29, 15, 19, 25}{12, 27, 8, 4} {28, 6, 7, 20} {13, 12, 31, 9} {25, 3, 20, 26} {12, 32, 15, 19} {30, 11, 18, 19}{13, 3, 15, 6} {3, 10, 22, 9} {14, 30, 16, 2} {25, 31, 14, 7} {15, 17, 28, 14} {32, 14, 20, 11}{13, 7, 4, 25} {3, 2, 14, 26} {16, 14, 3, 13} {26, 16, 4, 21} {16, 13, 28, 11} {25, 18, 15, 31}{14, 4, 30, 0} {3, 6, 18, 29} {17, 9, 19, 31} {26, 8, 11, 10} {16, 17, 20, 23} {17, 25, 28, 22}{18, 0, 21, 2} {30, 25, 9, 8} {18, 22, 28, 4} {27, 21, 22, 3} {21, 29, 11, 23}

3:8{0, 1, . . . , 41}þ�Eäk268�«|��ü(42, 10; 4, 2)-W¿§Ù¥É�{32, 33, . . . , 41}§¿�«|8dXeÄ«|ÏLgÓ�+G = 〈(0 8 · · · 24)(1 9 · · · 25) · · · (7 15 · · · 31)(32 33 34 35)(36 37 38 39)(40 41)〉Ðm��"

{5, 8, 30, 1} {9, 39, 1, 14} {30, 38, 26, 4} {15, 35, 22, 12} {32, 26, 21, 8} {21, 15, 30, 41}{12, 3, 4, 32} {28, 14, 40, 2} {4, 15, 27, 16} {15, 26, 25, 33} {22, 24, 2, 35} {36, 21, 15, 18}{32, 8, 28, 9} {17, 16, 2, 10} {3, 15, 31, 39} {13, 30, 29, 34} {31, 20, 40, 8} {37, 11, 25, 28}{18, 13, 2, 3} {27, 24, 1, 30} {14, 13, 0, 38} {24, 15, 23, 33} {25, 36, 9, 11} {40, 25, 16, 14}{3, 11, 5, 15} {31, 25, 2, 6} {2, 17, 21, 40} {17, 11, 40, 22} {12, 29, 7, 31} {11, 31, 18, 34}{11, 1, 34, 9} {4, 6, 18, 14} {19, 24, 3, 32} {13, 21, 25, 36} {18, 38, 17, 29} {28, 39, 16, 24}{19, 16, 5, 0} {0, 28, 5, 32} {19, 29, 9, 12} {31, 39, 24, 14} {24, 21, 18, 38} {15, 14, 35, 25}{28, 4, 5, 36} {2, 30, 7, 12} {15, 34, 9, 28} {33, 26, 17, 28} {15, 1, 41, 13} {38, 31, 22, 10}{33, 24, 9, 7} {11, 22, 32, 6} {8, 36, 27, 14} {30, 39, 16, 11} {35, 16, 26, 18} {22, 29, 33, 14}{0, 41, 8, 4} {4, 32, 5, 29} {23, 41, 5, 19} {28, 38, 17, 12} {10, 39, 23, 8} {10, 32, 19, 11}{5, 16, 23, 9} {18, 3, 5, 37} {38, 28, 3, 10} {41, 12, 27, 18} {12, 10, 15, 17} {23, 27, 22, 39}{3, 4, 29, 6}

3:8{0, 1, . . . , 44}þ�Eäk300�«|��ü(45, 13; 4, 2)-W¿§Ù¥É�{32, 33, 34, . . . , 44}"¤I«|8dXeÄ«|dgÓ�+G = 〈(0 8 16 24)(1 9 17 25) · · · (7 15 23 31)(32 33 34 35)(36 37 38 39)(40 41 42 43)(44)〉Ðm�


�"

{7, 11, 4, 9} {5, 4, 44, 19} {2, 21, 38, 31} {35, 19, 11, 6} {17, 31, 18, 34} {32, 26, 10, 31}{0, 32, 9, 27} {8, 19, 18, 7} {20, 2, 43, 31} {36, 8, 23, 20} {18, 30, 39, 24} {33, 15, 24, 16}{0, 4, 20, 34} {9, 38, 24, 5} {20, 7, 24, 41} {39, 21, 6, 26} {19, 42, 30, 15} {35, 28, 18, 10}{12, 9, 2, 40} {1, 17, 37, 21} {22, 15, 34, 2} {42, 6, 24, 11} {21, 28, 32, 23} {36, 12, 10, 16}{13, 3, 4, 39} {1, 31, 23, 42} {23, 40, 1, 30} {44, 14, 9, 18} {22, 23, 11, 38} {37, 31, 24, 25}{16, 5, 2, 39} {10, 29, 2, 27} {24, 40, 21, 4} {44, 15, 8, 11} {23, 38, 21, 14} {38, 18, 20, 12}{2, 9, 42, 24} {11, 37, 17, 2} {25, 44, 14, 4} {44, 29, 0, 23} {24, 13, 29, 32} {40, 21, 10, 13}{21, 42, 8, 1} {11, 5, 40, 29} {26, 8, 24, 34} {5, 26, 35, 30} {24, 17, 36, 29} {40, 30, 22, 26}{3, 1, 15, 35} {13, 12, 4, 23} {27, 2, 38, 19} {7, 23, 19, 39} {25, 17, 28, 32} {43, 12, 18, 19}{34, 28, 1, 6} {13, 7, 31, 42} {3, 13, 19, 34} {8, 43, 17, 11} {27, 17, 29, 35} {43, 30, 24, 12}{36, 14, 0, 1} {16, 14, 6, 34} {3, 35, 25, 12} {11, 30, 43, 28} {28, 22, 36, 19} {30, 13, 17, 14}{37, 7, 12, 6} {17, 41, 26, 9} {32, 12, 14, 5} {12, 11, 25, 39} {29, 30, 42, 12} {14, 11, 24, 16}{40, 0, 2, 11} {18, 23, 6, 17} {32, 30, 7, 21}

3©[159, Ún 5.3, Ún 5.5]¥§�ökXe(4, 2)-CRSSGDD�(J"

Ún 6.43 (Zhang§Ge[159]). é?¿u ∈ [4, 8]∪{16}∪ [20, 22]½u ≥ 24§�3�

�.�12u181�(4, 2)-CRSSGDD"é?¿(u,m) ∈ {(4, 30), (5, 18), (5, 30), (6, 18),

(6, 30), (7, 18), (7, 30)}§�3��.�24um1�(4, 2)-CRSSGDD"é?¿(u,m) ∈{(5, 42), (6, 18), (6, 30), (6, 78)}§�3��.�36um1�(4, 2)-CRSSGDD"

Ún 6.44. é?¿v ≡ 6, 9 (mod 12)§v ≥ 18§�3��`�ü(v, 4, 2)-W

¿"

y². év ∈ {18, 21, 30, 54, 57}§¤I�O3Ún6.40Ú6.41¥�E��"év ∈{33, 42, 45}§lÚn6.42��ü(v, w; 4, 2)-W¿§Ù¥(v, w) ∈ {(33, 7), (42, 10),

(45, 13)}§,�3|þ©OW\�`�ü(w, 4, 2)-W¿§w ∈ {7, 10, 13}£Ún6.35¤��¤I�O"

d�E6.5Ú6.6§XJ3Ún6.43¥�CRSSGDD�|þW\·��`�

ü(v, 4, 2)-W¿§v ∈ {12, 18, 24, 30, 36, 42, 78}£Ún6.39¤§·�Òé?¿t ≥4��I��`�ü(12t+18, 4, 2)-W¿¶XJéùCRSSGDDO\3�:§

¿3|þëÓÃ¡:W\T��ü(v, 3; 4, 2)-MIPD§v ∈ {15, 27, 39}£Ún6.38¤Ú�`�ü(v, 4, 2)-W¿§v ∈ {21, 33, 45, 81}§·�Òé?¿t ≥ 4��

�`�ü(12t+ 21, 4, 2)-W¿"

98 Aa|Ü?è¯KïÄ

(ÜÚn6.35§6.36§6.39Ú6.44§·��Xe(J"

½n 6.45. é?¿v ≥ 4§v 6∈ {4, 5, 6, 9}§D′3(v, 4, 2) = U(v, 4, 2)¶év ∈{4, 5, 6, 9}§D′3(v, 4, 2) = U(v, 4, 2)− 1"

Chapter 7

^Hananinnn��WWW¿¿¿��EEE��555��`õõõ��~~~èèè

7.1 ÚÚÚóóó999ÌÌÌ��(((JJJ

8c§éAq(n, d, w)�ïÄó�õ´�½dÚw§ïÄ,�(½�q§Ï~

´q ≤ 4"é?¿��ênÚq ≥ 2§Aq(n, d, w)Ñ(½��¹�k�(d, w) =

(3, 2) Ú(4, 3)�[23, 26]"

�C§Chee�3©[24]¥é?¿q ≥ 2§�Ñþ�w§ål�d = 2w −1�§�`q�~è�èi�ê�ìC(J"ù«ëêe�è�¡�´�5�

��§Ï�Aq(n, 2w − 1, w) = O(n) [24]"¯¢þ§¦�$^�«n��8�

í2§¿��Xe(Jµ

½n 7.1 (Chee�[24]). ew|(q − 1)n§é?¿n ≥ 2w(w(q − 1) − 1)2 + 1§

Aq(n, 2w − 1, w) = (q − 1)n/w"ew|n§é?¿n ≥ w((w − 1)(q − 2) + 1)§

Aq(n, 2w − 1, w) = (q − 1)n/w"

é�5��`q�~è§�kXe®�(Jµ

(1) �w = 2�§é?¿��ênÚq ≥ 2§Aq(n, 3, 2) = min{ ⌊

(q−1)n2

⌋,(n2

)}[26]"

(2) �q = 3�§é?¿w§A3(n, 2w − 1, w) = max{M |n ≥ dMw/2e +

max{bMw/2c −(M2

), 0}} [115]"

, §�w ≥ 3�q ≥ 4�§éAq(n, 2w − 1, w)Ø½n7.1¥�ìC(J

vk?Û®�(J"

3�Ù¥§·�òé?¿��ênÚq ≥ 2(½Aq(n, 5, 3)�O(�"·��

�{´ÄuHananin�W¿��E"·�ò3e�!¥�Ñ§�½Â9§��

`è�éX"

dÚn2.2¥�Johnson.§·��±éN´��Xe�5��~è�

.µ

100 Aa|Ü?è¯KïÄ

íØ 7.2. Aq(n, 2w − 1, w) ≤⌊n(q−1)w

⌋"

AO�§�(d, w) = (5, 3)�§·��Aq(n, 5, 3) ≤⌊(q−1)n

3

⌋"

ù�Ù�(�Xeµ317.2!¥§·�ò�ÑW¿�OÚHananin�W¿

�O��`è�éX¶317.3!¥§·��ErHananin�W¿��A�¤

k�Ý��`è¶17.4!¥§·�òé�{��Å�)û¶317.5!¥§·

�ò)ûHananin�W¿��35§¿317.6!¥§o(�Ù(J"

7.2 ��OOO��èèè��éééXXX

a. ��`WWW¿¿¿��`èèè��éééXXX

Chee�3©[24]¥ïá��(n, 2w − 1, w)qèC�Xe¿�^�µ

(C1) é?¿ØÓ�u, v ∈ C§|supp(u) ∩ supp(v)| ≤ 1"

(C2) é?¿ØÓ�u, v ∈ C§XJx ∈ supp(u) ∩ supp(v)§@oux 6= vx"

Ïd§·��±éN´��e¡(J"

Ún 7.3. -C ⊂ ZXq ´��(n, 2w − 1, w)qè§B = {supp(u) : u ∈ C}"@o(X,B)´��(n,w, 1)-W¿"

dÚn7.3§Aq(n, 2w − 1, w)Ø�LW¿êD(n,w, 2)"¯¢þ·��±y

²�q¿©��§Aq(n, 2w − 1, w) = D(n,w, 2)"

é��8ÜXÚ(X,B)§-P ⊂ B"é��êi§-C(P, i) := {uB :

B ∈ P}´��è§Ù¥èiuBXe½Âµ

uBx =

i§ ex ∈ B§

0§ ex 6∈ B"

Ún 7.4. é?¿q ≥⌊n−1w−1

⌋+ 1§Aq(n, 2w − 1, w) = D(n,w, 2)"

y². -(X,B)´��`(n,w, 1)-W¿§@oz�:�õÑy3⌊n−1w−1

⌋�«|

¥"�EèC(B, 1)§Ù¥z��I�õk⌊n−1w−1

⌋�1"·�rC(B, 1)�ù1©

O^1, 2, 3, · · ·O�§¦�z��I��"��ÑØÓ"@o·�Ò��kD(n,w, 2)�èi�(n, 2w− 1, w)b n−1

w−1c+1è"é?¿q ≥⌊n−1w−1

⌋+ 1§§�,

�´��(n, 2w − 1, w)qè"dÚn7.3��§·��E�è´�`�"

CHAPTER 7 ^HANANIn�W¿�E�5��`õ�~è 101

Ï��w ∈ {3, 4}�§�`(n,w, 1)-W¿�W¿ê®²��(½§¤±d

Ún7.4§·�U��éõ�`è"

íØ 7.5. é?¿q ≥ bn−12c+ 1§Aq(n, 5, 3) = D(n, 3, 2)"

b. Hananinnn��WWW¿¿¿��½½½ÂÂÂ

��Hananin�W¿£Hanani triple packing§HTP¤Ò´��`(n, 3, 1)-

W¿§§�«|8�±y©¤�PPC�8Ü§�Ø�õ��±§Ù

{Ñ´��§¿P�HTP(n)"Hananin�W¿´|Ü�Op�®�(

�§XHananin�X[140]§Kirkmann�X[41]�í2"�½��(n, k, 1)-W

¿(X,B)Ú��PPC P ⊂ B§·�^P5L«P�"�:§=µP = X \ {x :

x ∈ B,B ∈ P}"

é?¿n§-h =⌊n−12

⌋§b = bn/3c§t = bn/6c§c ≡ n (mod 3)§�0 ≤

c ≤ 2"@on = 3b+ c§�µ

Aq(n, 5, 3) ≤⌊

(q − 1)(3b+ c)

3

⌋= (q − 1)b+

⌊(q − 1)c

3

⌋"

·�3L7.1�Ñ��HTP(n)¤I«|�ê"·�òÌ�^HTP(n)5�E

�q ≤ h+ 1��`(n, 5, 3)qè"¯¢þ§·�I��r�½Â"

L 7.1: ��HTP(n)¤I�«|�ên D(n, 3, 2) = b · h+ ?

6t 6t2 − 2t = 2t(3t− 1) + 0

6t+ 1 6t2 + t = 2t · 3t+ t

6t+ 2 6t2 + 2t = 2t · 3t+ 2t

6t+ 3 6t2 + 5t+ 1 = (2t+ 1)(3t+ 1) + 0

6t+ 4 6t2 + 6t+ 1 = (2t+ 1)(3t+ 1) + t

6t+ 5 6t2 + 9t+ 2 = (2t+ 1)(3t+ 2) + 2t

½Â 7.6. �n 6≡ 0 (mod 3)�§b��HTP(n)kMPPC Pi§ i ∈ [h]§�

�PPC Ph+1"b�Pi = {ai,j : j = 1, . . . , c}§i ∈ [h]"�c = 2�§Pi¥�:´kS�"

3ù«�¹e§��HTP(n)�¡�´r�XJ¤k�PPC÷vXe^�§

é?¿s ∈ [t]µ

102 Aa|Ü?è¯KïÄ

(i) é?¿j = 1, . . . , c§{a3s−2,j, a3s−1,j, a3s,j}´��PPC�«|¶

(ii) ec = 2§c3s−1�MPPCÚ«|{a3s−2,1, a3s−2,2, a3s−1,1}/¤��(n, 3, 1)-

W¿§=µ{a3s−2,1, a3s−2,2, a3s−1,1}¥�?Û:éØÑy3⋃3s−1i=1 Pi�?Û

«|¥"

�n ≡ 0 (mod 3)�§��HTP(n)�´r�"

~ 7.7. ��rHTP(8)µ

-X = Z6∪{∞0,∞1}"e¡�ÑMPPC Pi§�A�kS8ÜPi§i ∈ [3]§Ú��PPC P4"

P1 = {{1, 5,∞0}, {2, 4,∞1}}, P1 = {0, 3}¶P2 = {{2, 3,∞0}, {0, 5,∞1}}, P2 = {1, 4}¶P3 = {{0, 4,∞0}, {1, 3,∞1}}, P3 = {2, 5}¶P4 = {{0, 1, 2}, {3, 4, 5}}"

~ 7.8. ��rHTP(10)µ

-X = Z6 ∪ {∞0,∞1,∞2,∞3}"e¡�ÑMPPC Pi§Pi§i ∈ [4]§Ú��PPC P5"

P1 = {{∞2, 0, 1}, {∞3, 2, 3}, {∞0, 4, 5}}, P1 = {∞1}¶P2 = {{∞1, 2, 4}, {∞3, 0, 5}, {∞0, 1, 3}}, P2 = {∞2}¶P3 = {{∞1, 1, 5}, {∞2, 3, 4}, {∞0, 0, 2}}, P3 = {∞3}¶P4 = {{∞1, 0, 3}, {∞2, 2, 5}, {∞3, 1, 4}}, P4 = {∞0}¶P5 = {{∞1,∞2,∞3}}"

~ 7.9. ��rHTP(17)µ

-X = Z12 ∪ {∞0,∞1,∞2,∞3,∞4}"e¡�ÑMPPC Pi§�A�kS8ÜPi§i ∈ [8]"��PPC P9 = {{0, 4, 8}+ i : i = 0, 1, 2, 3}"

P1 = {{2, 8,∞0}{6, 5,∞1}{4, 3,∞2}{9, 11,∞3}{10, 7,∞4}}, P1 = {0, 1}¶P2 = {{9, 3,∞0}{1, 7,∞1}{0, 10,∞2}{6, 8,∞3}{11, 2,∞4}}, P2 = {4, 5}¶P3 = {{7, 6,∞0}{3, 10,∞1}{11, 5,∞2}{0, 2,∞3}{1, 4,∞4}}, P3 = {8, 9}¶P4 = {{11, 10,∞0}{8, 9,∞1}{1, 6,∞2}{7, 4,∞3}{0, 5,∞4}}, P4 = {2, 3}¶P5 = {{5, 4,∞0}{0, 11,∞1}{2, 9,∞2}{1, 10,∞3}{3, 8,∞4}}, P5 = {6, 7}¶P6 = {{1, 0,∞0}{2, 4,∞1}{8, 7,∞2}{3, 5,∞3}{6, 9,∞4}}, P6 = {10, 11}¶P7 = {{0, 9, 7}{11, 4, 6}{2, 3, 1}{8, 5, 10}{∞0,∞1,∞2}}, P7 = {∞3,∞4}¶P8 = {{0, 6, 3}{4, 10, 9}{1, 8, 11}{2, 5, 7}{∞0,∞3,∞4}}, P8 = {∞1,∞2}"

c. Hananinnn��WWW¿¿¿��`èèè��éééXXX

XeÚn�Ñ��é?¿q ≥ 2§�E�`(n, 5, 3)qè��{"


Ún 7.10. XJ�3��rHTP(n)§@oé?¿q ≥ 2§Aq(n, 5, 3) = min{b(q−

1)n/3c, D(n, 3, 2)}"

y². Äk§·�y²q ≤ bn−12c+ 1��¹"

�n ≡ 0 (mod 3)�§��rHTP(n)TÐkh�MPPC P1,P2, . . . ,Ph"é?¿q ∈ [2, h+ 1]§-Cq =

⋃q−1i=1 C(Pi, i)"éN´�yCqÒ´�`(n, 5, 3)qè"

�n ≡ 1 (mod 3)§��rHTP(n)kh�MPPC Pi§i ∈ [h]§Ú��kt�

«|�PPC Ph+1"-{xi} = Pi§i ∈ [h]"é?¿s ∈ [h]§es�3�Ø§½

Â��| 8�{xs−2, xs−1, xs}�èius§Ù¥é?¿i ∈ [s − 2, s]§usxi = i"

-C2 = C(P1, 1)"é?¿q ∈ [3, h+ 1]§½Â

Cq =

Cq−1 ∪ C(Pq−1, q − 1)§ eq 6≡ 1 (mod 3)§

Cq−1 ∪ C(Pq−1, q − 1) ∪ {uq−1}§ ÄK"

drHananin�W¿�1��5�§z�CqÑ´��`(n, 5, 3)qè"

�n ≡ 2 (mod 3)§��rHTP(n)kh+ 1�MPPC Pi§i ∈ [h+ 1]§Ù¥X

Jn ≡ 2 (mod 6)§Ph+1´��§XJn ≡ 5 (mod 6)§Ph+1´��"b

�{ai,1, ai,2}´Pi§i ∈ [h]¥�kS:é"é?¿s ∈ [h]§½Â��èius§Ù

¥§�| 8�

supp(us) =

{as,1, as,2, as+1,1}§ es ≡ 1 (mod 3)§

{as−1,1, as,1, as+1,1}§ es ≡ 2 (mod 3)§

{as−2,2, as−1,2, as,2}§ es ≡ 0 (mod 3)"

�usai,j = i§i ∈ [s − 2, s + 1]"-C2 = C(P1, 1)"é?¿q ∈ [3, h + 1]§Ø

n ≡ 5 (mod 6)§q = h+ 1§½Â

Cq =

Cq−1 ∪ C(Pq−1, q − 1) ∪ {uq−2}§ eq ≡ 0 (mod 3)§

Cq−1 ∪ C(Pq−1, q − 1) ∪ {uq−2, uq−1} \ {uq−3}§ eq ≡ 1 (mod 3)§

Cq−1 ∪ C(Pq−1, q − 1)§ eq ≡ 2 (mod 3)"

én ≡ 5 (mod 6)§q = h + 1§½ÂCq = Cq−1 ∪ C(Pq−1, q − 1)"@oCqÒ´�`(n, 5, 3)qè"

d½Â§��rHTP(n)�´��`(n, 3, 1)-W¿§¿�¤k«|3�E

¥Ñ^�"¤±dÚn7.3§é?¿q > bn−12c+ 1§ÑØUk�õ�èi"

104 Aa|Ü?è¯KïÄ

3�!(å�c§·�ò|^~7.7–7.9�ÑÚn7.10�~f"

~ 7.11. én = 8§�µ

• C2 = C(P1, 1) = {01000110, 00101001}¶

• C3 = C2∪C(P2, 2)∪{u1}§Ù¥u1´��u10 = 1§u13 = 1§u11 = 2§é?¿Ù

{x ∈ X§u1x = 0�èi§=C3 = C2 ∪ {00220020, 20000202, 12010000}¶

• C4 = (C3\{u1})∪C(P3, 3)∪{u2, u3}§Ù¥C(P3, 3) = {30003030, 03030003}§u2 = 12300000§u3 = 00012300"

éN´�y§é?¿q ∈ [2, 4]§CqÑ´¤I�`(8, 5, 3)qè"

3e¡ü�~f¥§·�ò�ÑC(Pi, i)¥�èi"

~ 7.12. én = 10§�C2 = C(P1, 1)¶C3 = C2 ∪ C(P2, 2)¶C4 = C3 ∪ C(P3, 3) ∪{u3}§Ù¥u3 = 0000000123¶C5 = C4∪C(P4, 4)"éN´�y§é?¿q ∈ [2, 5]§

CqÒ´¤I�`(10, 5, 3)qè"

~ 7.13. én = 17§�C2 = C(P1, 1)¶C3 = C2 ∪ C(P2, 2) ∪ {u1}§Ù¥u1 =

11002000000000000¶C4 = (C3\{u1})∪C(P3, 3)∪{u2, u3}§Ù¥u2 = 10002000300

000000§u3 = 01000200030000000¶C5 = C4∪C(P4, 4)¶C6 = C5∪C(P5, 5)∪{u4}§Ù¥u4 = 00440050000000000¶C7 = (C6 \ {u4}) ∪ C(P6, 6) ∪ {u5, u6}§Ù¥u5 =

00400050006000000§u6 = 00040005000600000¶C8 = C7 ∪ C(P7, 7)¶C9 = C8 ∪C(P8, 8)"éN´�y§é?¿q ∈ [2, 9]§CqÒ´¤I�`(17, 5, 3)qè"

7.3 rrrHananinnn��WWW¿¿¿��333555

3�!¥§·�òïárHananin�W¿��35"w,§�n ≤ 5�§

rHTP(n)��35´²��"

��k-frameÒ´��{k}-GDD (X,G,B)§¦�B�±©¤�PPC�8

Ü§�z�PPC�Ö´GDD�,�|"

Ún 7.14 ([41]). e¡3-frameÑ�3µ

(1) .�hu§u ≥ 4§h ≡ 0 (mod 2)§h(u− 1) ≡ 0 (mod 3)§

(2) .�12um1§u ≥ 4§m ∈ {6, 18}"


a. ��n ≡ 0 (mod 3)��

w,§�n = 6t + 3§t ≥ 0�§��KTS(6t + 3)Ò´��HTP(6t + 3)¶

�n = 6t§t ≥ 3�§��.�23t�{3}-RGDDÒ´��HTP(6t)§¿�Ñ´r

�"dÚn2.12§·��e¡(J"

íØ 7.15. -n ≡ 0 (mod 3)"@o�3��HTP(n)��=�n 6∈ {6, 12}"

b. ��n ≡ 1 (mod 6)��

-v = 6t + 1§��ê�v�Hananin�W¿�´��.�16t+1�{3}-GDD§�«|�±y©¤3t�MPPC§Ú��äkt�«|�PPC"ù��

O�¡��Hananin�X"

Ún 7.16 (Vanstone�[140]). ��ê�n�Hananin�X�3��=�n ≡ 1

(mod 6)§n 6∈ {7, 13}"

e¡íØ`²��Hananin�XÒ´��rHananin�W¿"

íØ 7.17. -n ≡ 1 (mod 6)"��rHTP(n)�3��=�n 6∈ {7, 13}"

y². -(X,B)´��ê�6t + 1�Hananin�X§B�±y©¤3t�MPPC

Pi§i ∈ [3t]§Ú��kt�«|�PPC P3t+1"XJx´P3t+1¥�:§@ox�

½�TÐ��Pi§i ∈ [3t]�"§Ï�X¥�z�:ÑTÐÑy33t�«|¥"

Ïd§·��±#ü�Pi§i ∈ [3t]§¦�é?¿s ∈ [t]§P3s−2 ∪ P3s−1 ∪P3s´P3t+1¥��«|"

c. ��n ≡ 2 (mod 6)��

w,§��.�22t+1�3-frameÒ´��HTP(6t+2)"��rHTP(6t+

2)§·�I�e¡(J"

Ún 7.18. �3rHTP(20)"

y². -X = Z18 ∪ {∞0,∞1}§·�3Xþ�E��rHTP(20)"·�3e¡�ÑMPPC P1,P4,P7§Ú�A��"�:"é?¿i ∈ {2, 3, 5, 6, 8, 9}§

106 Aa|Ü?è¯KïÄ

MPPC Pi§�±ÏLéPi−13Z18¥\6��§Pi�´^Ó��{��"��PPC´P10 = {{0, 6, 12}+ i : i = 0, 1, . . . , 5}"

P1 = {{14, 7,∞0}, {10, 15,∞1}, {5, 16, 8}, {1, 3, 11}, {2, 12, 4}, {13, 6, 17}}§ P1 = {0, 9}¶P4 = {{16, 9,∞0}, {17, 1,∞1}, {14, 13, 10}, {0, 4, 5}, {7, 3, 6}, {8, 12, 15}}§ P4 = {2, 11}¶P7 = {{6, 5,∞0}, {12, 14,∞1}, {8, 9, 11}, {4, 17, 3}, {0, 16, 13}, {7, 2, 15}}§ P7 = {1, 10}"

Ún 7.19. é?¿t 6= 2§�3��rHTP(6t+ 2)"

y². ét ∈ {1, 3}§d~7.7ÚÚn7.18§�3rHTP(6t+ 2)"

é?¿t ≥ 4§lÚn7.14��.�6t�3-frame (X,G,B)§Ù¥X =

Z6 × [t]§G = {Z6 × {i} : i ∈ [t]}§B�±y©¤PPC Pi§i ∈ [3t]"b�P3i−2§

P3i−1§P3i´�"|Z6 × {i}§i ∈ [t]�PPC"O\ü�Ã¡:{∞0,∞1}"é?¿i ∈ [t]§3(Z6×{i})∪{∞0,∞1}þ�E��rHTP(8)§Ù¥MPPC�P i1§P i2§P i3§P i4§¦�P i4 = {∞0,∞1}"-P ′j+3(i−1) = Pj+3(i−1) ∪ P ij§i ∈ [t]§j ∈ [3]"

?�Ú§-P ′3t+1 = ∪ti=1P i4"@o(X ∪ {∞0,∞1},∪3t+1i=1 P ′i)Ò´¤I�O"

d. ��n ≡ 4 (mod 6)��

Ún 7.20. é?¿t ∈ [2, 7]§�3��rHTP(6t+ 4)"

y². é?¿t ∈ [2, 7]§·�3Z6t ∪ {∞0,∞1,∞2,∞3}þ�E��rHTP(6t +

4)"Ù¥§é?¿t§·�3e¡�ÑMPPC P1,P2,P3"é?¿i ∈ [3]§s ∈[t − 1]§MPPC Pi+3s´dPi3Z6t¥\6s��"P3t+1 = {{0, 2t, 4t} + i : i =

0, 1, . . . , 2t− 1} ∪ {{∞1,∞2,∞3}}"z�MPPC�¹��:§�P3i−2 ∪ P3i−1 ∪P3i§i ∈ [t]/¤��PPC P3t+2�t�«|"

t = 2µ

P1 = {9, 2,∞0}{3, 10,∞1}{7, 8,∞2}{4, 5,∞3}{6, 11, 1}¶P2 = {10, 7,∞0}{1, 0,∞1}{11, 4,∞2}{8, 6,∞3}{2, 3, 5}¶P3 = {0, 11,∞0}{5, 8,∞1}{9, 6,∞2}{3, 1,∞3}{2, 4, 7}"

t = 3µ

P1 = {5, 13,∞0}{2, 7,∞1}{3, 16,∞2}{14, 4,∞3}{8, 6, 1}{9, 10, 17}{15, 11, 12}¶P2 = {14, 15,∞0}{6, 3,∞1}{5, 8,∞2}{12, 17,∞3}{2, 1, 9}{4, 0, 7}{13, 16, 11}¶P3 = {10, 12,∞0}{17, 16,∞1}{6, 7,∞2}{13, 9,∞3}{1, 3, 5}{15, 2, 4}{11, 8, 0}"

t = 4µ


P1 = {16, 7,∞0}{18, 21,∞1}{6, 8,∞2}{4, 3,∞3}{9, 13, 23}{2, 20, 22}{12, 17, 19}{15, 1, 14}{10, 5, 11}¶P2 = {9, 18,∞0}{5, 16,∞1}{4, 11,∞2}{1, 12,∞3}{7, 2, 6}{13, 20, 17}{0, 19, 10}{15, 22, 8}{23, 14, 3}¶P3 = {14, 5,∞0}{19, 20,∞1}{17, 0, 6}{1, 3,∞2}{8, 11,∞3}{9, 15, 4}{10, 7, 13}{21, 12, 2}{22, 18, 16}"

t = 5µ

P1 = {20, 28,∞0}{17, 3,∞1}{25, 8,∞2}{19, 24,∞3}{11, 12, 29}{10, 16, 15}{22, 6, 18}{0, 1, 13}{26, 9, 27}{7, 5, 14}{21, 23, 4}¶P2 = {23, 1,∞0}{20, 22,∞1}{6, 17,∞2}{4, 15,∞3}{25, 26, 18}{21, 0, 27}{7, 11, 10}{9, 2, 13}{14, 8, 12}{16, 19, 28}{3, 29, 24}¶P3 = {21, 18,∞0}{13, 24,∞1}{22, 15,∞2}{29, 2,∞3}{1, 9, 7}{0, 28, 6}{26, 5, 12}{4, 20, 8}{19, 14, 3}{16, 11, 25}{17, 10, 23}"

t = 6µ

P1 = {27, 7,∞0}{35, 18,∞1}{26, 10,∞2}{14, 17,∞3}{13, 23, 15}{20, 9, 29}{34, 4, 11}{16, 6, 5}{32, 31, 2}{8, 12, 21}{1, 24, 30}{33, 22, 3}{28, 25, 19}¶P2 = {17, 26,∞0}{14, 1,∞1}{27, 28,∞3}{3, 25,∞2}{0, 8, 19}{35, 20, 34}{12, 10, 2}{5, 11, 7}{33, 18, 23}{16, 30, 31}{32, 29, 15}{24, 13, 21}{22, 6, 9}¶P3 = {4, 12,∞0}{22, 15,∞1}{6, 17,∞2}{31, 0,∞3}{21, 23, 30}{19, 34, 2}{29, 10, 13}{20, 5, 18}{7, 3, 35}{28, 26, 33}{24, 8, 9}{1, 32, 27}{16, 25, 11}"

t = 7µ

P1 = {29, 40,∞0}{36, 38,∞1}{25, 22,∞2}{19, 12,∞3}{27, 24, 30}{34, 1, 10}{39, 7, 17}{16, 32, 13}{11, 4, 6}{2, 31, 14}{33, 9, 26}{20, 28, 21}{5, 35, 8}{15, 23, 41}{0, 37, 18}¶P2 = {33, 14,∞0}{39, 41,∞1}{27, 32,∞2}{3, 16,∞3}{37, 38, 12}{26, 15, 31}{5, 36, 10}{25, 21, 19}{28, 13, 11}{40, 8, 30}{1, 9, 0}{24, 17, 4}{20, 22, 2}{7, 23, 29}{35, 18, 6}¶P3 = {7, 18,∞0}{10, 25,∞1}{36, 17,∞2}{8, 11,∞3}{23, 31, 19}{34, 15, 40}{13, 2, 37}{14, 5, 6}{27, 39, 12}{38, 0, 32}{4, 9, 3}{22, 41, 26}{20, 33, 29}{1, 21, 30}{28, 16, 24}"

Ún 7.21. é?¿��êt§�3��rHTP(6t+ 4)"

y². ét ≤ 7§¤IrHTP(6t + 4)d~7.8ÚÚn7.20�E��"ét ≥ 8§·

�©üÜ©�Eµ

�t = 2s§s ≥ 4�§lÚn7.14��.�12s�3-frame (X,G,B)§Ù

¥X = Z12 × [s]§G = {Z12 × {i} : i ∈ [s]}§B�±y©¤PPC Pi§i ∈ [6s]"

b�é?¿i ∈ [s]§Pj§j ∈ [6i − 5, 6i]´�"|Z12 × {i} �6�PPC"-Y =

{∞0,∞1,∞2,∞3}"é?¿i ∈ [s]§3(Z12 × {i}) ∪ Yþ�E��rHTP(16)§

Ù¥7�MPPC�P ij§j ∈ [7]§��PPC�P i8§¦�{∞1,∞2,∞3}´P i7¥�«|§�P i7 = {∞0}"-P ′j+6(i−1) = Pj+6(i−1) ∪ P ij§ i ∈ [s]§j ∈ [6]"��§

-P ′6s+1 = ∪si=1P i7§P ′6s+2 = ∪si=1P i8"@o§(X ∪ Y,∪6s+2i=1 P ′i)Ò´¤I�O"

108 Aa|Ü?è¯KïÄ

�t = 2s+1§s ≥ 4�§lÚn7.14��.�12s61�3-frame (X,G,B)§Ù

¥X = (Z12× [s])∪(Z6×{s+1})§G = {Z12×{i} : i ∈ [s]}∪{Z6×{s+1}}§B�±y©¤k4s−2�«|�PPC Pi§i ∈ [6s]§Úk4s�«|�PPC Pi§i ∈ [6s+

1, 6s+ 3]"b�é?¿i ∈ [s]§Pj§j ∈ [6i− 5, 6i]´�"|Z12 × {i}�PPC Pj§j ∈ [6s+ 1, 6s+ 3]´�"|Z6 × {s+ 1}�PPC"-Y = {∞0,∞1,∞2,∞3}"é?¿i ∈ [s]§3(Z12×{i})∪Yþ�E��rHTP(16)§Ù¥7�MPPC�P ij§j ∈[7]§��PPC�P i8§¦�{∞1,∞2,∞3}´P i7¥��«|§�P i7 = {∞0}"��§3(Z6×{s+1})∪Yþ�E��rHTP(10)§Ù¥4�MPPC�Ps+1

j §j ∈ [4]§

��PPC�Ps+15 = {∞1,∞2,∞3}"-P ′j+6(i−1) = Pj+6(i−1)∪P ij§i ∈ [s]§j ∈ [6]¶

P ′6s+j = P6s+j ∪ Ps+1j §j ∈ [3]¶P ′6s+4 = (∪si=1(P i7 \ {{∞1,∞2,∞3}})) ∪ Ps+1

4 §

P ′6s+5 = (∪si=1P i8) ∪ Ps+15 "@o(X ∪ Y,∪6s+5

i=1 P ′i) Ò´¤I�O"

e. ��n ≡ 5 (mod 6)��

Ún 7.22. �3��rHTP(23)"

y². -X = Z18 ∪ {∞0,∞1,∞2,∞3,∞4}"·�3e¡�ÑMPPC Pi§i ∈{1, 4, 7}Ú�A�8ÜPi"éi ∈ {2, 3, 5, 6, 8, 9}§MPPC Pi�±dPi−13Z18¥\6��§Pi�±^�Ó��{��"-P10 = {{14, 10, 13}+ 6i, {17, 9, 6}+ 6i :i = 0, 1, 2}∪{{∞0,∞1,∞2}}§P11 = {{8, 10, 0}+ 6i, {1, 3, 5}+ 6i : i = 0, 1, 2}∪{{∞0,∞3,∞4}}"��-P12 = {{0, 6, 12}+ i : i = 0, 1, . . . , 5}"

P1 = {{0, 1,∞0}, {16, 8,∞1}, {4, 5,∞2}, {11, 13,∞3}, {7, 12,∞4}, {14, 17, 3}, {15, 2, 6}}§P1 = {9, 10}¶P4 = {{14, 11,∞0}, {7, 3,∞1}, {9, 12,∞2}, {6, 8,∞3}, {10, 15,∞4}, {4, 17, 0}, {5, 13, 16}}§P4 = {1, 2}¶P7 = {{3, 4,∞0}, {0, 5,∞1}, {8, 13,∞2}, {16, 9,∞3}, {11, 2,∞4}, {12, 10, 1}, {7, 14, 15}}§P7 = {6, 17}"

Ún 7.23. é?¿��êt§t 6∈ {1, 4, 5, 6, 7, 9}§�3��rHTP(6t+ 5)"

y². ét ∈ {2, 3}§¤I�O3~7.9ÚÚn7.22¥�E��"ét ≥ 8§t 6= 9§

·�©ü«�¹�Eµ

ét = 2s§s ≥ 4§lÚn7.14��.�12s�3-frame (X,G,B)§Ù¥X =

Z12×[s]§G = {Z12×{i} : i ∈ [s]}§B�±y©¤PPC Pi§i ∈ [6s]"b�Pj§j ∈[6i−5, 6i]´�"|Z12×{i}§i ∈ [s]�6�PPC"-Y = {∞0,∞1,∞2,∞3,∞4}"


é?¿i ∈ [s]§3(Z12 × {i}) ∪ Yþ�E��rHTP(17)§Ù¥8�MPPC�P ij§j ∈ [8]§��PPC�P i9§¦�{∞0,∞1,∞2} ∈ P i7§P i7 = {∞3,∞4}§{∞0,∞3,∞4} ∈ P i8§P i8 = {∞1,∞2}"-P ′j+6(i−1) = Pj+6(i−1)∪P ij§i ∈ [s]§j ∈[6]§P ′6s+1 = ∪si=1P i7§P ′6s+2 = ∪si=1P i8§P ′6s+3 = ∪si=1P i9"@o(X∪Y,∪6s+3

i=1 P ′i)Ò´¤I�O"

ét = 2(s+1)+1§s ≥ 4§lÚn7.14��.�12s181�3-frame (X,G,B)§

Ù¥X = (Z12 × [s]) ∪ (Z18 × {s + 1})§G = {Z12 × {i} : i ∈ [s]} ∪ {Z18 ×{s + 1}}§B�±y©¤k4s + 2�«|�PPC Pi§i ∈ [6s]§Úk4s�«|

�PPC Pi§i ∈ [6s + 1, 6s + 9]"b�é?¿i ∈ [s]§Pj§j ∈ [6i − 5, 6i]´�"

|Z12×{i}�PPC Pj§j ∈ [6s+1, 6s+9]´�"|Z18×{s+1}�PPC"-Y =

{∞0,∞1,∞2,∞3,∞4}"é?¿i ∈ [s]§3(Z12×{i})∪Yþ�E��rHTP(17)§

Ù¥8�MPPC�P ij§j ∈ [8]§��PPC�P i9§¦�{∞0,∞1,∞2} ∈ P i7§P i7 = {∞3,∞4}§{∞0,∞3,∞4} ∈ P i8§P i8 = {∞1,∞2}"��3(Z18 × {s +

1}) ∪ Yþ�E��rHTP(23)§Ù¥11�MPPC�Ps+1j §j ∈ [11]§��

�PPC�Ps+112 ¦�{∞0,∞1,∞2} ∈ Ps+1

10 §Ps+110 = {∞3,∞4}§{∞0,∞3,∞4} ∈

Ps+111 §Ps+1

11 = {∞1,∞2}"-P ′j+6(i−1) = Pj+6(i−1)∪P ij§i ∈ [s]§j ∈ [6]¶P ′6s+j =

P6s+j ∪ Ps+1j §j ∈ [9]¶P ′6s+j = (∪si=1P ij−3) ∪ Ps+1

j §j ∈ {10, 11}¶P ′6s+12 =

∪si=0P i9 ∪ Ps+112 "@o(X ∪ Y,∪6s+12

i=1 P ′i)Ò´¤I�O"

·�^O�Å|¢(½Ø�3��HTP(11)"(ÜíØ7.15§Ún7.17§Ú

n7.19§Ún7.21ÚÚn7.23§·��rHananin�W¿��3(J"

½n 7.24. é?¿��ên§Ø(½��n ∈ {6, 7, 11, 12, 13}§ÚØ(½��n ∈ {14, 29, 35, 41, 47, 59}§Ñ�3��rHTP(n)"

7.4 Aq(n, 5, 3)��(((½½½

é½n7.24¥�(JA^Ún7.10§·�Ò��é?¿n 6∈ {6, 7, 11, 12, 13,

14, 29, 35, 41, 47, 59}§q ≥ 2§Aq(n, 5, 3) =⌊(q−1)n

3

⌋"3ù!¥§·�ò(½�

{�n"díØ7.5§·�ò�(½�2 ≤ q ≤ bn−12c�§Aq(n, 5, 3)��"

Ún 7.25. én ∈ {6, 12}§2 ≤ q ≤ bn−12c§Aq(n, 5, 3) = (q−1)n

3"

110 Aa|Ü?è¯KïÄ

y². én = 6§w,A2(6, 5, 3) = 2"én = 12§��Z12þ�(12, 3, 1)-W¿§Ù¥¤k«|�±y©¤Xeo�PC Pi§i ∈ [4]Xeµ

P1 = {{0, 8, 7}+ 6i, {9, 10, 11}+ 6i : i = 0, 1}¶P2 = {{0, 1, 9}+ 6i, {4, 8, 11}+ 6i : i = 0, 1}¶P3 = {{5, 7, 9}+ 6i, {6, 8, 10}+ 6i : i = 0, 1}¶P4 = {{1, 5, 8}+ 6i, {0, 3, 10}+ 6i : i = 0, 1}"

é?¿q ∈ [2, 5]§Cq = ∪q−1i=1C(Pi, i)´��`(12, 5, 3)qè"Ïdé?¿q ∈[2, 5]§Aq(12, 5, 3) = 4(q − 1)"

Ún 7.26. én ∈ {7, 13}§2 ≤ q ≤ bn−12c§Aq(n, 5, 3) = b (q−1)n

3c"

y². én = 7§w,A2(7, 5, 3) = 2"·��±�E��`(7, 5, 3)3è§C3 =

{1110000, 2000110, 0020021, 0200202}§¤±A3(7, 5, 3) = 4"

én = 13§3Z13þ�E��(13, 3, 1)-W¿§Ù¥¤k«|�±y©¤5�MPPC Pi§i ∈ [5]Ú��PPC P6 = {{0, 1, 2}}"5¿�Pi§i ∈ [3] �"�:©O�0, 1, 2"

P1 = {{5, 8, 9}, {1, 3, 6}, {2, 4, 7}, {12, 11, 10}}¶P2 = {{0, 3, 7}, {4, 12, 8}, {11, 6, 5}, {2, 9, 10}}¶P3 = {{3, 9, 12}, {1, 5, 7}, {8, 10, 6}, {11, 4, 0}}¶P4 = {{12, 6, 7}, {2, 8, 3}, {10, 0, 5}, {9, 1, 11}}¶P5 = {{9, 0, 6}, {11, 7, 8}, {5, 2, 12}, {1, 4, 10}}"

ùp§é?¿q ∈ {2, 3}§-Cq = ∪q−1i=1C(Pi, i)"é?¿q ∈ {4, 5, 6}§-Cq =

∪q−1i=1C(Pi, i)∪{u}§Ù¥u = 1230000000000"éN´�yé?¿q ∈ [2, 6]§CqÒ´�`(13, 5, 3)qè"

Ún 7.27. é?¿2 ≤ q ≤ 5§Aq(11, 5, 3) = b(q − 1)11/3c"

y². 3Z11þ�E��(11, 3, 1)-W¿§Ù¥¤k«|�±y©¤Xe4�MPPC"

P1 = {{1, 8, 0}, {3, 6, 9}, {5, 7, 10}}¶P2 = {{1, 6, 7}, {3, 8, 10}, {4, 9, 0}}¶P3 = {{1, 4, 5}, {2, 6, 8}, {3, 7, 0}}¶P4 = {{1, 2, 3}, {4, 7, 8}, {5, 6, 0}}"

-u1 = 00101200000§u2 = 00201000003§u3 = 00100200030"½ÂC2 =

C(P1, 1)§C3 = ∪2i=1C(Pi, i) ∪ {u1}¶éq ∈ {4, 5}§Cq = ∪q−1i=1C(Pi, i) ∪ {u2, u3}"

@oé?¿q ∈ [2, 5]§CqÒ´�`(11, 5, 3)qè"


Ún 7.28. é?¿2 ≤ q ≤ 6§Aq(14, 5, 3) = b(q − 1)14/3c"

y². 3Z14þ�E��(14, 3, 1)-W¿§Ù¥¤k«|�±y©¤Xe5�MPPCÚ��PPC P6 = {{0, 4, 8}, {6, 10, 2}}"

P1 = {{4, 11, 12}, {10, 5, 13}, {2, 9, 7}, {3, 8, 1}}¶P2 = {{0, 9, 12}, {1, 2, 13}, {5, 3, 6}, {11, 7, 8}}¶P3 = {{1, 6, 12}, {0, 7, 13}, {4, 5, 9}, {3, 11, 10}}¶P4 = {{10, 8, 12}, {3, 4, 13}, {6, 11, 9}, {0, 5, 2}}¶P5 = {{2, 3, 12}, {8, 9, 13}, {0, 1, 10}, {4, 6, 7}}"

éN´�y�E�(14, 3, 1)-W¿§÷vrHananin�W¿�ü�5�"¤±é

?¿q ∈ [2, 6]§·��`(14, 5, 3)qè"

Ún 7.29. é?¿n ≡ 5 (mod 6)§n ≥ 17§Aq(n, 5, 3) = b(q − 1)n/3c§Ù¥q = b(n− 1)/2c"

y². -n = 6t+5§Ù¥t ≥ 2"l©[41]¥��.�32t51�{3}-GDD (X,G,B)§

Ù¥X = Z3t∪{∞0, . . . ,∞4}§{∞0, . . . ,∞4}´��|"@oB∪{{∞0,∞1,

∞2}}Ò´��«|�ê�6t2 + 7t + 1 = b(q − 1)n/3c�(n, 3, 1)-W¿"@o�

`è�±^�Ún7.4¥�Ó��{�E§Ï�z�:�õÑy33t + 1�«|

¥"

Ún 7.30. é?¿n ∈ {29, 35, 47}§2 ≤ q ≤ b(n − 1)/2c§Aq(n, 5, 3) = b(q −1)n/3c"

y². -n = 6t + 5§t ∈ {4, 5, 7}"·�3X = Z6t ∪ {∞0, . . . ,∞4}þ�E��(n, 3, 1)-W¿§Ù¥¤k«|�±y©¤3t�MPPC Pi§i ∈ [3t]§Ú��PPC

P3t+1 = {{0, 2t, 4t}+ i : i = 0, 1, . . . , 2t− 1}"é?¿n§·�3e¡�Ñ1��MPPC P1"é?¿i ∈ [t− 1]§P3i+1�±

dP13Z6t¥\2i��§Pi+1�±dPi 3Z6t¥\2t��"Pi�dÓ��{��"

29 : {2, 4, 7}{3, 5, 6}{8, 17,∞0}{9, 15, 19}{10, 16, 20}{11, 22,∞1}{12, 23,∞2}{13, 18,∞3}{14, 21,∞4}

35 : {2, 4, 7}{8, 12, 19}{9, 20,∞0}{10, 23, 27}{11, 26,∞1}{3, 5, 6}{13, 18,∞2}{14, 22, 28}{15, 21, 29}{16, 25,∞3}{17, 24,∞4}

47 : {2, 4, 7}{8, 12, 18}{9, 13, 19}{10, 17, 32}{24, 39,∞0}{11, 27, 36}{14, 33,∞2}{15, 26, 34}{16, 28, 37}{20, 31, 38}{3, 5, 6}{21, 29, 41}{22, 35,∞3}{23, 40,∞4}{25, 30,∞1}

112 Aa|Ü?è¯KïÄ

éN´�y�E�(n, 3, 1)-W¿§n ∈ {29, 35, 41}÷vrHananin�W¿�ü

�5�"Ïd§·��±^Ún7.10¥��{5é?¿q ∈ [2, 3t + 1]§�E�

`(n, 5, 3)qèCq"�q = 3t+ 2§�`èdÚn7.29��"

Ún 7.31. é?¿2 ≤ q ≤ 20§Aq(41, 5, 3) = b(q − 1)41/3c"

y². -X = Z36 ∪ {∞0, . . . ,∞4}"·�3Xþ�E��(41, 3, 1)-W¿§Ù¥«|�±y©¤18�MPPC Pi§i ∈ [18]§Ú��PPC P19 = {{0, 12, 24} +i : i = 0, 1, . . . , 11}"·�3e¡�ÑMPPC P1§P10§Ù¥§�©O�"{0, 1}Ú{2, 3}"

P1 = {{∞0, 2, 4}, {∞1, 3, 5}, {∞2, 6, 9}, {∞3, 7, 8}, {∞4, 10, 15}, {11, 14, 18}, {12, 16, 19},{13, 17, 20}, {21, 26, 35}, {22, 29, 30}, {23, 28, 33}, {24, 32, 34}, {25, 27, 31}}¶

P10 = {{∞0, 1, 11}, {∞1, 0, 14}, {∞2, 12, 35}, {∞3, 17, 34}, {∞4, 4, 21}, {5, 18, 28},{6, 19, 27}, {7, 16, 25}, {8, 23, 29}, {9, 20, 31}, {10, 24, 30}, {13, 22, 33}, {15, 26, 32}}"

é?¿i ∈ {1, 2}§j ∈ {1, 10}§P3i+j´dPj3Z36¥\4i��"é?¿i ∈{0, 1, . . . , 5}§j ∈ {2, 3}§P3i+j´dP3i+j−13Z36¥\12��"é?¿i ∈ [18] \{1, 10}§Pi´dP1½P10�^Sû½"

éN´�yù�(41, 3, 1)-W¿÷vrHananin�W¿�ü�5�§Ïd

·��±^Ún7.30¥aq��{é?¿q ∈ [2, 19]§�E�`q�è"�q =

20�§�`èdÚn7.29��"

Ún 7.32. é?¿2 ≤ q ≤ 29§Aq(59, 5, 3) = b(q − 1)59/3c"

y². lÚn7.14��.�12461�3-frame (X,G,B)§Ù¥X = (Z12 × [4]) ∪(Z6 × {5})§G = {Z12 × {i} : i ∈ [4]} ∪ {Z6 × {5}}§B�±y©¤k14�

«|�PPC Pi§i ∈ [24]§Úk16�«|�PPC Pi§i ∈ [25, 27]"b�é?

¿i ∈ [4]§Pj§j ∈ [6i − 5, 6i]´�"|Z12 × {i}�PPC§Pj§j ∈ [25, 27]´�

"|Z6 × {5}�PPC"-Y = {∞0,∞1,∞2,∞3,∞4}"é?¿i ∈ [4]§3(Z12 ×{i}) ∪ Yþ�E��rHTP(17)§Ù¥8�MPPC�P ij§j ∈ [8]§��PPC�P i9§�{∞0,∞1,∞2} ∈ P i7§P i7 = {∞3,∞4}§{∞0,∞3,∞4} ∈ P i8§P i8 = {∞1,∞2}"

-P ′j+6(i−1) = Pj+6(i−1) ∪ P ij§i ∈ [4]§j ∈ [6]"P ′25 = ∪si=1P i9"@o(X ∪Y,∪25

i=1P ′i)Ò´��÷vrHananin�W¿5��(59, 3, 1)-W¿"Ïd§é?

¿q ∈ [2, 25]§·��±��`q�èCq"


y3·�3(Z6 × {5}) ∪ Yþ�E�Ý�11��`k�è§P�Dk§k ∈[2, 4]"éDk�z��"�\24Ò�Dk+24§Ù¥�"��[25, 27]"éq ∈[26, 28]§Cq = C25∪ (∪q−1i=25C(Pi, i))∪DqÒ´��`(41, 5, 3)qè"éq = 29§�

`èdÚn7.29 ��"

7.5 Hananinnn��WWW¿¿¿��333555

éN´y²�n ≡ 0, 1 (mod 3)�§��HTP(n)�´��rHTP(n)"Ï

d§�(J��5§·�òy²HTP(n)��35"

Ún 7.33. �3��HTP(29)"

y². -X = Z24 ∪ {∞0, . . . ,∞4}"·�3Xþ�E��HTP(29)§Ù¥«|�±y©¤12�MPPC Pi§i ∈ [12]Ú3�PPC Pi§i ∈ [13, 15]"·�3e¡�ÑPi§i ∈ [4]ÚP15"Pi§i ∈ [5, 12]�±dPi−43Z24þ\8��"P13 = {B + 8 :B ∈ P15} ∪ {{∞0,∞1,∞2}}§P14 = {B + 16 : B ∈ P15} ∪ {{∞0,∞3,∞4}}"

P1 = {∞0, 2, 3}{∞1, 4, 5}{∞2, 6, 7}{∞3, 8, 9}{∞4, 10, 12}{11, 13, 14}{15, 16, 19}{17, 20, 22}{18, 21, 23}¶P2 = {∞0, 0, 4}{∞1, 1, 7}{∞2, 5, 8}{∞3, 6, 10}{∞4, 9, 11}{12, 15, 20}{13, 17, 21}{14, 19, 23}{16, 18, 22}¶P3 = {∞0, 1, 13}{∞1, 10, 22}{∞2, 11, 18}{∞3, 12, 23}{∞4, 15, 21}{0, 7, 19}{2, 9, 16}{3, 14, 20}{6, 8, 17}¶P4 = {∞0, 15, 22}{∞1, 3, 16}{∞2, 1, 20}{∞3, 5, 19}{∞4, 0, 14}{2, 11, 12}{4, 10, 21}{8, 13, 23}{9, 17, 18}¶P15 = {{0, 8, 20}{1, 11, 19}{2, 15, 18}{3, 12, 21}{4, 14, 17}{5, 10, 16}{6, 13, 22}{7, 9, 23}"

Ún 7.34. �3��HTP(41)"

y². -X = Z36 ∪ {∞0, . . . ,∞4}"·�3Xþ�E��HTP(41)§Ù¥«|�±y©¤20�MPPC Pi§i ∈ [20]Ú��PPC P21"·�3e¡�ÑPi§i ∈[2]"Pi§i ∈ [3, 18]�±dPi−23Z36¥\4��"-D = {{0, 15, 28}, {1, 14, 29},{6, 20, 34}, {7, 21, 35}}"P21 = {B + 12i : B ∈ D, i = 0, 1, 2}§P19 = {B + 4 :B ∈ P21} ∪ {{∞0,∞1,∞2}}§P20 = {B + 8 : B ∈ P21} ∪ {{∞0,∞3,∞4}}"

P1 = {{∞0, 2, 3}, {∞1, 4, 5}, {∞2, 6, 8}, {∞3, 7, 9}, {∞4, 10, 13}, {11, 12, 14}, {15, 19, 24},{16, 20, 23}, {17, 27, 33}, {18, 29, 31}, {21, 25, 32}, {22, 28, 34}, {26, 30, 35}}¶P2 = {{∞0, 1, 16}, {∞1, 7, 18}, {0, 5, 24}, {6, 23, 35}, {∞2, 13, 31}, {∞3, 4, 22}, {∞4, 15, 32},{8, 19, 34}, {9, 14, 33}, {10, 17, 26}, {11, 21, 27}{12, 25, 28}, {20, 29, 30}}"

Ún 7.35. é?¿n ∈ {35, 47, 59}§�3��HTP(n)"

114 Aa|Ü?è¯KïÄ

y². -n = 6t + 5§t ∈ {5, 7}§X = Z6t ∪ {∞0, . . . ,∞4}"·�3Xþ�EHTP(n)§Ù¥«|�±y©¤3t + 2�MPPC Pi§i ∈ [3t + 2] Ú��PPCP3t+3"é?¿n§·�3e¡�ÑP1"é?¿i ∈ [2, 3t]§Pi�±dP13Z3t¥\2(i − 1)��"-D = {{0, 1, 2}, {3, 5, 10}}"P3t+3 = {B + 6i : B ∈ D, i =0, 1, . . . , t − 1}§P3t+1 = {B + 2 : B ∈ P3t+3} ∪ {{∞0,∞1,∞2}}§P3t+2 ={B + 4 : B ∈ P3t+3} ∪ {{∞0,∞3,∞4}}"

35 : {∞0, 2, 5}{∞1, 3, 6}{∞2, 4, 13}{∞3, 7, 20}{∞4, 16, 21}{8, 18, 24}{9, 15, 25}{10, 17, 29}{11, 19, 28}{12, 23, 27}{14, 22, 26}

47 : {∞0, 2, 5}{∞1, 3, 6}{∞2, 4, 9}{∞3, 7, 16}{∞4, 8, 17}{10, 21, 34}{11, 27, 38}{12, 24, 37}{13, 28, 32}{14, 31, 35}{15, 23, 33}{18, 26, 40}{19, 25, 39}{20, 30, 36}{22, 29, 41}

59 : {∞0, 9, 32}{∞1, 8, 37}{∞2, 35, 16}{∞3, 36, 19}{42, 2, 6}{∞4, 31, 40}{44, 53, 50}{38, 30, 51}{29, 21, 33}{17, 3, 46}{52, 25, 20}{12, 28, 45}{24, 14, 48}{43, 49, 27}{0, 15, 26}{41, 18, 5}{22, 10, 7}{4, 39, 11}{23, 13, 47}

(ÜrHananin�W¿�(J§·��Xe(Jµ

½n 7.36. é?¿��ên§Ø(½��n ∈ {6, 7, 11, 12, 13}§Ñ�3��HTP(n)"

7.6 (((ØØØ

3�Ù¥§·�ÏLïÄHananin�W¿��E§é?¿��ênÚq ≥2(½�`(n, 5, 3)qè�èi�ê"3d�c§Aq(n, 5, 3)�k�q ∈ {2, 3}§Ú��q3w|(q − 1)n�n¿©��â�(½"

½n 7.37. Aq(n, 5, 3) = min{b (q−1)n

3c, D(n, 3, 2)

}§Ù¥

D(n, 3, 2) =

bn3 bn−12 cc − 1§ en ≡ 5 (mod 6)§

bn3bn−1

2cc§ ÄK"

(7.1)

Hananin�W¿´Hananin�X��í2"·��(½Hananin

�W¿��35"

Chapter 8

^��©©©|||èèè��EEE��`~~~EEEÜÜÜèèè


~EÜè£constant-composition code§CCC¤´~è��«AÏ�

¹§§3?ènØ¥u��^"~EÜè¥��è3ÃPÁ&�

¥"Ø��½�"Uå�(½[138]§õ�¯Ï&[54]§¥/èN�[55]§

DNAè[27, 101, 110]§>å�Ï&[37, 43]§aª[38]§ªÇü��[94]Úk�

�°�&�?è[46]��¡Ñk�A^"

þVÊ�c�"§é~EÜèÒkXÚ�ïÄ[14, 17, 133]"y3§

<��(½~EÜè��U�èi�êÚ\�«��{§XO

�Å|¢�{[16]§W¿�O[38, 49, 50, 93, 144, 153, 154, 157]§¿m�O[158]§

õ�ªÚ��5¼ê[38, 47, 48, 51, 52]§PBD4��{[25, 28]Ú�Ù¦��

{[105, 106, 134, 149]�"

þ�3�~EÜè�èi�êdChee§GeÚLing3©[25]¥��(½"

GaoÚGe3©[61]¥��(½þ�4§ål�5��`n�~EÜè�è

i�ê"ZhuÚGe3©[164]¥(½þ�4§ål�5½6��`o�~E

Üè�èi�ê"©[24, 25]¥�(½�AÏëê��`~EÜè�èi

�ê"

Ún 8.1 (Chee�[25]).

Aq(n, d, [w1, . . . , wq−1]) =

(

n∑q−1i=1 wi

)( ∑q−1i=1 wi

w1,...,wq−1

)§ ed ≤ 2§⌊

n∑q−1i=1 wi

⌋§ ed = 2

∑q−1i=1 wi§

1§ ed ≥ 2∑q−1

i=1 wi + 1"

ÏÚn2.3§·��µ

íØ 8.2. A3(n, 6, [2, 2]) ≤⌊n2A3(n− 1, 6, [2, 1])

⌋=⌊n2

⌊n−13

⌋⌋:= U(n, 6, [2, 2])"

116 Aa|Ü?è¯KïÄ

3�Ù¥§·�ò�Eþ�4§ål�6§.�[2, 2]��`n�~E

Üè§=(n, 6, [2, 2])3è�èi�ê"·�òé?¿�Ýn 6≡ 5 (mod 6)§n 6∈{13, 14, 16, 22, 76, 88, 94, 124, 142, 166, 184}§(½A3(n, 6, [2, 2])3��"én ≡ 5

(mod 6)§·��Ð�e."¤^��{´13Ù¥�Ñ�éGDC�Ì��

E�{"Ï�3�Ù¥§·��Ä�Ñ´[2, 2]-GDC(6)"¤±3e©¥§·�

ò[2, 2]-GDC(6)Ñ{P�GDC"

ù�Ù�(�Xeµ318.2!¥§·�ò0��^�Room frame�

EGDC��{¶318.3!¥§·�ò©�¹�E�`(n, 6, [2, 2])3è¶318.4!

¥§òé�Ù�Ì�(J?1o("

8.2 ��Room frame��EEE{{{

Ún 8.3. XJ�3��.�tu��Room frame§@o�3��.�(6t)u§�

��6t2u(u− 1)�GDC"

y². -F��½�.�tu��Room frame"·�3|8{{(i + k, j) : 0 ≤i ≤ t − 1, j ∈ Z6} : k = 0, t, . . . , t(u − 1)}þ�E��.�(6t)u�GDC§§�¹

¤k�èi〈(x, j), (y, j), (c, 1 + j), (r, 4 + j)〉§〈(c, 4 + j), (r, 1 + j), (x, j), (y, j)〉§Ù¥j ∈ Z6§{x, y}´3F�1c�1r1��"

nÜÚn6.13ÚÚn8.3§·��µ

½n 8.4. -u ≥ 4§t(u− 1) ≡ 0 (mod 2)§Ø(t, u) ∈ {(1, 5), (2, 4)}§ÚXe�U�(t, u)§�3.�(6t)u§��6t2u(u− 1)�GDCµ

(i) u = 4§t ≡ 2 (mod 4)§

(ii) u = 5§t ∈ {17, 19, 23, 29, 31}"


a. ��GDC

Ún 8.5. �3��.�210§��60�GDC"

CHAPTER 8 ^�©|è�E�`~EÜè 117

y². -X = Z20§G = {{i, i + 10} : 0 ≤ i ≤ 9}"@o(X,G, C)´��.�210�GDC§XJC´dèi〈0, 5, 3, 7〉§ 〈0, 4, 1, 13〉§ 〈0, 8, 14, 19〉3Z20¥+1

(mod 20)Ðm��"

Ún 8.6. é?¿5 ≤ t ≤ 11§�3��.�6t§��6t(t− 1)�GDC"

y². ét ∈ {5, 8}§-Xt = Z6t§Gt = {{i, i+ t, i+ 2t, i+ 3t, i+ 4t, i+ 5t} : 0 ≤i ≤ t− 1}"@o(Xt,Gt, Ct)´��.�6t§��6t(t− 1)�GDC§XJCtdXeèi+1 (mod 6t)Ðm��"Ù¥C5�〈0, 24, 1, 13〉§〈0, 3, 17, 26〉§〈0, 9, 8, 11〉§〈0, 12, 4, 28〉¶C8�〈0, 18, 13, 3〉§〈0, 6, 5, 7〉§〈0, 46, 27, 9〉§〈0, 4, 19, 39〉§〈0, 10, 22,

36〉§〈0, 14, 31, 37〉§〈0, 28, 21, 25〉"ét = 6§-X6 = Z12×Z3§G6 = {{(i, 0), (i+6, 0), (i, 1), (i+6, 1), (i, 2), (i+

6, 2)} : 0 ≤ i ≤ 5}"@o(X6,G6, C6)´��.�66§��180�GDC§XJC6dXeèi3Z12 × Z3¥(+1 (mod 12),−)Ðm��"

〈(0, 0)(2, 0)(7, 2)(4, 2)〉〈(0, 1)(9, 1)(10, 0)(5, 0)〉〈(0, 1)(3, 0)(4, 2)(2, 2)〉〈(0, 1)(11, 0)(8, 2)(7, 2)〉〈(0, 2)(2, 2)(4, 1)(7, 0)〉〈(0, 1)(7, 0)(10, 2)(5, 2)〉〈(0, 2)(9, 1)(5, 1)(8, 2)〉〈(0, 2)(9, 2)(1, 0)(11, 0)〉〈(0, 1)(9, 2)(1, 2)(4, 1)〉〈(0, 1)(9, 0)(11, 1)(1, 1)〉〈(0, 0)(5, 0)(8, 0)(9, 0)〉〈(0, 0)(11, 0)(7, 1)(10, 1)〉〈(0, 2)(5, 2)(8, 0)(1, 1)〉〈(0, 2)(11, 2)(10, 1)(9, 0)〉〈(0, 1)(2, 1)(4, 0)(7, 1)〉

ét ∈ {7, 9, 11}§¤IGDCd½n8.4��"ét = 10§¤IGDCdé.

�210�GDC^3)ä��"

-P = [9, 19] ∪ [21, 23] ∪ [26, 28] ∪ [31, 33]"

Ún 8.7. é?¿�t ≥ 9§t 6∈ P§�3��.�24i30j36k42l48m�GDC§Ù

¥i§j§k§l§m´�K�ê�4i+ 5j + 6k + 7l + 8m = t"

y². é?¿t ≥ 9§t 6∈ P§lÚn2.4��(t + 1, {5, 6, 7, 8, 9}, 1)-PBD"l

ù�PBD�:8�K��:��.�4i5j6k7l8m�{5, 6, 7, 8, 9}-GDD§Ù

¥4i+ 5j + 6k+ 7l+ 8m = t"éù�GDD^Ä��E{\�6§¿Ñ\.�6u§

u ∈ {5, 6, 7, 8, 9}�GDC£Ún8.6¤§��.�24i30j36k42l48m�GDC"

Ún 8.8. XeGDCþ�3µ

i) .�18u§��54u(u− 1)§u ∈ {4, 5, 6, 7, 9, 11}¶

ii) .�24u§��96u(u− 1)§u ∈ {4, 7, 8}¶

118 Aa|Ü?è¯KïÄ

iii) .�24u361§��96u(u+ 2)§u ∈ {4, 5}¶

iv) .�188421§��2520¶

v) .�304181§��1260¶

vi) .�244181§��864"

y². i)��.�184�GDCdÚn8.32��E��"é?¿t ∈ {5, 6, 7, 9,11}§lÚn8.6��.�6t�GDC§¿^3)ä��.�18t�GDC" ii)¤

IGDCd½n8.4��" iii)�.�6u91§u ∈ {4, 5}�{4}-GDD£�[77, ½n

1.6]¤"^Ä��E{\�4��I��GDC"ùpÑ\�´.�44�GDC£Ú

n8.32¤" iv)��.�3871�{5}-GDD£�[71]¤"^Ä��E{\�6��.

�188421�GDC"ùp§Ñ\�´.�65�GDC£Ún8.6¤"v)��TD(5, 5)§

^Ä��E{éco�|�¤k:§Ú��|�1�:\�6§Ù{

:\�3§��.�304181�GDC"ùpÑ\�´.�65Ú6431�GDC£Ú

n8.6Ú8.33¤" vi)��TD(5, 4)§^Ä��E{éco�|�¤k:§�

��|�2�:\�6§Ù{:\�3��.�244181�GDC"ùpÑ\.

�65Ú6431�GDC£Ún8.6Ú8.33¤"

b. ��ÝÝÝn ≡ 0, 1 (mod 6)��

Ún 8.9. A3(7, 6, [2, 2]) = 3§A3(13, 6, [2, 2]) ≥ 21"

y². én = 7§d©[159]¥(7, 6, 4)3è�(J��A3(7, 6, [2, 2]) ≤ 3"éN´�

E{0, 1, 2, 3, 4, 5, 6}þ�3�èi〈0, 1, 2, 3〉§〈0, 4, 5, 6〉§〈2, 5, 1, 4〉"én = 13§-:8�{0, 1, 2, . . . , 12}"¤I�21�èiXeµ

〈4, 11, 2, 9〉〈11, 7, 1, 6〉〈2, 5, 0, 7〉〈9, 12, 11, 7〉〈1, 4, 7, 10〉〈10, 5, 8, 4〉〈9, 3, 5, 6〉〈0, 1, 2, 3〉〈0, 12, 10, 5〉〈1, 6, 9, 12〉〈0, 6, 4, 11〉〈8, 6, 3, 7〉〈10, 7, 12, 2〉〈2, 3, 10, 11〉〈8, 4, 0, 12〉〈2, 12, 6, 8〉〈0, 7, 8, 9〉〈5, 11, 3, 12〉〈3, 12, 1, 4〉〈1, 8, 11, 5〉〈9, 10, 0, 1〉

Ún 8.10. é?¿t ∈ [3, 11]∪{13, 14, 17}§A3(6t+1, 6, [2, 2]) = U(6t+1, 6, [2, 2])"


L 8.1: Ún8.10¥�`(6t+ 1, 6, [2, 2])3è�Äèt èi

3 〈0, 1, 4, 16〉〈0, 7, 9, 17〉〈0, 8, 13, 14〉

4〈(0, 3), (2, 0), (3, 0), (4, 3)〉〈(0, 2), (2, 3), (2, 1), (0, 4)〉〈(0, 2), (1, 0), (1, 4), (0, 3)〉〈(0, 0), (1, 1), (2, 0), (4, 1)〉

5 〈0, 3, 11, 23〉〈0, 7, 2, 5〉〈0, 6, 15, 22〉〈0, 14, 4, 10〉〈0, 12, 13, 30〉6 〈0, 23, 27, 35〉〈0, 8, 11, 17〉〈0, 5, 25, 1〉〈0, 6, 22, 36〉〈0, 18, 2, 7〉〈0, 13, 10, 28〉

7〈0, 35, 26, 23〉〈0, 16, 33, 37〉〈0, 29, 22, 38〉〈0, 3, 10, 18〉〈0, 30, 11, 12〉〈0, 1, 6, 20〉〈0, 4, 2, 32〉

8〈0, 36, 40, 45〉〈0, 28, 14, 47〉〈0, 42, 10, 32〉〈0, 33, 25, 18〉〈0, 44, 15, 26〉〈0, 11, 48, 12〉〈0, 22, 3, 46〉〈0, 43, 23, 2〉

9〈0, 8, 1, 21〉〈0, 43, 19, 42〉〈0, 32, 34, 39〉〈0, 20, 30, 45〉〈0, 28, 9, 26〉〈0, 17, 41, 14〉〈0, 5, 16, 49〉〈0, 22, 4, 51〉〈0, 15, 6, 18〉

10〈0, 27, 36, 41〉〈0, 11, 21, 39〉〈0, 3, 22, 26〉〈0, 1, 47, 53〉〈0, 5, 35, 37〉〈0, 17, 24, 25〉〈0, 4, 33, 16〉〈0, 6, 51, 54〉〈0, 18, 38, 49〉〈0, 2, 15, 42〉

11〈0, 20, 33, 44〉〈0, 1, 32, 41〉〈0, 10, 49, 53〉〈0, 30, 48, 51〉〈0, 3, 28, 65〉〈0, 7, 26, 36〉〈0, 8, 23, 35〉〈0, 9, 54, 61〉〈0, 4, 42, 50〉〈0, 12, 14, 34〉〈0, 11, 16, 17〉

13

〈0, 11, 54, 56〉〈0, 5, 49, 60〉〈0, 18, 35, 64〉〈0, 63, 66, 76〉〈0, 29, 65, 71〉〈0, 10, 24, 33〉〈0, 7, 15, 19〉〈0, 1, 22, 40〉〈0, 9, 57, 62〉〈0, 4, 34, 41〉〈0, 2, 27, 28〉〈0, 6, 38, 58〉〈0, 20, 51, 67〉

14

〈0, 9, 41, 53〉〈0, 1, 50, 83〉〈0, 45, 47, 63〉〈0, 46, 61, 70〉〈0, 4, 25, 59〉〈0, 17, 27, 28〉〈0, 8, 56, 75〉〈0, 16, 74, 80〉〈0, 19, 22, 62〉〈0, 6, 29, 36〉〈0, 7, 38, 42〉〈0, 12, 26, 72〉〈0, 34, 54, 71〉〈0, 52, 57, 65〉

17

〈0, 20, 62, 63〉〈0, 10, 58, 88〉〈0, 3, 59, 95〉〈0, 17, 24, 50〉〈0, 1, 76, 97〉〈0, 12, 46, 66〉〈0, 36, 64, 81〉〈0, 74, 85, 99〉〈0, 5, 40, 49〉〈0, 19, 87, 90〉〈0, 9, 47, 70〉〈0, 21, 53, 72〉〈0, 14, 69, 79〉〈0, 2, 6, 18〉〈0, 23, 31, 60〉〈0, 26, 39, 41〉〈0, 30, 52, 57〉

y². -X4 = Z5×Z5"@o(X4, C4)´¤I��`(25, 6, [2, 2])3è§XJC4´dL8.1¥�èi3Z5 × Z5þ(+1 (mod 5),+1 (mod 5))Ðm��"

ét 6= 4§-Xt = Z6t+1"@o(Xt, Ct)´¤I��`(6t + 1, 6, [2, 2])3è§X

JCtdL8.1¥�èi3Z6t+1þ+1 (mod 6t+ 1)Ðm��"

Ún 8.11. é?¿t ≥ 12§ t 6∈ {13, 14, 17}§A3(6t + 1, 6, [2, 2]) = U(6t +

1, 6, [2, 2])"

y². é?¿t ≥ 9§t 6∈ P§lÚn8.8��.�24i30j36k42l48m�GDC"O

\��Ã¡:§¿3|þëÓÃ¡:W\�A��`è��I��è"é

120 Aa|Ü?è¯KïÄ

?¿t ∈ {12, 15, 16, 18, 19, 21, 22, 23, 26, 27, 28, 31, 32, 33}§�Ún8.8¥�GDC"

O\��Ã¡:§¿3|þëÓÃ¡:W\�A��`è��I��è"

½n 8.12. é?¿t ≥ 3§A3(6t+1, 6, [2, 2]) = U(6t+1, 6, [2, 2])¶A3(7, 6, [2, 2]) =

3¶A3(13, 6, [2, 2]) ≥ 21"

c. ��ÝÝÝn ≡ 0 (mod 6)��

½n 8.13. é?¿t ≥ 1§A3(6t, 6, [2, 2]) = U(6t, 6, [2, 2])"

y². ét ∈ {1, 2}§-Xt = Z6t"@o¤I�èCt´dXeèi38ÜZ6tþ+2

(mod 6t)Ðm��"Ù¥C1�〈0, 1, 2, 3〉§C2�〈0, 2, 8, 5〉§〈0, 9, 7, 11〉§〈1, 5, 0, 10〉"é?¿t ≥ 3§l½n8.12¥��`(6t + 1, 6, [2, 2])3è�K��IÚ3ù��

Iþ�Ø�"�¤kèi��I��è"

d. ��ÝÝÝn ≡ 2 (mod 6)��

Ún 8.14. A3(8, 6, [2, 2]) = 5§A3(14, 6, [2, 2]) ≥ 27"

y². én = 8§d©[159]¥(8, 6, 4)3è�(J��A3(8, 6, [2, 2]) ≤ 5"·�38

Ü{0, 1, 2, 3, 4, 5, 6, 7}þ�E¤I�èiXeµ〈0, 1, 2, 3〉§〈0, 4, 5, 6〉§〈1, 5, 4, 7〉§〈2, 3, 6, 7〉§〈6, 7, 0, 1〉"én = 14§-:8�{0, 1, 2, . . . , 13}§¤I�èiXeµ

〈0, 2, 5, 9〉〈8, 13, 0, 2〉〈6, 10, 3, 4〉〈0, 7, 11, 12〉〈5, 9, 6, 10〉〈5, 7, 1, 13〉〈3, 5, 2, 11〉〈7, 9, 2, 4〉〈4, 11, 5, 7〉〈3, 7, 8, 10〉〈10, 13, 5, 12〉〈9, 11, 0, 3〉〈9, 12, 8, 13〉〈2, 11, 6, 8〉〈1, 8, 3, 5〉〈1, 10, 0, 7〉〈6, 13, 7, 9〉〈1, 13, 4, 11〉〈3, 4, 9, 12〉〈2, 12, 3, 7〉〈5, 12, 0, 4〉〈0, 4, 1, 8〉〈0, 3, 6, 13〉〈2, 4, 10, 13〉〈8, 10, 9, 11〉〈1, 6, 2, 12〉〈11, 12, 1, 10〉

éN´�±wÑXJ�3��.�23t+1§��2t(3t+ 1)�GDC§@o�

3��`(6t+ 2, 6, [2, 2])3è"¤±·�òÏL�E.�23t+1�GDC�E�`

è"

Ún 8.15. é?¿3 ≤ t ≤ 11§t ∈ {14, 17}§�3��.�23t+1§��2t(3t+

1)�GDC"


L 8.2: Ún8.15¥.�23t+1�GDC�Äèt èi

4 〈0, 4, 3, 11〉〈0, 5, 6, 15〉〈0, 9, 2, 23〉〈0, 8, 20, 24〉

5〈0, 28, 9, 29〉〈1, 7, 2, 12〉〈0, 15, 24, 7〉〈0, 10, 30, 12〉〈0, 18, 23, 21〉〈1, 14, 22, 9〉〈1, 15, 11, 5〉〈1, 3, 0, 26〉〈0, 26, 25, 11〉〈1, 21, 4, 8〉

6 〈0, 7, 20, 5〉〈0, 21, 11, 27〉〈0, 23, 14, 35〉〈0, 30, 24, 25〉〈0, 22, 18, 26〉〈0, 1, 3, 10〉

7〈0, 5, 21, 33〉〈0, 32, 34, 42〉〈0, 14, 23, 29〉〈0, 6, 7, 25〉〈0, 36, 40, 35〉〈0, 20, 17, 3〉〈0, 18, 11, 31〉

8〈0, 40, 29, 34〉〈0, 30, 2, 11〉〈0, 37, 3, 33〉〈0, 45, 12, 27〉〈0, 35, 21, 8〉〈0, 49, 18, 42〉〈0, 24, 28, 38〉〈0, 9, 7, 6〉

9〈0, 39, 33, 45〉〈0, 12, 48, 53〉〈0, 22, 35, 9〉〈0, 5, 3, 37〉〈0, 29, 31, 47〉〈0, 4, 25, 42〉〈0, 16, 23, 24〉〈0, 26, 19, 46〉〈0, 1, 11, 15〉

10〈0, 9, 38, 57〉〈0, 22, 30, 33〉〈0, 15, 39, 5〉〈0, 20, 12, 21〉〈0, 58, 23, 45〉〈0, 3, 2, 17〉〈0, 19, 51, 55〉〈0, 6, 13, 50〉〈0, 28, 46, 26〉〈0, 25, 35, 41〉

11〈0, 55, 33, 19〉〈0, 37, 65, 49〉〈0, 52, 8, 63〉〈0, 67, 60, 2〉〈0, 10, 4, 35〉〈0, 59, 66, 48〉〈0, 50, 22, 21〉〈0, 23, 38, 64〉〈0, 26, 5, 56〉〈0, 14, 20, 43〉〈0, 17, 44, 53〉

14

〈0, 1, 20, 31〉〈0, 7, 47, 68〉〈0, 11, 15, 78〉〈0, 14, 71, 80〉〈0, 23, 82, 83〉〈0, 5, 29, 41〉〈0, 2, 46, 56〉〈0, 9, 35, 42〉〈0, 12, 18, 70〉〈0, 17, 25, 39〉〈0, 34, 37, 50〉〈0, 13, 62, 64〉〈0, 10, 38, 55〉〈0, 21, 48, 53〉

17

〈0, 1, 81, 86〉〈0, 4, 37, 58〉〈0, 8, 65, 74〉〈0, 12, 59, 76〉〈0, 22, 71, 72〉〈0, 51, 78, 90〉〈0, 2, 40, 62〉〈0, 5, 11, 46〉〈0, 9, 32, 88〉〈0, 17, 30, 61〉〈0, 31, 45, 98〉〈0, 69, 89, 93〉〈0, 7, 75, 77〉〈0, 48, 84, 91〉〈0, 21, 55, 63〉〈0, 3, 19, 29〉〈0, 10, 25, 28〉

y². ét = 3§¤I�è3Ún8.5¥�E"é?¿4 ≤ t ≤ 11½t ∈ {14, 17}§-Xt = Z6t+2§Gt = {{i, i + 3t + 1} : 0 ≤ i ≤ 3t}"@o§ (Xt,Gt, Ct)Ò´.�23t+1§��2t(3t + 1)�GDC§XJC5´dL8.2¥�èi3Z32¥+2

(mod 32)Ðm��§�t 6= 5�§CtdL8.2¥èi3Z6t+2¥+1 (mod 6t + 2)Ð

m��"

Ún 8.16. é?¿t ≥ 12§t 6∈ {14, 17}§�3��.�23t+1§��2t(3t +

1)�GDC"

y². é?¿t ≥ 9§t 6∈ P§lÚn8.8��.�24i30j36k42l48m�GDC"O

\��Ã¡:§3|þëÓÃ¡:W\.�2u§u ∈ {13, 16, 19, 22, 25}�GDC�

�¤IGDC"é?¿t ∈ {12, 15, 16, 18, 19, 21, 22, 23, 26, 27, 28, 31, 32, 33}§�Ún8.8¥�GDC§O\��Ã¡:§3|þëÓÃ¡:W\.�2u§u ∈{10, 13, 19}�GDC��¤IGDC"ét = 13§��TD(4, 5)§^Ä��E{

122 Aa|Ü?è¯KïÄ

\�4��.�204�GDC§23|þW\.�210�GDC§Ò��¤IGDC"

½n 8.17. é?¿t ≥ 3§A3(6t+2, 6, [2, 2]) = U(6t+2, 6, [2, 2])¶A3(8, 6, [2, 2]) =

5¶A3(14, 6, [2, 2]) ≥ 27"

e. ��ÝÝÝn ≡ 5 (mod 6)��

Ún 8.18. A3(5, 6, [2, 2]) = 1§A3(11, 6, [2, 2]) = 15§A3(17, 6, [2, 2]) ≥ 40"

y². 1��ªéw,"d©[159]¥��`(11, 6, 4)3è�(J§·��

�A3(11, 6, [2, 2]) ≤ A3(11, 6, 4) = 15"��`(11, 6, [2, 2])33©[159]¥�E

��"

én = 40§-:8�Z16 ∪ {∞}"¤I�40�èidèi〈0, 3, 9, 5〉§〈0, 11,

12, 10〉§〈1, 6,∞, 12〉§〈1, 5, 8, 14〉§〈1, 2, 15, 9〉§〈0,∞, 13, 15〉§〈3, 13, 7, 12〉§〈3,10, 15, 6〉§〈2, 4, 5, 6〉§〈0, 6, 7, 4〉3Z16¥+4 (mod 16)Ðm��"

Ún 8.19. é?¿3 ≤ t ≤ 8§�3��.�23t51§��2t(3t+ 4)�GDC"

y². -Xt = Z6t+5§Gt = {{i, i+ 3t} : 0 ≤ i ≤ 3t− 1}∪ {{6t, 6t+ 1, 6t+ 2, 6t+

3, 6t+ 4}}"@o(Xt,Gt, Ct)´��.�23t51�GDC§XJC3´dL8.3¥�èi

dgÓ�+G = 〈(0 3 6 · · · 15)(1 4 7 · · · 16)(2 5 8 · · · 17)(18 19 20) (21 22)〉Ðm��¶�t > 3§Ct´dL8.3¥�èidgÓ�+G = 〈(0 3 6 · · · 6t −3)(1 4 7 · · · 6t− 2)(2 5 8 · · · 6t− 1)(6t)(6t+ 1)(6t+ 2)(6t+ 3)(6t+ 4)〉Ðm��"

Ún 8.20. é?¿t ≥ 12§t 6∈ {13, 14, 17}§�3��.�23t51§��2t(3t+

4)�GDC"

y². é?¿t ≥ 9§t 6∈ P§lÚn8.7¥��.�24i30j36k42l48m�GDC§

Ù¥4i + 5j + 6k + 7l + 8m = t"O\��Ã¡:§3|þëÓÃ¡:W\.

�23s51§s ∈ {4, 5, 6, 7, 8}�GDC§��I��GDC"é?¿t ∈ {12, 15, 16, 18,

19, 21, 22, 23, 26, 27, 28, 31, 32, 33}§�Ún8.8 ¥�GDC"O\��Ã¡:§3

|þëÓÃ¡:W\.�23s51§s ∈ {3, 4, 5, 6, 7}�GDC��¤IGDC"


L 8.3: Ún8.19¥.�23t51�GDC�Äèt èi

3〈1, 5, 3, 22〉〈0, 21, 7, 11〉〈0, 22, 1, 17〉〈1, 20, 2, 15〉〈0, 12, 15, 19〉〈2, 8, 5, 19〉〈0, 20, 4, 14〉〈2, 18, 1, 6〉〈1, 11, 0, 12〉〈2, 4, 12, 22〉〈0, 5, 10, 16〉〈1, 13, 16, 19〉〈0, 13, 2, 8〉

4

〈2, 25, 18, 22〉〈1, 23, 20, 2〉〈0, 22, 25, 5〉〈0, 19, 23, 26〉〈2, 0, 10, 28〉〈2, 27, 16, 21〉〈2, 26, 15, 7〉〈1, 9, 17, 27〉〈2, 8, 19, 17〉〈1, 19, 10, 12〉〈0, 4, 1, 7〉〈0, 18, 11, 9〉〈1, 28, 0, 14〉〈2, 6, 9, 3〉〈0, 14, 13, 24〉〈1, 24, 11, 15〉

5

〈2, 18, 21, 9〉〈0, 19, 2, 33〉〈2, 26, 0, 4〉〈1, 6, 31, 11〉〈1, 15, 34, 8〉〈2, 32, 19, 24〉〈2, 33, 25, 27〉〈0, 13, 11, 32〉〈1, 27, 24, 4〉〈0, 10, 26, 30〉〈0, 1, 9, 22〉〈1, 19, 23, 28〉〈0, 12, 20, 6〉〈2, 20, 29, 23〉〈1, 3, 2, 20〉〈2, 30, 28, 12〉〈1, 26, 7, 25〉〈2, 31, 15, 16〉〈2, 34, 22, 3〉

6

〈2, 23, 4, 28〉〈0, 28, 23, 37〉〈2, 26, 22, 0〉〈2, 37, 24, 13〉〈1, 0, 4, 30〉〈0, 22, 36, 20〉〈2, 38, 15, 31〉〈0, 31, 27, 2〉〈0, 11, 15, 12〉〈1, 21, 39, 26〉〈1, 10, 2, 23〉〈2, 36, 33, 25〉〈2, 40, 16, 21〉〈2, 8, 35, 11〉〈2, 39, 9, 10〉〈0, 8, 24, 7〉〈1, 25, 31, 22〉〈0, 3, 17, 9〉〈1, 24, 20, 38〉〈0, 19, 34, 21〉〈1, 12, 11, 5〉〈0, 10, 26, 40〉

7

〈2, 15, 38, 0〉〈0, 24, 28, 22〉〈2, 29, 35, 19〉〈2, 36, 9, 41〉〈2, 33, 11, 26〉〈1, 16, 13, 4〉〈2, 18, 31, 43〉〈1, 29, 5, 30〉〈2, 30, 42, 22〉〈1, 7, 20, 8〉〈2, 12, 45, 1〉〈0, 30, 7, 25〉〈1, 45, 0, 26〉〈1, 42, 6, 35〉〈1, 43, 11, 3〉〈2, 27, 46, 37〉〈0, 44, 16, 2〉〈1, 17, 39, 44〉〈2, 4, 21, 39〉〈0, 38, 1, 41〉〈2, 32, 13, 40〉〈0, 36, 33, 3〉〈1, 10, 21, 32〉〈1, 46, 38, 24〉〈1, 25, 9, 15〉

8

〈0, 45, 18, 30〉〈2, 50, 42, 25〉〈2, 52, 40, 27〉〈1, 3, 45, 7〉〈0, 1, 15, 16〉〈1, 4, 34, 22〉〈0, 19, 47, 41〉〈2, 32, 34, 12〉〈0, 43, 32, 27〉〈2, 29, 36, 28〉〈0, 22, 48, 17〉〈0, 7, 34, 20〉〈0, 5, 9, 6〉〈1, 9, 51, 32〉〈2, 48, 0, 43〉〈0, 36, 44, 14〉〈2, 41, 22, 15〉〈1, 21, 10, 12〉〈16, 39, 26, 50〉〈16, 4, 23, 20〉〈16, 2, 10, 7〉〈2, 49, 18, 37〉〈0, 13, 38, 49〉〈3, 13, 5, 14〉〈2, 8, 19, 39〉〈2, 5, 38, 17〉〈2, 51, 21, 46〉〈0, 31, 29, 52〉

Ún 8.21. é?¿t ≥ 3§t 6∈ {9, 10, 11, 14}§A3(6t+5, 6, [2, 2]) ≥ U(6t+5, 3)−1¶

ét = 14§A3(6t+ 5, 6, [2, 2]) ≥ U(6t+ 5, 3)− 2"

y². é?¿t 6∈ {9, 10, 11, 13, 14, 17}§lÚn8.19Ú8.20�.�23t51§t ≥ 3§

t 6∈ {9, 10, 11, 13, 14, 17}�GDC"3��5�|þW\�k��èi��

`(5, 5, [2, 2])3��I��è"é?¿t ∈ {13, 14, 17}§lÚn8.35©O�.

�18461§184121½24461�GDC"O\5�Ã¡:§3��18½24�|þëÓ

Ã¡:W\.�2951½21251�GDC§¿3��|þëÓÃ¡:W\Ú

n8.18¥�Ý�11½17�èÒ��¤I�è"

½n 8.22. é?¿t ≥ 3§t 6∈ {9, 10, 11, 14}§A3(6t + 5, 6, [2, 2]) ≥ U(6t +

5, 6, [2, 2])−1¶A3(5, 6, [2, 2]) = 1¶A3(11, 6, [2, 2]) = 15¶ét ∈ {2, 14}§A3(6t+

5, 6, [2, 2]) ≥ U(6t+ 5, 6, [2, 2])− 2"

124 Aa|Ü?è¯KïÄ

L 8.4: Ún8.23¥��`(6t+ 4, 6, [2, 2])3è�Äèt èi

1 〈0, 3, 9, 7〉〈0, 4, 2, 5〉〈1, 3, 2, 6〉

4〈0, 15, 21, 24〉〈0, 6, 9, 11〉〈1, 22, 26, 21〉〈1, 3, 11, 15〉〈0, 1, 18, 19〉〈1, 6, 23, 16〉〈1, 4, 17, 2〉〈0, 12, 14, 20〉〈1, 5, 24, 12〉

5〈0, 11, 17, 21〉〈1, 19, 10, 18〉〈1, 9, 12, 23〉〈1, 33, 30, 20〉〈0, 7, 14, 20〉〈0, 24, 23, 15〉〈0, 8, 1, 3〉〈0, 19, 31, 13〉〈1, 31, 2, 21〉〈0, 28, 12, 16〉〈0, 4, 2, 9〉

6〈0, 19, 10, 12〉〈1, 2, 13, 31〉〈1, 26, 0, 3〉〈0, 25, 5, 8〉〈0, 4, 32, 13〉〈0, 6, 22, 7〉〈0, 3, 27, 31〉〈1, 19, 11, 17〉〈1, 5, 18, 12〉〈1, 20, 6, 10〉〈0, 2, 20, 35〉〈1, 4, 27, 28〉〈1, 7, 15, 36〉

7

〈1, 35, 40, 24〉〈1, 31, 0, 29〉〈1, 9, 10, 12〉〈0, 16, 43, 21〉〈0, 22, 17, 7〉〈0, 32, 9, 25〉〈1, 43, 26, 38〉〈1, 18, 7, 21〉〈1, 23, 3, 41〉〈0, 34, 26, 8〉〈0, 18, 37, 33〉〈0, 40, 4, 36〉〈1, 33, 14, 20〉〈0, 2, 13, 1〉〈1, 11, 8, 32〉

8

〈0, 50, 3, 11〉〈0, 49, 38, 21〉〈1, 33, 37, 43〉〈1, 18, 40, 38〉〈1, 9, 30, 2〉〈1, 12, 13, 27〉〈0, 17, 12, 39〉〈1, 39, 3, 14〉〈1, 20, 51, 19〉〈1, 34, 41, 44〉〈1, 7, 35, 16〉〈0, 45, 42, 23〉〈1, 24, 32, 49〉〈0, 4, 18, 47〉〈0, 6, 36, 40〉〈0, 24, 16, 9〉〈0, 27, 26, 32〉

9

〈1, 31, 49, 55〉〈0, 1, 17, 39〉〈1, 14, 41, 46〉〈1, 18, 36, 11〉〈0, 31, 52, 15〉〈1, 6, 35, 4〉〈0, 5, 37, 9〉〈0, 10, 13, 22〉〈1, 26, 51, 52〉〈1, 2, 40, 45〉〈1, 10, 24, 16〉〈1, 23, 21, 34〉〈1, 48, 20, 30〉〈0, 8, 54, 55〉〈1, 47, 15, 32〉〈1, 7, 56, 9〉〈0, 34, 20, 36〉〈0, 21, 4, 28〉〈0, 16, 23, 35〉

f. ��ÝÝÝn ≡ 4 (mod 6)��

Ún 8.23. é?¿t = 1½4 ≤ t ≤ 9§A3(6t+ 4, 6, [2, 2]) = U(6t+ 4, 6, [2, 2])"

y². -Xt = Z6t+4"@o(Xt, Ct)´¤I��`(6t + 4, 6, [2, 2])3§XJCt´dL8.4¥�èi3Z6t+4¥+2 (mod 6t+ 4)Ðm��"

Ún 8.24. é?¿t ≥ 142§A3(6t+ 4, 6, [2, 2]) = U(6t+ 4, 6, [2, 2])"

y². lÚn2.9��TD(8, k)"^Ä��E{éc6�|�¤k:§17�

|�x�:\�6§17�|�3�:\�3§Ù{:\�0"ùp§Ñ\�O

�.�6s§s ∈ {6, 7}�GDC£Ún8.6¤§Ú.�6s31§s ∈ {6, 7}�GDC£Ú

n8.33¤"·��.�(6k)6(6x)191�GDC§Ù¥x = 0 ½3 ≤ x ≤ k"O\

��Ã¡:§3��6k½6x�|þëÓÃ¡:W\�Ý�6k + 1½6x + 1�

�`è£½n8.12¤§3��9�|ëÓÃ¡:W\�`(10, 6, [2, 2])3è"·�

Ò��Ý�n = 36k + 6x+ 10 = 6(6k + x+ 1) + 4��`è"�k ≥ 23§·

��Ý�n = 6t + 4��`è§Ù¥t = 6k + x + 1�±�?Ût ≥ 142�

��ê"


Ún 8.25. é?¿t = 43§½46 ≤ t ≤ 141§t 6= 51§A3(6t + 4, 6, [2, 2]) =

U(6t+ 4, 6, [2, 2])"

y². lÚn2.9¥��TD(m, k)§k ∈ {7, 8, 9, 13}§8 ≤ m ≤ 12§m ≤ k+1"

^Ä��E{éc6�|�¤k:\�6§��|�3�:\�3§é�

{�m − 7�|�xi�:§1 ≤ i ≤ m − 7\�6"Ù{:\�0"Ù¥xi = 0½

ö3 ≤ xi ≤ k"ùpÑ\è�.�6s§6 ≤ s ≤ 11�GDC£Ún8.6¤Ú.�6s31§

6 ≤ s ≤ 11�GDC£Ún8.33¤"·�Ò��.�(6k)6x11x12 . . . x

1m−79

1�GDC"

O\��:§3cm− 1�|þëÓÃ¡:W\�Ý�6k + 1½6xi + 1��`è

£½n8.12¤§3�Ý�9�|ëÓÃ¡:W\�`(10, 6, [2, 2])3è§Ò��

Ý�n = 36k + 6∑m−7

i=1 xi + 10 = 6(6k +∑m−7

i=1 xi + 1) + 4��`è"

-t = 6k +∑m−7

i=1 xi + 1"�k = 7§m = 8§·�kt ∈ {43} ∪ [46, 50]¶

�k = 8§m = 9§·�kt ∈ [52, 65]¶�k = 9§m = 10§·�kt ∈ [66, 82]¶

�k = 13§m = 12§·�kt ∈ [83, 141]"

Ún 8.26. é?¿t ∈ {24} ∪ [32, 34] ∪ [36, 44]§A3(6t + 4, 6, [2, 2]) = U(6t +

4, 6, [2, 2])"

y². lÚn2.9��TD(5, k)§k ∈ {5, 7, 8, 9}"^Ä��E{éc4�|�¤

k:§��|�x�:\�6§é��|�y�:\�3§Ù¥x+y = k"

ùp§Ñ\�´.�65�GDC£Ún8.6¤Ú.�6431�GDC£Ún8.33¤"·

��.�(6k)4(6x + 3y)1�GDC"O\��Ã¡:§3co�|þëÓ

Ã¡:W\�Ý�6k + 1��`è£½n8.12¤§3��6x + 3y�|þë

ÓÃ¡:W\�`(6x + 3y + 1, 6, [2, 2])3è£Ún8.23¤§Ò��Ý�n =

24k + 6x + 3y + 1 = 6(4k + x + y−12

) + 4��`è"-t = 4k + x + y−12"é?

¿t§ëê(k, x, y)ÚI�W\�è�s = 6x+ 3y + 13L8.5 ¥�Ñ"

L 8.5: Ún8.26¥¤Iëêt (k, x, y) s t (k, x, y) s t (k, x, y) s

24 (5, 4, 1) 28 32 (7, 2, 5) 28 33 (7, 4, 3) 34

34 (7, 6, 1) 40 36 (8, 1, 7) 28 37 (8, 3, 5) 34

38 (8, 5, 3) 40 39 (8, 7, 1) 46 40 (9, 0, 9) 28

41 (9, 2, 7) 34 42 (9, 4, 5) 40 43 (9, 6, 3) 46

44 (9, 8, 1) 52

126 Aa|Ü?è¯KïÄ

Ún 8.27. é?¿t ∈ {10, 11, 13, 16, 17, 18, 19, 21, 22, 25, 26, 28, 31, 45, 51}§A3(6t

+ 4, 6, [2, 2]) = U(6t+ 4, 6, [2, 2])"

y². ét = 11§lÚn8.32�.�107�GDC§3|þW\�`(10, 6, [2, 2])3è

��¤I�è"ét = 17§lÚn8.35��.�24491�GDC§O\��Ã¡

:§3|þëÓÃ¡:W\�`(25, 6, [2, 2])3è½�`(10, 6, [2, 2])3è��¤I

�è"é?¿t ∈ {10, 13, 16, 18, 19, 21, 22, 25, 26, 28, 31, 45, 51}§·�r��t¤I�ëê�3L8.6¥"·�r��.�gum1�GDC^w)ä§O\a�:§,

�3|þëÓÃ¡:W\�Ý�s ∈ S�è��¤I�è"

L 8.6: Ún8.27¥¤Iëêt n gum1 × w 5 a S

10 64 37 × 3 Ún8.32 1 10

13 82 6431 × 3 Ún8.33 1 19, 10

16 100 311 × 3 Ún8.32 1 10

18 112 44 × 7 Ún8.32 0 28

19 118 313 × 3 Ún8.32 1 10

21 130 213 × 5 Ún8.15 0 10

22 136 6731 × 3 Ún8.33 1 19, 10

25 154 6791 × 3 Ún8.34 1 19, 28

26 160 44 × 10 Ún8.32 0 40

28 172 6891 × 3 Ún8.34 1 19, 28

31 190 37 × 9 Ún8.32 1 28

45 274 37 × 13 Ún8.32 1 40

51 310 131 × 10 Ún8.10 0 10

Ún 8.28. ét ∈ {29, 35}§A3(6t+ 4, 6, [2, 2]) = U(6t+ 4, 6, [2, 2])"

y². ét = 29§��.�67�GDC£Ún8.6¤§¿^4)ä"ùp�Ñ\

�O´.�44�{4}-MGDD"·�Ò��GDC§�| 8/¤�ê

�2�.�(24, 64)7�DGDD"O\9�Ã¡:§W\.�6791�GDC��.

�24791�GDC"O\��Ã¡:§3|þëÓÃ¡:W\�Ý�25½10��

`è��I��è"ét = 35§�.�67�GDC§¿�K��|�¤k

:Ú�¹ù�:�èi§¿^5)ä"ùpÑ\�´.�54�{4}-MGDDÚ.

�53��©){3}-MGDD"��GDC�| 8�¤��{3, 4}-DGDD§¿�

¤k�n�|/¤24�²1a"O\24�:§Ö�²1a"2O\9�Ã¡:§


¿W\.�6691�GDCÒ��.�306331�GDC"2O\��Ã¡:§¿3

|þëÓÃ¡:W\�Ý�31½34��`èÒ��¤I��`è"

nÜþãÚn§·��µ

½n 8.29. é?¿t ≥ 1§t 6∈ {2, 3, 12, 14, 15, 20, 23, 27, 30}§A3(6t+4, 6, [2, 2]) =

U(6t+ 4, 6, [2, 2])"

g. ��ÝÝÝn ≡ 3 (mod 6)��

½n 8.30. é?¿t ≥ 1§A3(6t+ 3, 6, [2, 2]) = U(6t+ 3, 3)"

y². é?¿t ≥ 1§t 6∈ {2, 3, 12, 14, 15, 20, 23, 27, 30}§l½n8.29¥��`(6t+

4, 6, [2, 2])3è�K��IÚ3ù��IØ�"�èi��A�è"ét = 3§

w,Ún8.32¥�.�37�GDCÒ´¤I�è"ét ∈ {12, 15, 27, 30}§©O�.�184£Ún8.32¤§185§189£½n8.4¤½1810£é��.�110�GDC§Ú

n8.23§^18)ä��¤�GDC"O\3�Ã¡:§3|þëÓÃ¡:W\.

�37�GDC§Ò©O��.�325§331§355½361�GDC"éN´�y§�Ò

´¤I�è"ét = 23§lÚn2.9��TD(5, 5)"^Ä��E{éc4�|�

¤k:§��|�2�:\�6§��|�3�:\�3§Ò��.

�304211�GDC"3|þW\�Ý�21½30��`èÒ��¤I�è"

ét ∈ {2, 14, 20}§-Xt = Z6t+3"@o(Xt, Ct)Ò´�`(6t + 3, 6, [2, 2])3è§

XJCt´de¡èi3Z6t+3¥+1 (mod 6t+ 3)Ðm��"

t = 2µ〈0, 3, 4, 14〉〈0, 5, 7, 13〉

t = 14µ

〈0, 1, 68, 8〉〈0, 24, 47, 61〉〈0, 41, 16, 36〉〈0, 73, 57, 40〉〈0, 14, 49, 5〉〈0, 29, 40, 38〉〈0, 58, 68, 27〉〈0, 20, 54, 84〉〈0, 60, 62, 9〉〈0, 57, 51, 63〉〈0, 22, 25, 53〉〈0, 43, 33, 32〉〈0, 60, 6, 45〉〈0, 28, 71, 60〉

t = 20µ

〈0, 16, 46, 63〉〈0, 92, 85, 7〉〈0, 34, 122, 9〉〈0, 10, 43, 74〉〈0, 5, 109, 82〉〈0, 6, 21, 79〉〈0, 24, 22, 94〉〈0, 23, 37, 55〉〈0, 13, 42, 62〉〈0, 17, 2, 8〉〈0, 56, 81, 68〉〈0, 20, 95, 39〉〈0, 45, 93, 97〉〈0, 4, 57, 80〉〈0, 18, 54, 59〉〈0, 11, 102, 3〉〈0, 72, 35, 69〉〈0, 58, 1, 84〉〈0, 27, 71, 87〉〈0, 83, 50, 61〉

128 Aa|Ü?è¯KïÄ

8.4 (((ØØØ

3�Ù¥§·�ïÄ�`(n, 6, [2, 2])3è�èi�ê"·�r(Jo(X

eµ

½n 8.31. é?¿�ên ≥ 4§

A3(n, 6, [2, 2]) =

1§ �n ≤ 5�

3§ �n = 7�

5§ �n = 8�

15§ �n = 11�⌊n2

⌊n−13

⌋⌋§ �n ≥ 6§n 6≡ 5 (mod 6)§n 6∈ {7, 8, 16, 22,

76, 88, 94, 124, 142, 166, 184}�

A3(13, 6, 4) ∈ [21, 26]§A3(14, 6, 4) ∈ [27, 28]"

�n ≡ 5 (mod 6)�§é?¿t ≥ 3§t 6∈ {9, 10, 11, 14}§

A3(6t+ 5, 6, [2, 2]) ∈ [2(3t+ 1)(t+ 1)− 1, 2(3t+ 1)(t+ 1)]§

ét ∈ {2, 14}§

A3(6t+ 5, 6, [2, 2]) ∈ [2(3t+ 1)(t+ 1)− 2, 2(3t+ 1)(t+ 1)]"

8.5 NNN¹¹¹

Ún 8.32. ©O�3.�37§311§313§44§107Ú184�GDC"

y². é?¿.�gu�GDC§-X = Zgu§G = {{i, i+u, i+2u, . . . , i+(g−1)u} :

0 ≤ i ≤ u− 1}"@o§(X,G, C)Ò´¤I�GDC§XJC´de¡èi3Zgu¥Ðm��"

37µ+1 (mod 21) 〈7, 3, 18, 13〉〈4, 3, 12, 16〉〈0, 5, 2, 3〉

311µ+1 (mod 33) 〈13, 27, 15, 25〉〈25, 15, 0, 7〉〈2, 15, 18, 32〉〈10, 5, 1, 14〉〈0, 7, 1, 6〉

313µ+1 (mod 39) 〈20, 2, 25, 10〉〈21, 6, 23, 17〉〈6, 12, 5, 26〉〈30, 3, 28, 19〉〈21, 24, 31, 4〉〈0, 9, 1, 4〉

44µ+8 (mod 16)

〈1, 7, 2, 4〉〈3, 13, 0, 6〉〈6, 12, 5, 7〉〈2, 4, 5, 11〉〈2, 8, 3, 9〉〈3, 5, 2, 12〉〈0, 14, 3, 5〉〈4, 6, 1, 3〉〈8, 14, 1, 7〉〈15, 5, 4, 6〉〈4, 10, 7, 9〉〈9, 7, 8, 6〉〈3, 9, 4, 14〉〈9, 11, 0, 2〉〈8, 10, 5, 15〉〈13, 15, 2, 8〉


107µ+1 (mod 70)

〈0, 17, 27, 58〉〈0, 61, 24, 46〉〈0, 25, 6, 68〉〈0, 3, 2, 40〉〈0, 32, 16, 52〉〈0, 34, 47, 64〉〈0, 11, 26, 29〉〈0, 62, 57, 66〉〈0, 48, 1, 60〉〈0, 31, 50, 5〉

184µ+2 (mod 72)

〈1, 3, 8, 22〉〈0, 34, 1, 15〉〈0, 26, 11, 13〉〈1, 35, 32, 58〉〈1, 7, 16, 54〉〈1, 15, 50, 56〉〈0, 2, 21, 71〉〈0, 7, 18, 25〉〈0, 30, 23, 61〉〈1, 11, 62, 40〉〈1, 31, 34, 44〉〈0, 14, 43, 17〉〈0, 9, 54, 63〉〈0, 10, 5, 47〉〈0, 22, 49, 55〉〈1, 27, 26, 28〉〈0, 6, 51, 41〉〈1, 23, 60, 18〉

Ún 8.33. é?¿4 ≤ t ≤ 11§t 6= 5§�3��.�6t31§��3t2�GDC"

y². -Xt = Z6t+3§Gt = {{i, i + t, i + 2t, . . . , i + 5t} : 0 ≤ i ≤ t − 1} ∪{{6t, 6t + 1, 6t + 2}}"@o§(Xt,Gt, Ct)´��.�6t31�GDC§XJC4´de¡èi3Z27¥dgÓ�+G = 〈(0 2 4 · · · 22)(1 3 5 · · · 23)(24 25 26)〉Ðm��§ 6 ≤ t ≤ 11�§Ct´de¡èi3Z6t+3¥dgÓ�+G = 〈(0 1 2 · · · 6t−1)(6t 6t+ 1 6t+ 2)〉Ðm��"t = 4µ〈1, 25, 8, 6〉〈1, 3, 18, 0〉〈0, 10, 24, 23〉〈1, 7, 10, 20〉〈0, 22, 7, 1〉〈0, 6, 21, 11〉〈1, 11, 12, 26〉〈8, 24, 3, 1〉

t = 6µ〈0, 34, 27, 5〉〈0, 28, 3, 13〉〈0, 10, 19, 35〉〈0, 20, 17, 15〉〈0, 37, 32, 4〉〈0, 14, 1, 38〉

t = 7µ〈0, 8, 12, 38〉〈0, 26, 25, 44〉〈0, 43, 20, 22〉〈0, 6, 23, 33〉〈0, 40, 1, 9〉〈0, 10, 15, 39〉〈0, 18, 31, 37〉

t = 8µ

〈0, 10, 12, 46〉〈0, 4, 21, 31〉〈0, 26, 41, 45〉〈0, 28, 5, 23〉〈0, 48, 11, 37〉〈0, 42, 1, 29〉〈0, 14, 13, 49〉〈0, 18, 3, 9〉

t = 9µ

〈0, 52, 5, 21〉〈0, 54, 49, 53〉〈0, 38, 3, 51〉〈0, 34, 12, 22〉〈0, 4, 28, 30〉〈0, 6, 17, 37〉〈0, 14, 1, 47〉〈0, 44, 15, 29〉〈0, 8, 43, 55〉

t = 10µ

〈0, 12, 36, 37〉〈0, 54, 45, 57〉〈0, 8, 29, 35〉〈0, 4, 9, 17〉〈0, 34, 23, 61〉〈0, 22, 53, 55〉〈0, 28, 7, 11〉〈0, 58, 16, 42〉〈0, 62, 19, 41〉〈0, 1, 15, 47〉

t = 11µ

〈0, 10, 1, 13〉〈0, 67, 7, 17〉〈0, 12, 30, 38〉〈0, 64, 25, 46〉〈0, 4, 23, 39〉〈0, 34, 5, 65〉〈0, 16, 36, 40〉〈0, 21, 49, 63〉〈0, 6, 15, 47〉〈0, 8, 59, 61〉〈0, 14, 43, 68〉

Ún 8.34. é?¿t ∈ {6, 7, 8}§�3��.�6t91§��6t(t+ 2)�GDC"

y². -Xt = Z6t+9§Gt = {{i, i+ t, i+2t, . . . , i+5t} : 0 ≤ i ≤ t−1}∪{{6t, 6t+1, 6t+ 2, . . . , 6t+ 8}}"@o§(Xt,Gt, Ct)´.�6t91�GDC§XJCt´de¡èi3Z6t+9¥dgÓ�+G = 〈(0 1 2 · · · 6t−1)(6t 6t+1 6t+2 6t+3 6t+4 6t+5)

(6t+ 6 6t+ 7 6t+ 8)〉Ðm��"

130 Aa|Ü?è¯KïÄ

t = 6µ

〈0, 22, 13, 29〉〈0, 34, 21, 38〉〈0, 16, 19, 37〉〈0, 36, 5, 15〉〈0, 43, 17, 31〉〈0, 41, 33, 25〉〈0, 8, 4, 44〉〈0, 10, 9, 11〉

t = 7µ

〈0, 36, 34, 10〉〈0, 45, 8, 39〉〈0, 23, 25, 49〉〈0, 44, 3, 22〉〈0, 29, 33, 42〉〈0, 48, 38, 1〉〈0, 24, 41, 12〉〈0, 11, 20, 26〉〈0, 5, 46, 32〉

t = 8µ

〈0, 10, 3, 15〉〈0, 14, 54, 37〉〈0, 49, 27, 29〉〈0, 36, 31, 1〉〈0, 46, 42, 4〉〈0, 22, 51, 33〉〈0, 56, 47, 19〉〈0, 18, 9, 35〉〈0, 48, 7, 21〉〈0, 20, 52, 45〉

Ún 8.35. ©O�3.�18461§184121§24461Ú24491�GDC"

y². é?¿.�gum1�GDC§-X = Zgu+m§G = {{0, u, 2u, . . . , (g − 1)u}+

i : 0 ≤ i ≤ u − 1} ∪ {{gu, gu + 1, gu + 2, . . . , gu + m − 1}}"@o(X,G, C)´I��GDC§XJC´de¡èi3Zgu+m¥dgÓ�+GÐm��"18461µG = 〈(0 1 2 · · · 71)(72 73 74 75 76 77)〉"

〈0, 30, 1, 7〉〈0, 50, 53, 76〉〈0, 54, 21, 59〉〈0, 72, 29, 51〉〈0, 26, 9, 19〉〈0, 10, 25, 67〉〈0, 2, 13, 71〉〈0, 66, 17, 35〉〈0, 58, 31, 33〉〈0, 77, 37, 63〉〈0, 34, 61, 73〉

184121µG = 〈(0 1 2 · · · 71)(72 73 74 75 76 77) (78 79 80 81 82 83)〉"

〈0, 46, 7, 72〉〈0, 77, 57, 71〉〈0, 34, 63, 79〉〈0, 22, 49, 78〉〈0, 6, 19, 41〉〈0, 76, 15, 25〉〈0, 14, 1, 31〉〈0, 18, 23, 61〉〈0, 42, 39, 45〉〈0, 82, 21, 67〉〈0, 83, 9, 11〉〈0, 10, 47, 65〉〈0, 70, 51, 73〉

24461µG = 〈(0 1 2 · · · 95)(96 97 98 99 100 101)〉"

〈0, 86, 25, 75〉〈0, 62, 5, 19〉〈0, 42, 23, 29〉〈0, 82, 81, 97〉〈0, 46, 41, 79〉〈0, 6, 7, 61〉〈0, 38, 11, 89〉〈0, 26, 3, 98〉〈0, 18, 27, 49〉〈0, 74, 37, 71〉〈0, 30, 21, 47〉〈0, 2, 15, 45〉〈0, 101, 63, 65〉〈0, 100, 57, 67〉

24491µG = 〈(0 1 2 · · · 95)(96 97 98)(99 100 101) (103 104 102)〉"

〈0, 82, 15, 73〉〈0, 38, 3, 49〉〈0, 46, 47, 98〉〈0, 102, 35, 37〉〈0, 99, 53, 79〉〈0, 54, 41, 59〉〈0, 18, 45, 31〉〈0, 6, 39, 69〉〈0, 30, 51, 9〉〈0, 22, 17, 104〉〈0, 2, 57, 67〉〈0, 34, 23, 101〉〈0, 96, 43, 77〉〈0, 10, 81, 7〉〈0, 26, 19, 25〉

Chapter 9

^��©©©|||CCCXXX��EEE��`õõõ��~~~CCCXXXèèè


-X´��k�8"|X| = n"¡ZXq ¥��i"Zqþ�(n,w, t, d)~

CXè´ZXq ¥��þ�w�f8§¦�z�þ�t�iu ∈ ZXq ��èi�ål�d"Pù�è��U�èi�ê�Kq(n,w, t, d)§��ù�

èi�ê�è¡�´�`�"

ïÄCXè��Ñu:´du§3êâØ �{¥�A^§�[39, 80]"

�Ä��?èì§§�¹�~CXè§z�è�þÑØÓ"?¿�

�Ñ\�þx�?èìØ ¤��:é(i, j)§Ù¥iL«�âx�þÀJ�è§

jL«ÀJ�è¥�x��C�èi"ØùA^§Kq(n,w, t, d)�(½�

��´|ÜnØ¥��Ä�¯K§3L��8�c¥kéõêÆ[ïÄ"3

|Ü�OnØ¥§kéõaq�|Ü(�[41]§~XTuran�O§ç¦�YÚ

CX�O�"�w − t ≥ 0�§��Zqþ�(n,w, t, w − t)~CXè�du��©|CX�O"�q = 2 �§kéõ©ÙïÄKq(n,w, t, d)�þ.Úe.§

X[20, 57]", §��(½��kAaAÏ��§~XµK2(n, 2, t, t−2) [139]§

K2(n, 3, 2, 1) [58]§K2(n, 4, 3, 1) [88]§K2(n,w, 2, w − 2) [90] �"�q ≥ 3�§

©[89]¥�k�(J"

3�!¥§·�òéq = 3§4½q = 2m + 1§m ≥ 2§Ø(q, n) = (3, 5)§

��(½Kq(n, 4, 3, 1)��"¯K�)û�6u^|Üóä�E��d�|

Ü¯K§=�©|CX§Ú�«�'�9Ï�OH-frame§§é�E�©|3-�

OåX�Ç�.�O3�E3²ï�O�aq��^"

ù�Ù�(�Xeµ319.2!¥§·�ò0��Ä�VgÚÄ��E�

{¶319.3!¥§·�ò�E�`n�~CXè¶319.4!¥§·�ò�

E�`o�~CXè¶319.5!¥§·�òé?¿q = 2m + 1£m ≥ 2¤§�

E�`q�~CXè¶319.6!¥§òé�Ù�Ì�(J?1o("

132 Aa|Ü?è¯KïÄ

9.2 OOO��£££ÚÚÚ��EEE��{{{

-vÚt��ê§K´��ê8"��ê�v§«|��K��

©|tCX£group divisible covering§GDC¤§P�GDC(t,K, v)§´��n�

|(X,G,B)§Ù¥µ

1. X´��v��8Ü£¡�:¤¶

2. G = {G1, G2, . . . }´X��y©£¡�|¤¶

3. B´G��|î�£¡�«|¤�8Ü§¦�?¿«|��K¥��§Ù¥z��î�´X��f8��z�|�õ�u��:¶

4. G¥�?¿t��î��¹3��«|¥"

��t-GDC��{´G�t�î�T�¤�õ8§Ù¥ê�|{B ∈ B :

T ⊂ B}| − 1"GDC(t,K, v)�.½Â�{|G||G ∈ G}"XJ��GDCkni��

��gi�|§1 ≤ i ≤ r§@o·�^�êÎÒgn11 g

n22 · · · gnr

r 5L«|�."�

�t-GDC�¡´��XJ¤k�|��Ñ�Ó"�K = {k}�§·�rK{P�k"

��t-GDC�¡��©|t-�O£t-GDD¤§P�GDD(t, k, v)§XJG¥�?¿t�î�TÑTÐ�¹3��«|¥"Mills3©[113]¥^H(n, g, 4, 3)�

O5L«.�gn�GDD(3, 4, ng)"��.�1n�GDD(t,K, n)�¡��ê

�n§|��K�t²ï�O§P�S(t,K, n)"�t = 3§K = {4}�§�¡�Steinero�X§P�SQS(n)"Hanani3©[84]¥�Ñµ��SQS(n)�3��

=�n ≡ 2, 4 (mod 6)"

½n 9.1 (Mills [113]§Ji [96]). é?¿n > 3§n 6= 5§��.�gn�GDD(3, 4, gn)

�3��=�ng´óê�g(n− 1)(n− 2) �±�3 �Ø"én = 5§�g´óê§

g 6= 2�g 6≡ 10, 26 (mod 48)�§�3��.�g5�GDD(3, 4, 5g)"

-C(n, g, k, t)L«?¿.�gn�GDC(t, k, ng)��U�«|�ê"�

�.�gn�GDC(t, k, ng) (X,G,B)´�`�£OGDC¤XJ|B| = C(n, g, k, t)"

w,§XJ�3��.�gn�GDC(t, k, ng)§@o§Ò´�`�"

-L(n, g, k, t) = dgnkdg(n−1)

k−1 · · · dg(n−t+1)k−t+1

e · · · ee"Schonheim3©[121]¥�Ñ

é?¿n ≥ k ≥ t ≥ 1§C(n, g, k, t) ≥ L(n, g, k, t)"

CHAPTER 9 ^�©|CX�E�`õ�~CXè 133

ét = 3§k = 4§g = 1§Mills3©[111]¥�Ñé?¿n 6≡ 7 (mod 12)§

C(n, 1, 4, 3) = L(n, 1, 4, 3)"Kalbfleisch�[98]§Swift[136]�ÑC(7, 1, 4, 3) = L(7,

1, 4, 3)+1 = 12"Mills [112]y²C(499, 1, 4, 3) = L(499, 1, 4, 3)"Hartman�3

©[88]¥�Ñé?¿n ≥ 52423§C(n, 1, 4, 3) = L(n, 1, 4, 3)"�CJi [95]U?¦

��(J��é?¿n§Øn = 7§Ú�U�Ø(½��n = 12k + 7§k ∈ {1,2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 16, 21, 23, 25, 29}§C(n, 1, 4, 3) = L(n, 1, 4, 3)"

Ún 9.2. ��.�gn�OGDC(t, k, ng)��35�du��Zg+1þ��`(n, k,

t, k − t)~CXè§=µC(n, g, k, t) = Kg+1(n, k, t, k − t)"

y². b�·�k��.�gn�OGDC(t, k, ng)§(In×Ig, {{i}×Ig : i ∈ In},B)§

Ù¥Is = {1, 2, . . . , s}"é?¿«|{(a1, b1), (a2, b2), . . . , (ak, bk)} ∈ B§ÏLé?¿1 ≤ j ≤ k§31bj ��aj§Ù{ ��0§·��Ý�n�è

i"éN´wÑ¤k�èi/¤��Zg+1þ��`(n, k, t, k − t)~CXè"

��§b�·�k��Zg+1þ��`(n, k, t, k − t)~CXèC"éz��èiu ∈ C§XJu��" ��a1, a2, . . . , ak§�A ��´ba, b2, . . . , bk§

@o·��±��«|{(a1, b1), (a2, b2), . . . , (ak, bk)}"éN´�y¤k�«|/¤��In × Igþ§|8�{{i} × Ig : i ∈ In}þ�OGDC(t, k, ng)"

y3§·��ÑH-frame�½Â§§´©[88]¥��í2"��H(t,K, v)

frame´��äkXe5��kSo�|(X,G,B,F)µ

1. X´��v�:�8Ü¶

2. G = {G1, G2, . . .}rXy©¤|¶

3. F´G��xf8{Fi}§¡�É"z�Fi ∈ F§Fi = {Gi1 , Gi2 , . . . , Gis}§�XJFiÚFj´É§@oFi ∩ Fj�´É"É¥|��ê¡�§��"

4. B´G��î��8Ü£¡�«|¤§z�«|��áu8ÜK§¦�G�?¿t�î�§XJØ´,�ÉFi ∈ F�t�î�§TÐ�¹3��«|¥§vk«|�¹É¥�t�î�"

XJG¥¤k|��þ�g§@o(X,G,B,F)Ò´��H(v/g, g,K, t)

frame§{P�HF(v/g, g,K, t)§ù�©[88]¥½Â�Ó"XJ��HF(q, g,K, 3)

134 Aa|Ü?è¯KïÄ

kni��mi + s�É§i = 1, 2, . . . , r§��u��ú��s�É§

@o·�rù��OP�K-HFg(mn11 m

n22 · · ·mnr

r : s)"�g = 1�§��K-

HF1(mn11 m

n22 · · ·mnr

r : s)�¡��Ç�.�O§P�K-CS(mn11 m

n22 · · ·mnr

r :

s) (X,S,Γ,B)§Ù¥S´��ú��É§¡�Z§Γ = {F \S : F ∈ F}´��|�8Ü"�K = {4}§�¡�Ç�.o�X£CQS¤"XJ��HF(q, g,K, 3)�

k��s�É§@o·�¡§��Ø��©|�O§P�IGDD((q :

s), g,K, 3)"

XJ��H(3, K, v) frame§(X,G,B,F)äk5�µG¥¤k�|ØG1�

��g − 1§��þ�g§ �Fkni��mi + s�É§i = 1, 2, . . . , r§

§��u��ú��s�É§G1áuù�ú��É§@o·�¡ù

�H(3, K, v)�modified H(3, K, v) frame§P�K-MHFg(mn11 m

n22 · · ·mnr

r : s)"

Ún 9.3 (Mills [111]). é?¿n ≥ 0§�3��CQS(6n : 0)"

Ún 9.4. é?¿n ≥ 3§�3��{4, 6}-CS(2n : 2)"

y². é?¿n ≡ 0, 1 (mod 3)§n ≥ 3§�±lSQS(2n + 2)��CQS(2n :

2)"é?¿n ≡ 2 (mod 3)§n ≥ 5§�±lCQS(6(n+1)/3 : 0)�ØÓ|�ü�:

�¤Z��{4, 6}-CS(2n : 2)"

Ún 9.5. b�(X,S,Γ,A)´��K-CS(mn : s)§∞ ∈ S"-K1 = {|A| :

∞ ∈ A ∈ A}§K2 = {|A| : ∞ 6∈ A ∈ A}"XJé?¿k1 ∈ K1§�3

��4-HFg(tk1−1 : a)£½4-MHFg(t

k1−1 : a)¤§é?¿k2 ∈ K2§�3��.

�(gt)k2�GDD(3, 4, gtk2)§@o©O�3��4-HFg((tm)n : t(s − 1) + a)Ú

£½4-MHFg((tm)n : t(s− 1) + a)¤"

y². b��½�K-CS(mn : s)�|8�Γ = {G1, . . . , Gn}"·�Äk�Ñ4-

HFg((tm)n : t(s−1)+a)��E"½ÂG′x,j = x×{j}×Zg"-X ′ = ((X \{∞})×Zt×Zg)∪({∞}×Za×Zg)§G ′ = {G′x,j : x ∈ X \{∞}, j ∈ Zt}∪{G′∞,j : j ∈ Za}§F = {Fi : 0 ≤ i ≤ n}§Ù¥F0 = {G′x,j : x ∈ S\{∞}, j ∈ Zt}∪{G′∞,j : j ∈ Za}´��t(s−1)+a�ú��É§�Fi = {G′x,j : x ∈ Gi, j ∈ Zt}∪F0§1 ≤ i ≤ n"

é?¿B ∈ A§∞ ∈ B§�E��4-HFg(t|B|−1 : a)§Ù¥:8�((B \

{∞})×Zt×Zg)∪({∞}×Za×Zg)§|8�{G′x,j : x ∈ B\{∞}, j ∈ Zt}∪{G′∞,j :


j ∈ Za}§É�Fx = {G′x,j : j ∈ Zt} ∪ F∞§x ∈ B \ {∞}§�u��aú��ÉF∞ = {G′∞,j : j ∈ Za}"P«|8�AB"

é?¿B ∈ A§∞ 6∈ B§�E��.�(gt)k2�GDD(3, 4, gtk2)§Ù¥:8

�B × Zt × Zg§|8�{x× Zt × Zg : x ∈ B}"P«|8�CB"

-A′ = (∪B∈A,∞∈BAB) ∪ (∪B∈A,∞/∈BCB)"N´�yA′´X ′þ|8�G ′§É�F�4-HFg((tm)n : t(s− 1) + a)�«|8"

�E4-MHFg((tm)n : t(s − 1) + a)�y²�þ¡aq"PZ∗a = Za \ {0}"-X ′′ = ((X \{∞})×Zt×Zg)∪ ({∞}× ((Za×Zg)\{(0, 0)}))§G ′′ = {G′x,j : x ∈X\{∞}, j ∈ Zt}∪{G′∞,j : j ∈ Z∗a}∪{G′∞,0\{(∞, 0, 0)}}§F ′ = {F ′i : 0 ≤ i ≤ n}§Ù¥F ′0 = {G′x,j : x ∈ S \{∞}, j ∈ Zt}∪{G′∞,j : j ∈ Z∗a}∪{G′∞,0 \{(∞, 0, 0)}}´��t(s − 1) + a�ú��É§ �F ′i = {G′x,j : x ∈ Gi, j ∈ Zt} ∪ F ′0§1 ≤ i ≤ n"éN´ÏL�E4-HFg((tm)n : t(s− 1) + a)aq��{�EX ′′þ|

8�G ′′§É8�F ′�4-MHFg((tm)n : t(s− 1) + a)"

9.3 ��`nnn��~~~CCCXXXèèè

3�!¥§·�ò(½K3(n, 4, 3, 1)§=C(n, 2, 4, 3)��"d½n9.1§X

Jn ≡ 1, 2 (mod 3)§n 6= 5§�3.�2n�GDD(3, 4, 2n)§Ò´é?¿ù�

�n§C(n, 2, 4, 3) = L(n, 2, 4, 3)"·�ò�ÑC(5, 2, 4, 3)�e.Úþ."

Ún 9.6. L(5, 2, 4, 3) + 2 ≤ C(5, 2, 4, 3) ≤ 24"

y². �±éN´�EÑ5äk24�«|�.�25�GDC(3, 4, 10)"-X = Z8§

G = {{i, i + 4} : i = 0, 1, 2, 3}}"@o©O�3:8�X§|8�Gþ�.�24�GDD(3, 4, 8)ÚGDD(2, 3, 8)§PÙ«|8©O�BÚT§§�þk8�«

|"-X ′ = X ∪ {∞1,∞2}§G ′ = G ∪ {{∞1,∞2}}"é?¿i = 1, 2§-Ci =

{T ∪ {∞i} : T ∈ T }"w,§B ∪ C1 ∪ C2´��:8X ′§|8G ′þ�k24�«|

�.�25�GDC(3, 4, 10)"Ïd§C(5, 2, 4, 3) ≤ 24"

ée.§·��I�y²äk21�«|�.�25�GDC(3, 4, 10)Ø�3"

b�(X ′,G ′,A)´��.�25�GDC(3, 4, 10)§Ù¥|A| = 21§�{�E"@

o|E| = 4§E�¹12��§Ù¥��n�´ØÓ�"b�E�¹5�½�õ

�:"@oE�¹��15��§Ï�é�{¥�z�:§�{¥�¹ù

�:�n�|êÑ�3�Ø"b�ETÐ�¹3�ØÓ�:§��a, b, c§@o

136 Aa|Ü?è¯KïÄ

§�3E¥Ñy�gê�U´3, 3, 6"�´ù�Ø�U/¤{a, b, c} þ�o�|"¤±E�½�¹o�ØÓ�:§��a§b§c§d§�z��¹3E�n

�n�|¥"¤±EAT´d{a, b, c}§{a, b, d}§{a, c, d}§{b, c, d}|¤�" �{a, b, c, d} /∈ A"Ï�ÄK§·�Ul«|8¥�K{a, b, c, d}��.�25

�GDD(3, 4, 10)"ù�§��35gñ"Ïd§{a, b, c}§{a, b, d}§{a, c, d}§{b, c, d}�½�¹A �l�«|¥"Ï�{{a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}}¥�?¿ü�n�|�uü�ú��:§�{a, b, c}§{a, b, d}§{a, c, d}§{b, c, d}´�{¥=k�o�n�|§A¥�l�«|��:AT´üüØÓ�"Ïd§·�ÒkX ′¥��12�:§ùÒ�|X ′| = 10�gñ"

�n ≡ 0 (mod 3)�§L(n, 2, 4, 3) = n(n2 − 3n + 3)/3"·�¡��.

�2n�GDD(3, {4, 6}, 2n)´Ð�XJ§TÐkn/3��6�«|"

Ún 9.7. e�3.�2nÐ�GDD(3, {4, 6}, 2n)§KC(n, 2, 4, 3) = L(n, 2, 4, 3)"

y². .�2n�Ð�GDD(3, {4, 6}, 2n)¥��4�«|ê�n(n2 − 3n − 3)/3"

r?¿��6�«|^.�16�OGDC(3, 4, 6)O�"Ï�C(6, 1, 4, 3) = 6§·

�Ò��¤I(J"

5¿�©[142]¥é¤kn ≡ 0 (mod 3)�E.�2n�GDD(3, {4, 6}, 2n)§

�Ø´¤kÑ´Ð�"

Ún 9.8 (Wang§Ji [142]). é?¿n ∈ {6, 9, 15}§�3��.�2n�Ð�GDD(3,

{4, 6}, 2n)"

Ún 9.9. é?¿n ≡ 0 (mod 6)§�3��.�2n�Ð�GDD(3, {4, 6}, 2n)"

y². é?¿n = 6k§k ≥ 1§lÚn9.3��CQS(6k : 0)"éz�:\�2§

¿3z�«|þ�E��.�24�GDD(3, 4, 8)§Ò��4-HF2(6k : 0)"

34-HF2(6k : 0)�ÉþW\.�26�Ð�GDD(3, {4, 6}, 12)��¤I�.

�26k�Ð�GDD(3, {4, 6}, 12k)"

Ún 9.10. �3kü��6�«|�IGDD((9 : 3), 2, {4, 6}, 3)"

y². -:8�Z18§|8�{{i, i + 9} : i = 0, 1, . . . , 8}§É�{{i, i + 9} :

i = 6, 7, 8}"ü��6�«|´{0, 1, 2, 3, 4, 5} Ú{9, 10, 11, 12, 13, 14}"�{


�52×3��4�«|dXe52�Ä«|dgÓ�〈(0 1 2)(3 4 5)(6 7 8)(9 10

11)(12 13 14)(15 16 17)〉Ðm��"{3, 5, 6, 7} {5, 6, 11, 17} {0, 6, 12, 17} {0, 5, 10, 17} {1, 3, 9, 13} {3, 10, 13, 15}{0, 3, 13, 16} {1, 4, 9, 15} {6, 7, 9, 11} {0, 4, 10, 12} {3, 6, 13, 14} {3, 4, 6, 11}{0, 11, 14, 15} {2, 6, 14, 17} {0, 1, 6, 7} {1, 2, 13, 14} {0, 10, 14, 16} {0, 6, 11, 13}{5, 8, 11, 16} {2, 12, 13, 16} {3, 5, 9, 11} {3, 7, 11, 13} {0, 1, 8, 14} {0, 4, 8, 11}{1, 2, 15, 16} {0, 4, 15, 16} {0, 2, 10, 15} {2, 7, 9, 12} {6, 8, 12, 14} {0, 4, 6, 14}{2, 4, 8, 15} {6, 8, 9, 13} {0, 5, 13, 15} {5, 9, 13, 16} {3, 4, 14, 15} {2, 4, 6, 7}{4, 9, 11, 17} {5, 7, 11, 15} {1, 4, 7, 11} {1, 4, 8, 12} {6, 11, 12, 16} {0, 3, 8, 15}{6, 10, 13, 16} {0, 4, 7, 17} {1, 8, 9, 11} {4, 6, 9, 12} {9, 11, 15, 16} {6, 9, 14, 16}{3, 4, 16, 17} {11, 13, 16, 17} {12, 13, 15, 17} {2, 7, 14, 15}

Ún 9.11. �3��4-HF2(35 : 0)"

y². -:8�Z30§|8�{Gi = {i, i + 15} : i = 0, 1, . . . , 4}§Ê�É�Fi =

{Gi, Gi+5, Gi+10}§i = 0, 1, . . . , 4"¤I�OdXeÄ«|3Z30þ+1 (mod 30)Ð

m��"{0, 6, 13, 27} {0, 8, 11, 29} {0, 3, 12, 17} {0, 4, 10, 23} {0, 1, 4, 7} {0, 21, 28, 29}{0, 19, 24, 28} {0, 2, 7, 16} {0, 1, 20, 26} {0, 7, 14, 26} {0, 18, 23, 26} {0, 2, 12, 24}{0, 4, 26, 29} {0, 6, 7, 12} {0, 13, 17, 29} {0, 11, 12, 19} {0, 16, 18, 22} {0, 2, 10, 18}{0, 3, 20, 23} {0, 1, 3, 28} {0, 5, 7, 13} {0, 5, 17, 26} {0, 2, 4, 13} {0, 2, 9, 22}{0, 13, 19, 22} {0, 9, 10, 19} {0, 5, 6, 22} {0, 10, 14, 24} {0, 5, 16, 19} {0, 1, 11, 13}

e¡(JÄuHartman3[86, 14!]¥éCQS((6n)3 : 2s)��E§Ì��9

Ï�O´�a¡�A-pairing£P�A(n, 2s)¤��O"

½n 9.12. é?¿3n ≥ s ≥ 0§�3��4-HF2((3n)3 : s)"

y². é?¿3n ≥ s ≥ 0§(n, s) 6= (1, 1)§Hartman3[86, 14!]¥�E�

aCQS((6n)3 : 2s)§Ù¥:8X = {ai : a ∈ Z6n, i ∈ Z3} ∪ {∞1,∞2, . . . ,∞2s}§|8{{ai : a ∈ Z6n} : i ∈ Z3}§Z�{∞1,∞2, . . . ,∞2s}"-«|8�B§Ù¥k�a«|µ

φ = {{ai, bi, ci+1, di+1} : {a, b} ∈ F (k)i , {c, d} ∈ F (k)

i+1,

1 ≤ k ≤ 6n− 1− 2r − 2h, i ∈ Z3},

ùp§F(1)i |F

(2)i | . . . |F

(6n−1−2r−2h)i ´dA(n, 2s)½Â�ãZ6n×{i}��Ïf©)"

d[86, 15!]¥A(n, 2s)�äN�E§·��6n− 1− 2r − 2h ≥ 1"

138 Aa|Ü?è¯KïÄ

¤I�4-HF2((3n)3 : s)ò3:8Xþ�E§Ù¥|8G = {{ai, bi} : {a, b} ∈F

(1)i , i ∈ Z3} ∪ {{∞i,∞i+s} : 1 ≤ i ≤ s}§n�ÉFi+1 = {{ai, bi} : {a, b} ∈F

(1)i } ∪ F0§i ∈ Z3§�u��ú��ÉF0 = {{∞i,∞i+s} : 1 ≤ i ≤ s}"

-

φ1 = {{ai, bi, ci+1, di+1} : {a, b} ∈ F (1)i , {c, d} ∈ F (1)

i+1, i ∈ Z3}.

5¿�φ1 ⊂ φ§ �φ1¥�z�«|�uG¥ü�ØÓÉ�|"éN´�yB \ φ1Ò´¤I4-HF2((3n)3 : s)�«|8"

é(n, s) = (1, 1)§��4-HF2(33 : 1)�±dÚn9.5§¿^©[84]¥�CQS(33 :

1)Ú��.�24�GDD(3, 4, 8)��"

Ún 9.13. é¤kn ≡ 3 (mod 6)§n ≥ 9§�3.�2nÐ�GDD(3, {4, 6}, 2n)"

y². én ∈ {9, 15}§¤I�OdÚn9.8��"é?¿n = 6m+ 3§m ≥ 3§d

Ún9.4�3��{4, 6}-CS(2m : 2)"A^Ún9.5§�g = 2§t = 3§a = 0Ò�

�4-HF2(6m : 3)"ùpÑ\�O�4-HF2(3

k−1 : 0)Ú.�6k�GDD(3, 4, 6k)§

k ∈ {4, 6}£½n9.1§9.12ÚÚn9.11¤"é4-HF2(6m : 3)�cm − 1�É§W

\IGDD((9 : 3), 2, {4, 6}, 3)£Ún9.10¤"3��ÉW\��.�29�Ð

�GDD(3, {4, 6}, 18)"·�Ò��.�2n�Ð�GDD(3, {4, 6}, 2n)"

nÜÚn9.6§9.7§9.9Ú9.13§·��µ

½n 9.14. é?¿n ≥ 4§n 6= 5§C(n, 2, 4, 3) = L(n, 2, 4, 3)"

9.4 ��`ooo��~~~CCCXXXèèè

3�!¥§·�ò(½K4(n, 4, 3, 1)§=C(n, 3, 4, 3)��"d½n9.1§é?

¿n ≡ 0 (mod 2)§�3.�3n�GDD(3, 4, 3n)§=C(n, 3, 4, 3) = L(n, 3, 4, 3)"

�n ≡ 1 ( mod 2)�§L(n, 3, 4, 3) = 3n(n−1)(3n−5)/8".�3n�GDC(3, 4, 3n)

�¡�´Ð�§XJ§��{/¤��.�3n�GDD(2, 3, 3n)"N´�y��

.�3n�Ð�GDC(3, 4, 3n)�«|�êTÐ�L(n, 3, 4, 3)"·��µ

Ún 9.15. e�3.�3nÐ�GDC(3, 4, 3n)§KC(n, 3, 4, 3) = L(n, 3, 4, 3)"


Ún 9.16. é?¿n ∈ {5, 7, 9, 11}§�3��.�3n�Ð�GDC(3, 4, 3n)"

y². é?¿n ∈ {5, 7, 9, 11}§��.�3n�Ð�GDC(3, 4, 3n)´3:8Z3n§

|8{{i, i + n, i + 2n} : i = 0, 1, . . . , n − 1}þ�E§¤I«|dXeÄ«|3Z3nþÐm��"

n = 5µ{0, 1, 2, 4} {0, 1, 4, 13} {0, 1, 7, 9} {0, 1, 8, 12} {0, 2, 8, 11}

n = 7µ

{0, 11, 15, 17} {0, 12, 18, 20} {0, 8, 9, 11} {0, 1, 6, 11} {0, 16, 17, 20} {0, 1, 12, 16}{0, 2, 8, 18} {0, 3, 5, 8} {0, 9, 12, 15} {0, 8, 19, 20} {0, 4, 8, 16} {0, 2, 4, 13}

n = 9µ

{0, 5, 21, 25} {0, 6, 13, 17} {0, 10, 11, 22} {0, 4, 7, 17} {0, 21, 23, 26} {0, 2, 4, 19}{0, 2, 12, 13} {0, 3, 15, 19} {0, 12, 22, 25} {0, 13, 24, 26} {0, 12, 19, 20} {0, 1, 7, 13}{0, 5, 19, 24} {0, 1, 3, 11} {0, 4, 12, 23} {0, 5, 7, 20} {0, 3, 25, 26} {0, 6, 13, 16}{0, 5, 22, 26} {0, 5, 11, 12} {0, 6, 8, 14} {0, 3, 6, 10}

n = 11µ

{0, 1, 10, 14} {0, 17, 18, 21} {0, 21, 23, 29} {0, 25, 29, 32} {0, 9, 19, 25} {0, 4, 10, 19}{0, 9, 14, 26} {0, 1, 13, 17} {0, 8, 16, 23} {0, 2, 6, 19} {0, 17, 20, 30} {0, 1, 6, 20}{0, 1, 15, 16} {0, 21, 28, 31} {0, 2, 5, 14} {0, 4, 6, 9} {0, 3, 8, 24} {0, 20, 26, 27}{0, 8, 10, 29} {0, 13, 21, 26} {0, 7, 9, 24} {0, 5, 6, 8} {0, 4, 12, 13} {0, 7, 14, 30}{0, 14, 17, 23} {0, 2, 10, 28} {0, 1, 5, 31} {0, 13, 15, 28} {0, 2, 16, 31} {0, 23, 28, 32}{0, 9, 15, 21} {0, 8, 18, 28} {0, 6, 13, 25} {0, 15, 27, 30} {0, 1, 8, 32}

��ÑÌ��E§·�I�½Â�«Ø��Ð��©|CX"-X´

��3n�:8§G´X��©¤n��3�|�y©"b�H ⊂G´��m�É§�Ò´m�|�8Ü"B´�xî�o�|£«|¤¦�vk«|�¹É¥�n�|§z�Ø3É¥�î�n�|�¹3��«

|¥§¿��{/¤��.�3n−m(3m)1�GDD(2, 3, 3n)"@o(X,G,H,B)Ò

¡��.�(3n : 3m)�Ø��Ð�GDC(3, 4, 3n)"

Ún 9.17. b��3��4-MHF3(mn : s)"XJ�3��.�(3m+s : 3s)�Ø�

��Ð�GDC(3, 4, 3(m+s))§@o©O�3.�(3mn+s : 3s)Ú(3mn+s : 3m+s)�

Ø��Ð�GDC(3, 4, 3(mn + s))"?�Ú§XJ�3��.�3m+s�Ð

�GDC(3, 4, 3(m+ s))§@o�3��.�3mn+s�Ð�GDC(3, 4, 3(mn+ s))"

y². -(X,G,B,F)´�½�4-MHF3(mn : s)§Ù¥G1 = {α, β}´��ÉF0¥

��2�AÏ|"-G′1 = G1 ∪ {∞}§∞ /∈ X"-X ′ = X ∪ {∞}§G ′ =

140 Aa|Ü?è¯KïÄ

G∪{G′1}\{G1}§F ′ = {F ∪{G ′∞}\{G∞} : F ∈ F}"-Tα = {B \{α} : α ∈ B ∈B}"@oTα/¤.�(3m)n�GDD(2, 3, 3mn)§Ù¥:8�X \ (∪G∈F0G)§|8

�{∪G∈(F\F0)G : F ∈ F , F 6= F0}"-B∞ = {T ∪ {∞} : T ∈ Tα}§B′ = B ∪B∞"é?¿��m + s�ÉF ′ ∈ F ′§�E��.�(3m+s : 3s)§É�F ′0 = F0 ∪{G′1} \ {G1}�Ø��GDC(3, 4, 3(m + s))"P«|8�AF ′§�{/¤�|�∪G∈F ′0G�.�3m(3s)1�GDD(2, 3, 3(m + s))"-C = B′ ∪ (∪F ′∈F ′,F ′ 6=F ′0AF ′)"éN´�yC��{/¤�|�∪G∈F ′0G�.�3mn(3s)1�GDD(2, 3, 3(mn+ s))"

Ïd§C´X ′þ|8�G ′§É�F ′0§.�(3mn+s : 3s)�Ð�GDC(3, 4, 3(mn +

s))"XJ·��3��É§½öW\��.�3m+s�GDC(3, 4, 3(m+ s))§

·�Ò©O��.�(3mn+s : 3m+s)Ø��Ð�GDC(3, 4, 3(mn+ s))½

��.�3mn+s�Ð�GDC(3, 4, 3(mn+ s))"

Ún 9.18. �3��.�(37 : 33)�Ø��Ð�GDC(3, 4, 21)"

y². -:8�Z21§|8�{{i, i+7, i+14} : i = 0, 1, . . . , 6}§É�{{i, i+7, i+

14} : i = 4, 5, 6}"¤I�OdXeÄ«|dgÓ�+〈(0 7 14)(1 2 3 8 9 10 15 16

17)(4 5 6 11 12 13 18 19 20)〉Ðm��"{1, 7, 13, 18} {11, 1, 7, 12} {18, 8, 9, 12} {0, 8, 10, 13} {0, 4, 6, 16} {0, 2, 3, 15}{0, 10, 11, 16} {16, 4, 8, 14} {6, 14, 17, 19} {2, 5, 15, 18} {9, 0, 6, 8} {1, 6, 9, 19}{2, 3, 5, 20} {0, 3, 8, 11} {1, 10, 12, 16} {6, 7, 8, 19} {1, 5, 13, 17} {1, 2, 12, 17}{6, 8, 11, 14} {6, 7, 12, 16} {10, 13, 16, 18} {3, 13, 15, 19} {8, 10, 19, 20} {0, 2, 6, 19}{7, 11, 13, 16} {2, 6, 7, 17} {6, 2, 10, 11}

Ún 9.19. �3��.�(39 : 33)�Ø��Ð�GDC(3, 4, 27)"

y². -:8�Z27§|8�{{i, i+9, i+18} : i = 0, 1, . . . , 8}§É�{{i, i+9, i+18} : i = 6, 7, 8}"¤I�OdXeÄ«|dgÓ�+〈(0 1 2 9 10 11 18 19 20)(3 4 5 12 13 14 21 22 23)(6 7 8)(15 16 17)(24 25 26)〉Ðm��"{5, 7, 18, 19} {10, 4, 11, 26} {5, 9, 12, 24} {6, 11, 19, 25} {4, 11, 23, 25} {0, 15, 19, 20}{2, 5, 8, 15} {0, 11, 25, 26} {10, 11, 12, 23} {8, 11, 22, 24} {0, 8, 13, 16} {0, 3, 15, 17}{1, 8, 15, 18} {7, 13, 21, 23} {8, 14, 18, 25} {4, 7, 8, 19} {0, 5, 13, 15} {0, 2, 16, 23}{22, 9, 20, 21} {7, 19, 21, 24} {10, 18, 20, 22} {22, 14, 16, 18} {5, 15, 18, 20} {13, 16, 19, 21}{3, 9, 13, 26} {5, 7, 13, 26} {4, 7, 20, 21} {2, 3, 16, 22} {7, 11, 14, 18} {4, 1, 14, 26}{9, 16, 17, 19} {0, 5, 7, 24} {8, 0, 2, 7} {11, 12, 17, 24} {10, 18, 23, 26} {22, 2, 5, 12}{3, 9, 20, 24} {1, 2, 13, 16} {10, 11, 14, 25} {2, 13, 18, 19} {2, 7, 14, 19} {8, 11, 16, 21}{14, 1, 22, 24} {3, 4, 6, 10} {13, 14, 15, 17} {2, 3, 6, 23} {1, 17, 23, 24} {0, 4, 8, 20}{5, 16, 18, 24} {4, 17, 18, 24} {3, 6, 17, 20} {0, 3, 4, 11} {11, 21, 22, 23} {5, 8, 11, 19}{11, 14, 19, 22} {1, 12, 17, 22} {25, 0, 17, 20} {0, 2, 22, 24} {8, 3, 6, 13} {0, 3, 7, 10}{10, 22, 25, 26} {3, 7, 17, 23} {4, 5, 24, 25} {2, 4, 14, 16} {5, 15, 21, 26}


Ún 9.20. �3��4-MHF3(23 : 1)"

y². -:8�Z20§|8�{Gi = {i, i + 6, i + 12} : i = 0, 1, . . . , 5}§n�É{Gi, Gi+3} ∪ S§i = 0, 1, 2�u��ú��ÉS = {{18, 19}}"¤I�OdXeÄ«|dgÓ�+〈(0 6 12)(1 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(18)(19)〉Ðm��"

{3, 5, 6, 8} {0, 10, 11, 18} {2, 10, 13, 17} {0, 2, 9, 10} {4, 8, 12, 13} {2, 3, 4, 18}{0, 8, 16, 18} {0, 3, 4, 7} {2, 3, 10, 19} {0, 14, 16, 19} {9, 10, 11, 19} {1, 3, 8, 18}{6, 9, 11, 16} {0, 7, 9, 11} {1, 9, 17, 18} {4, 9, 17, 19} {6, 11, 13, 19} {0, 13, 14, 18}{0, 7, 15, 16} {0, 1, 4, 15} {5, 8, 10, 13} {0, 5, 13, 16} {1, 5, 9, 19} {0, 2, 7, 18}{0, 2, 4, 5} {2, 9, 16, 18} {7, 10, 14, 15} {1, 11, 15, 18} {4, 5, 6, 9} {0, 2, 13, 19}{5, 10, 12, 14} {0, 4, 11, 13} {6, 7, 17, 19} {0, 4, 17, 18} {1, 2, 4, 12} {12, 13, 17, 18}

{1, 2, 5, 10} {4, 7, 9, 12} {1, 8, 10, 15} {2, 3, 5, 7} {2, 5, 9, 12} {2, 3, 6, 11}{3, 8, 12, 16} {12, 13, 14, 15} {5, 9, 14, 16} {1, 8, 11, 12} {2, 5, 15, 16} {3, 10, 12, 13}{0, 10, 17, 19} {3, 7, 8, 19} {0, 9, 13, 17} {0, 1, 8, 17} {2, 7, 9, 19} {10, 15, 17, 18}{5, 8, 12, 15} {0, 4, 8, 19} {10, 11, 13, 14} {1, 2, 9, 11} {11, 13, 15, 16} {1, 4, 5, 14}{2, 3, 12, 17} {0, 3, 8, 13} {4, 5, 13, 15}

Ún 9.21. �3��4-MHF3(25 : 1)"

y². -:8�Z32§|8�{Gi = {i, i + 10, i + 20} : i = 0, 1, . . . , 9}§Ê�É{Gi, Gi+5}∪S§i = 0, 1, 2, 3, 4�u��ú��ÉS = {{30, 31}}"¤I�OdXeÄ«|dgÓ�+〈(0 1 2 3 4 5 6 · · · 21 22 23 24 25 26 27 28 29)(30 31)〉Ðm��"

{12, 17, 26, 28} {15, 24, 27, 31} {10, 18, 22, 24} {2, 6, 17, 19} {6, 12, 19, 31} {2, 9, 15, 21}{7, 19, 22, 25} {1, 9, 23, 27} {2, 7, 19, 24} {6, 7, 23, 24} {3, 6, 7, 22} {0, 6, 14, 21}{0, 4, 18, 31} {2, 3, 5, 10} {18, 25, 26, 29} {2, 11, 18, 20} {6, 14, 23, 31} {0, 1, 7, 12}{1, 18, 22, 27} {2, 3, 4, 31} {3, 18, 24, 25} {6, 11, 14, 27} {7, 8, 26, 29} {13, 15, 19, 22}{19, 22, 24, 26} {4, 23, 25, 28} {12, 15, 26, 30} {2, 6, 13, 30} {18, 19, 23, 24} {12, 14, 15, 29}{7, 12, 19, 26} {2, 13, 19, 21} {11, 13, 19, 31}

Ún 9.22. é¤kn ≡ 3 (mod 4)§n ≥ 7§�3.�3n�Ð�GDC(3, 4, 3n)"

y². én ∈ {7, 11}§¤I�O3Ún9.16¥�E��"én = 4m+ 3§m ≥ 3§

lÚn9.4��{4, 6}-CS(2m : 2)"A^Ún9.5§ùp^4-MHF3(2k−1 : 1)Ú.

�6k�GDD(3, 4, 6k)§k ∈ {4, 6}��Ñ\�O§·�Ò��4-MHF3(4m : 3)"

,�^Ún9.17^m − 1�Ø��.�(37 : 33)�Ð�GDC(3, 4, 21)Ú��.

�37�Ð�GDC(3, 4, 21)��Ñ\�OÒ��¤I(J"ùp�Ñ\�O5

uÚn9.18§9.20Ú9.21"

142 Aa|Ü?è¯KïÄ

Xe�4-MHF3((2n)3 : s)�´ÄuHartman3©[86, 14!]éCQS((6n)3 :

2s)��E"Äk§·�0��Vg"éx ∈ Zn§-|x|�x§e0 ≤ x ≤ n/2§

�n − x§en/2 < x < n"éãZn�?¿>8E§·�^LE5L«E¥�>��Ý�8Ü§=µLE = {|x−y| : {x, y} ∈ E}"én ≥ 2§L ⊆ {1, 2, . . . , bn/2c}§-G(n, L)´:8Znþ§>8�E��Kã§¦�{x, y} ∈ E��=�|x − y| ∈L"

½n 9.23. é?¿��ênÚÛês§6n ≥ 3s−1§�3��4-MHF3((2n)3 : s)"

y². é(n, s) = (1, 1)§¤I�OdÚn9.20��"é?¿��ênÚÛês§

6n ≥ 3s − 1§(n, s) 6= (1, 1)§·�3Z6nþ�E�©[86]¥�A-pairingaq�9

Ï�O(D,H,R0, R1, R2)(n,s)"ùp§�Ý�2n�>ØÑy"

1. �n = 2§s = 1�§-D = {4, 10}§H = {{1,−1}{2, 7}}§R0 = {{3, 6}}§R1 = {{5, 8}}§R2 = {{0, 9}}"

2. �n ≥ 3§s = 1�§-D = {2n, 4n − 1}§H = {{1,−1}, {2,−2}}§R0 =

{{0, 2n−1}}∪{{k, 2n−k+1} : k = 3, 4, . . . , n}§R1 = {{2n+k, 4n−k−1} :

k = 1, 2, . . . , n− 1}§R2 = {{4n+ k, 6n− 3− k} : k = 0, 1, . . . , n− 2}"

3. �s ≥ 3§6n ≥ 3s−1�§(D,H,R0, R1, R2)(n,s)d(D′, H ′, R′0, R′1, R

′2)(n,s−2)

Ì��E��"-ri´R′i¥��"@oD = D′ ∪ (∪i=2

i=0ri)§H = H ′§

Ri = R′i \ {ri}§i = 0, 1, 2"

éXþ�?¿:é(n, s)§éN´�yãG(6n, LH∪LRi∪{2n})�Öã�3�Ïf©)F

(1)i |F

(2)i | . . . |F

(4n+s−6)i §i = 0, 1, 2"y3¤I�4-MHF3((2n)3 : s)X

e�Eµ-X = {ai : a ∈ Z6n, i ∈ Z3} ∪ {∞1,∞2, . . . ,∞3s−1}§Gi,j = {ji, (j +

2n)i, (j+ 4n)i}§i = 0, 1, 2§j = 0, . . . , 2n− 1§G∞,j = {∞j,∞j+s,∞j+2s}§j =

1, 2, . . . , s − 1§G∞,s = {∞s,∞2s}"ùpkn�ÉFi = {Gi,j : j = 0, . . . , 2n −1} ∪ S§i = 0, 1, 2�u��ú��ÉS = {G∞,j : j = 1, 2, . . . , s}"-«|8


�B§§�¹XeÊ�«|8µ

δ = {{∞j, (a+ d)0, (b− d)1, (c+ d)2} : a+ b+ c ≡ 0 (mod 6n),

D´d�1j��, 1 ≤ j ≤ 3s− 1}§

ρ = {{(a+ q)i, (a+ t)i, bi+1, ci+2} : a+ b+ c ≡ 0 (mod 6n),

{q, t} ∈ Ri, i ∈ Z3}§

φ = {{ai, bi, ci+1, di+1} : {a, b} ∈ F (k)i , {c, d} ∈ F (k)

i+1,

1 ≤ k ≤ 4n+ s− 6, i ∈ Z3}§

χ1 = {{ai+1, (a+ 3ε)i+2, (x− 2a− 3ε)i, (y − 2a− 3ε)i} :

a ∈ Z6n, i ∈ Z3, ε ∈ Z2n, {x, y} ∈ H}§

χ2 = {{ai, (a+ |x− y|)i, (a+ 3ε)i+1, (a+ 3ε+ |x− y|)i+1} :

a ∈ Z6n, i ∈ Z3, ε ∈ Z2n, {x, y} ∈ H}"

�e�y²�©[86]¥�y²aq§·�òÑ�"

Ún 9.24. é?¿n ≡ 5 (mod 8)§�3��.�3n�Ð�GDC(3, 4, 3n)"

y². én = 5§¤I�OdÚn9.16��"é?¿n ≡ 5, 13 (mod 24)§n ≥13§lSQS((n+ 3)/4)��CQS(1(n−1)/4 : 1)"^Ún9.5§^4-MHF3(4

3 : 1)Ú�

�.�124�GDD(3, 4, 48)��Ñ\�O§·�Ò��4-MHF3(4(n−1)/4 :

1)",�^Ún9.17§¿ò.�35�Ð�GDC(3, 4, 15)��Ñ\�O"ùp�Ñ

\�O4-MHF3(43 : 1)5g½n9.23"

én = 21§l½n9.23��4-MHF3(63 : 3)"A^Ún9.17§¿rØ��

�.�(39 : 33)�Ð�GDC(3, 4, 27)£Ún9.19¤Ú.�39�Ð�GDC(3, 4, 27)

£Ún9.16¤��Ñ\�O§Ò��.�321�Ð�GDC(3, 4, 63)Ú.�(321 :

39)�Ø��Ð�GDC(3, 4, 63)"

én = 45§l½n9.23��4-MHF3(123 : 9)"A^Ún9.17§¿r.

�321�Ð�GDC(3, 4, 63)§Ú.�(321 : 39)�Ø��Ð�GDC(3, 4, 63)��

Ñ\�O§Ò��.�345�Ð�GDC(3, 4, 135)Ú.�(345 : 321)�Ø��

Ð�GDC(3, 4, 135)"

én = 69§l½n9.23��4-MHF3(83 : 1)"^Ún9.17§¿r.�39�

Ð�GDC(3, 4, 27)Ú.�(39 : 33)�Ø��Ð�GDC(3, 4, 27)��Ñ\�

144 Aa|Ü?è¯KïÄ

OÒ��.�(325 : 33)�Ø��Ð�GDC(3, 4, 75)Ú��.�325�Ð

�GDC(3, 4, 75)",�l½n9.23��4-MHF3(223 : 3)§¿A^Ún9.17§r

.�(325 : 33)�Ø��Ð�GDC(3, 4, 75)§Ú.�325�Ð�GDC(3, 4, 75)�

�Ñ\�OÒ��.�369�Ð�GDC(3, 4, 207)"

é?¿n = 24k + 21§k ≥ 3§·�k`²�3CQS(6k : 6)"¯¢þ§b

�(X,G,B)´��.�6k+1�GDD(3, 4, 6(k+1))§Ù¥G = {Gi : i = 1, 2, . . . , k+

1}"é?¿i = 1, 2, . . . , k§:8�Gi��ã�3�Ïf©)F(1)i |F

(2)i | · · · |F

(5)i "

é?¿:é{i, j} ⊂ {1, 2, . . . , k}§-Ai,j = {{a, b, c, d} : {a, b} ∈ F(l)i , {c, d} ∈

F(l)j , l = 1, 2, . . . , 5}"@oB ∪ (∪{i,j}⊂{1,2,...,k}Ai,j)Ò´Xþ§|8�G \ {Gk+1}§Z�Gk+1�CQS(6k : 6)�«|8"ùp��CQS(6k : 6)§A^Ún9.5§

¿r4-MHF3(43 : 1)Ú��.�124�GDD(3, 4, 48)��Ñ\�OÒ��4-

MHF3(24k : 21)",�A^Ún9.17§r.�345�Ð�GDC(3, 4, 135)Ú.

�(345 : 321)�Ð�GDC(3, 4, 135)��Ñ\�OÒ��¤I�O"

½n 9.25. é?¿��ênÚÛês§6n ≥ 3s−1§�3��4-MHF3((2n)4 : s)"

y². é?¿��ênÚÛês§6n ≥ 3s − 1§GranvilleÚHartman3©[79]¥

�E�aCQS((6n)4 : 3s − 1)§Ù¥:8X = {ai : a ∈ Z6n, i ∈ Z4} ∪{∞1,∞2, . . . ,∞3s−1}§|�{{ai : a ∈ Z6n} : i ∈ Z4}§Z�{∞1,∞2, . . . ,∞3s−1}"¦�½ÂHanani©)§Ò´o�|(D,E,G,H)§¦�D ⊂ {1, 3, 5, . . . , 6n −1}§E ⊂ {0, 2, 4, . . . , 6n−2}§|D| = |E| = (3s−1)/2§G = {G0, G1, . . . , G3n−1}´:8Z6nþ��ã�Ü©�Ïf§Ù¥|Gi| = 3n− (3s− 1)/2 CXZ6n \ ((D ∪E)+2i)§i ∈ {0, 1, . . . , 3n−1}§H´��Ïf�8Ü§¦�G∪H´:8Z6nþ

��ã��y©"y3·�?U¦��E5��4-MHF3((2n)4 : s)"

½ÂΓ´��CXG¥¤k>�ã"d©[79, ½n6.1]¥Hanani©)��

E§Γ´Ì��Ø�¹�Ý�2n�>"-Υ´k2n�Ü§{i, i + 2n, i + 4n}§i = 0, 1, . . . , 2n − 1��õÜã"ØJ�yΓ3Υ ¥�Ök��Ïf©)§

P�H′"3CQS((6n)4 : 3s− 1)��E¥§^H′O�H",§r��ãZ6n�

Ñy3«|

{{hi, hi, aj, aj} : {i, j} ∈ {{0, 1}, {2, 3}}, {h, h}, {a, a} ∈ Jk, 0 ≤ k ≤ 6n− 2}

¥��Ïf©)J0|J1| . . . |J6n−2^ΥO�"


-Gi,j = {ji, (j + 2n)i, (j + 4n)i}§i = 0, 1, 2, 3§j = 0, . . . , 2n− 1§G∞,j =

{∞j,∞j+s,∞j+2s}§j = 1, 2, . . . , s − 1§G∞,s = {∞s,∞2s}"@oXþ�E�«|8�¤��:8X§|8�{Gi,j : i = 0, 1, 2, 3, j = 0, . . . , 2n− 1} ∪ {G∞,j :

j = 1, 2, . . . , s}§o�ÉFi = {Gi,j : j = 0, . . . , 2n− 1} ∪ S§i = 0, 1, 2, 3§�u

��ú��ÉS = {G∞,j : j = 1, 2, . . . , s}�4-MHF3((2n)4 : s)"

Ún 9.26. é¤kn ≡ 1 (mod 8)§n ≥ 9§�3.�3n�Ð�GDC(3, 4, 3n)"

y². é?¿n = 8k + 1§k ≥ 1§y²d48��"ék = 1§��.�39�

Ð�GDC(3, 4, 27)dÚn9.16��"�k > 1�§b�é?¿i < k§�3.

�38i+1�Ð�GDC(3, 4, 3(8i + 1))"dÚn9.22Ú9.24§·��é?¿Û

êj < 8k + 1§Ñ�3.�3j�Ð�GDC(3, 4, 3j)"A^Ún9.17§¿r�

�4-MHF3((2k)4 : 1).�32k+1�Ð�GDC(3, 4, 3(2k + 1))��Ñ\�O§Ò�

�.�38k+1�Ð�GDC(3, 4, 3(8k + 1))"

(ÜÚn9.22§9.24Ú9.26§·��µ

½n 9.27. é?¿n ≥ 4§C(n, 3, 4, 3) = L(n, 3, 4, 3)"

9.5 Z2m+1þþþ��`~~~CCCXXXèèè

3�!¥§·�ò�Ñ��Z2m+1þ�`(n, 4, 3, 1)~CXè��(J§

=µ�E|��2m§m ≥ 2��`�©|CX"l½n9.1, é?¿g ≡ 2, 4

(mod 6)§g 6≡ 10, 26 ( mod 48)§n ≡ 1, 2 ( mod 3)§�3.�gn�GDD(3, 4, gn)§

�Ò´é?¿ù��gÚn§C(n, g, 4, 3) = L(n, g, 4, 3)"y3·��Äg ≡ 2, 4

(mod 6)§n ≡ 0 (mod 3)��¹"N´O�L(n, g, 4, 3) = g3n(n−1)(n−2)/24+

gn/6"¤±XJ�3TÐkng6��6�«|�.gn�GDD(3, {4, 6}, gn)§·

�^.�16�OGDC(3, 4, 6)O��6�«|§Ò��kL(n, g, 4, 3)�«|�

.�gn�GDC(3, 4, gn)"

�13!¥�½Âaq§·�¡��.�gn�GDD(3, {4, 6}, gn)´Ð�§X

J§TÐ�¹ng6��6�«|"

Ún 9.28. XJ�3��.�gn�Ð�GDD(3, {4, 6}, gn)§@o�3��.

�(2g)n�Ð�GDD(3, {4, 6}, 2gn)"

146 Aa|Ü?è¯KïÄ

y². -(X,G,B)´�½�.�gn�Ð�GDD(3, {4, 6}, gn)§Ù¥TÐkgn/6�

��6�«|"-X ′ = X × Z2§G ′ = {G × Z2 : G ∈ G}"é?¿��4�«|B ∈ B§3B × Z2þ�E.�24�GDD(3, 4, 8)§Ù¥|�{x} × Z2§

x ∈ B"PÙ«|8�AB"é?¿��6�«|B ∈ B§3B × Z2þ�E.

�26�GDD(3, 4, 12)§Ù¥|8�{x} × Z2§x ∈ B"PÙ«|8�CB"N´�y(X ′,G ′, (∪B∈B,|B|=4AB)∪ (∪B∈B,|B|=6CB))Ò´.�(2g)n�GDD(3, {4, 6}, 2gn)§

Ù¥gn/3�«|��6"

é?¿�êm ≥ 2§w,2m ≡ 2, 4 (mod 6)§2m 6≡ 10, 26 (mod 48)"(Ü

½n9.1§Ún9.28§ÚÚn9.9§9.13§·��Xe(Jµ

½n 9.29. é?¿m ≥ 2§n ≥ 4§C(n, 2m, 4, 3) = L(n, 2m, 4, 3)"

9.6 (((ØØØ

3�Ù¥§·�é¤kn ≥ 4§q = 3, 4½öq = 2m + 1§m ≥ 2§Ø��

Ø(½��(q, n) = (3, 5)§(½Zqþ��`(n, 4, 3, 1)~CXè�èi�

ê"

Chapter 10

��EEEäääkkk©©©��555��@@@yyyèèè

10.1 ÚÚÚóóó

3@yè�IO�.¥[122–124, 127]§��u�ìI�3��ØS�&�

Dx&E��Âì§ ��'<�¯ù�&�¿�Ð¢�Âì"'<�±

3ù�&�\\�#�&E§½ölu�ì:�&E¿?U¤gC�&E"

3?¿�«�¹e§'<�8IÑ´Ð¢�Âì¦��&#�&E´�&�

£=5 uu�ì¤"1��ôÂ´Äu\\#�&E§¡��°�[§1��

ôÂ´Äu?U5gu�ì�&E§¡��°O�"

��/§-SL«¤k G��8Ü§M�¤k&E�8Ü§E´¤k?è5K�8Ü"ùÑ´k�8"�� G�ú�ì��D��Âì�&

E"��?è5K´��lS�2M��ü�"u�ìÚ�Âì¯k�½��

��?è5Ke ∈ E"�D4�� G�s ∈ S§D4ì(½M = e(s) £ù

pM ⊆M¤§¿ÀJ��&Em ∈Mux��Âì"�Âì@�Â��&E´�&�XJe��¥�3��M�¹Â��&E"�¦�ÂìU¡E G�§

z�?è5KI�÷v^�µ

e(s) ∩ e(s′) = ∅§éØÓ�s, s′ ∈ S"

n�|(S,M, E)�¡�´��@yè§½ö{¡�A-è"

��A-è(S,M, E)�±L«¤��|E| × |S|Ý§Ù¥1´d?è5KIP§�´d G�IP§¦�31e ∈ E1§s ∈ S��é(s)"

'u@yè�ïÄÌ�´8¥3z�?è5KÑ´lS�(Mc

)�ü��

¹§Ù¥c�,��ê"ù��A-code�¡�ć-©�A-è"��1-©

�A-è�¡�Ã©�A-è"�c ≥ 2§��c-©�A-codeè�¡��k©�A-

è"��k©�A-èéSimmons3©[125, 126]¥JÑ�@yè�í2��.�

~k^"

148 Aa|Ü?è¯KïÄ

3��ê�i�Ð¢ôÂ [108]¥§'<3�S�&�¥*�du�ì

uÑ�3�Ó?è5Ke�i�ØÓ�&E"'<\\��#�&E£�®u

x�i�&EØÓ¤§¿F"��Âì@�´�&�"3ù«µee§�éA-è

��°�[Ú�°O�ôÂ�´�ê�0Ú1�Ð¢ôÂ�§®²kéõïÄ

ó�§, ��êi ≥ 2�§cÙ´�c ≥ 2�§éc-©�A-è�ïÄ%é�"

G�8ÜSþ�VÇ©Ù�)(Si

)§i ≥ 0�VÇ©Ù"�½ù

VÇ©Ù§u�ìÚ�ÂìÀJEþ�VÇ©Ù§¡�?èüÑ"é?¿s ∈ SÚe ∈ E§u�ìÀJe(s)þ�VÇ©Ù§¡�©�üÑ"b�'<®²��?èÚ©�üÑ"u�ìÚ�ÂìÏLÀJ?èÚ©�üÑ5¦�'<

Ð¢�VÇ��"·�r'<^��ê�i�Ð¢ôÂØ��Âì�VÇP

�Pdi"®�Xe'uPdi�e."

·K 10.1 (Huber [92]). 3��c-©�A-è(S,M, E)¥§é?¿i ≥ 0§Pdi ≥c · |S|−i|M|−i"

��c-©�A-è�¡�´(t − 1)-�S��Ð¢�XJé?¿i§0 ≤ i < t§

Pdi = c(|S| − i)/(|M| − i)"�{²§·�¡ù«è�(t, c)-©�A-è"

Huber3©[92]¥�ÑXJ��A-è´(t− 1)-�S��Ð¢�§?è5K

�ê8�½�v�"

·K 10.2 (Huber [92]). 3��(t, c)-©�A-è(S,M, E)¥§|E| ≥ 1ct· (|M|t )

(|S|t )"

�Ä��Ç§·�I��A-è¥�?è5K�ê8��Ð"·�¡�

�(t, c)-©�A-è´�`�§XJ§U��·K10.2¥�e."

�Ù¥�Ì�(J´éc ≥ 2§t ∈ {2, 3}§�Eäkn� G��`(t, c)-©�A-è"AO�§·�y²�3Xeü�#�Ã¡aµ

(i) é?¿v ≡ 1 mod 150§v 6= 301§�3äkn� G�Úv�&E�(2, 5)-

©�A-è"

(ii) é?¿v ≡ 2 mod 8§�3äkn� G�Úv�&E�(3, 2)-©�A-è"

·��(3, 2)-©�A-è´�t > 2Úc > 1�§(t, c)-©�A-è�1��

®��Ã¡a"·��y²äkk� G�Úv�&E�(2, c)-©�A-èé?

¿¿©��v£�kÚc�½�¤Ñ´�3�"

CHAPTER 10 �Eäk©�5��@yè 149

ù�Ù�(�Xeµ3110.2!¥§·�ò0��Ä�VgÚÄ�(

J¶3110.3!¥§·��Ñ��ìC5��3(J¶3110.4!¥§·�ò

í?t = 2�§©�Aè�(J¶3110.5!¥§·��Ñt = 3�§��#�©

�Aè�Ã¡a¶3110.6!¥§òé�Ù�Ì�(J?1o("

10.2 OOO��£££

�!¥·�ò0�3e¡A!�E¥^��Ä�½ÂÚ(J"

Huber3©[92]¥½Â©�t-�O§í2Ogata�3[114]¥½Â�©

�2-�O"

½Â 10.3. -t§v§k§cÚλ´��ê§�t ≤ k§ck ≤ v"��©�t-�O§½

ö©�t-(v, k × c, λ)�O§´��|(X,A)¦�

(i) X´��v��8Ü§¡�:¶

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)§TÐkλ�«|§¦�xi§1 ≤ i ≤ t§TÐ

Ñy3z�«|�t�ØÓ�1"

5¿�©�t-(v, k × 1, λ)�O�t-(v, k, λ)�O�²;½Â��"Huber3

©[92]¥y²©�t-�O��`©�A-è��d5"

½n 10.4 (Huber [92]). �3��©�t-(v, k × c, 1)�O��=��3��ä

kk� G�§v�&EÚ(vt

)/ct(kt

)�?è5K��`(t, c)-©�A-è"

��©�t-�O�3�7�^�Xeµ

·K 10.5 (Huber [92]). �3��©�t-(v, k × c, λ)�O�7�^�´§é?

¿s§0 ≤ s ≤ t§

λ

(v − st− s

)≡ 0 mod ct−s

(k − st− s

)"

k�§��©�t-�O(X,A)�:�±�\{+Γ¥��§¦�X = Γ"

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150 Aa|Ü?è¯KïÄ

~ 10.6. -X = Z151"re¡��

A =

0 1 2 3 4

5 13 59 105 118

28 67 73 112 134

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e¡½no(�λ = 1�§©�t-�O�3��®�(J"

½n 10.7 (Du [53]§Ge�[72]§Wang [141]§Wang�[143]). 3e¡�¹e§�

�©�2-(v, k × c, 1)�O�3�7�^�£·K10.5¤�´¿©�µ

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s L«

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�^"e¡·��ÑGDD�Wilson’sÄ��E{[145, 146]��í2"

½n 10.8 (Ä��E{). -(X,G,A)´��GDD(t, k, v)"b�é?¿«|A ∈A§�3��.�ck�©�GDD(t, k′ × c, kc) (XA,GA,BA)§Ù¥XA = A ×{1, . . . , c}§GA = {{x} × {1, . . . , c} : x ∈ A}"@o�3��.�{c|G| : G ∈G}�©�GDD(t, k′ × c, vc) (X ′,G ′,A′)§Ù¥X ′ = X × {1, . . . , c}§G ′ = {G ×{1, . . . , c} : G ∈ G}§A′ = ∪A∈ABA"

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íØ 10.9. e�3.�gn11 . . . gns

s �GDD(t, k, v)§K�3.�(cg1)n1 . . . (cgs)

ns�

©�GDD(t, k × c, vc)"

XGe�3©[72]¥¤ã§·��±3©�GDD�|þW\©�2-�O5

��#�©�2-�O"

·K 10.10 (W|). -(X,G,A)´��©�GDD(2, k×c, v)"XJé?¿G ∈ G§�3��©�2-(|G| + 1, k × c, 1)�O§@o�3��©�2-(v + 1, k × c, 1)�

O"

10.3 ��333555ÚÚÚìììCCC(((JJJ

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º:TÐkλ^>�ã"-G´��vk�á:�{üã"��ê�v§�ê

�λ�G-�O´rλKv�>8y©¤�GÓ��fã"XJe(G)L«G¥>��

ê§d(G)L«G¥º:Ýê��úÏf§@oÏL{ü�O��±�Ñµ

(i) λv(v − 1) ≡ 0 mod 2e(G)§

(ii) λ(v − 1) ≡ 0 mod d(G)§

´��ê�v§�ê�λ�G�O�3�7�^�"Wilson3©[148]y²ù

7�^��´ìC¿©�"

152 Aa|Ü?è¯KïÄ

½n 10.11 (Wilson [148]). -G´��vk�á:�{üã"@o�3��

�6GÚλ�~êv0§¦�é?¿v ≥ v0§�3��ê�v�ê�λ�G-�O§

÷vλv(v − 1) ≡ 0 mod 2e(G)§λ(v − 1) ≡ 0 mod d(G)"

-Kk×cL«��k-Üã§z�Ü©kc�º:"��©�2-(v, k ×c, λ)�O(X,A)�du��ê�v§�ê�λ�Kk×c-�O§ÏLïáXeé

Xµ

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��©�2-(v, k × c, λ)�O§÷vµλv(v − 1) ≡ 0 mod c2k(k − 1)§λ(v − 1) ≡0 mod c(k − 1)"

·�^��35(J(å�!"Huang3©[91]¥�Ñy©Kv�>8

��kÜã��ê��´d(v − 1)/(k − 1)e"

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1)/k��kÜfã"dHuang�(Jù´Ø�U�§Ïdd(k − 1)c2/(k − 1)e =

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10.4 ©©©��2-��OOO

3�!¥§·�òïá©�2-(v, 3 × 5, 1)�O��Ã¡a§¿�K½

n10.7(v)¥�Ø(½��v = 385"


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�2-(v, 3× 5, 1)�O"

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Ø10.95��.�150m�©�GDD(2, 3× 5, 150m)"3ù�©�GDD�|

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·K 10.15. �3��©�2-(385, 4× 2, 1)�O"

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10.5 ©©©��3-��OOO

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a"ùp§·�k0�©�Ç��O�½Â"

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5�µ

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(v) é?¿{xi : 1 ≤ i ≤ t} ∈(Xt

)§Ù¥é?¿i§|T ∩ (S ∪ Gi)| < t§TÐ�

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��©�(t, k)Ç��O(X,S,G,A)�.´��õ8{|G| : G ∈ G}"��.�gn1

1 · · · gnrr §Z��s�©�(t, k)Ç��O§P�(t, k)-CS(gn1

1 · · · gnrr :

s)"

154 Aa|Ü?è¯KïÄ

Xe½n´©[87]¥é��Ç��O�Hartman’sÄ��E{í2�©

�(3, k × c)Ç��O��¹"

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r : s)§��©�(3, k′ ×c)-CS(mk−1 : a)Ú��.�mk�©�GDD(3, k′ × c,mk)§@o�3��©

�(3, k′ × c)-CS((g1m)n1 · · · (grm)nr : m(s− 1) + a)"

y². -(X,S,G,A)´��(3, k)-CS(gn11 · · · gnr

r : s)§�∞ ∈ S"éY ⊆ X§½

Â8Ü

P (Y ) = ((Y \ {∞})× Zm) ∪ ({∞} × Za)"

?�Ú½Âµ

S ′ = ((S \ {∞})× Zm) ∪ ({∞} × Za)§

G ′ = {G× Zm : G ∈ G}"

é?¿��¹:∞�A ∈ A§-

(P (A), {∞} × Za, {{x} × Zm : x ∈ A \ {∞}},BA)

´��©�(3, k′ × c)-CS(mk−1 : a)§�é?¿Ø�¹:∞�A ∈ A§-

(A× Zm, {{x} × Zm : x ∈ A}, CA)

´��.�mk�©�GDD(3, k′ × c,mk)"

éN´�y(P (X), S ′,G ′,A′)§Ù¥

A′ =

( ⋃A∈A:∞∈A

BA

)∪

( ⋃A∈A:∞6∈A

CA

)§

´¤I�©�(3, k′ × c)-CS((g1m)n1 · · · (grm)nr : m(s− 1) + a)"

·��±3©�Ç��O�|þW\©�3-�O��©�3-�

O"

·K 10.17. XJ�3��©�(3, k × c)-CS(gn11 · · · gnr

r : s)§Ù¥s ≤ 2§¿

�é?¿1 ≤ i ≤ r§�3��©�3-(gi + s, k × c, 1)�O§@o�3��©

�3-(s+∑r

i=1 gini, k × c, 1)�O"


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r : s)§Ù¥s ≤ 2"é

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∑ri=1 gini, k × c, 1)�O"

�A^½n10.16Ú·K10.17§·�I��©�Ç��O"

Ún 10.18. �3��©�(3, 3× 2)-CS(82 : 0)Ú��©�(3, 3× 2)-CS(82 : 2)"

y². -X = Z16§G = {{2i+ j : 0 ≤ i ≤ 7} : j ∈ {0, 1}}"-

B =

0 4

6 9

7 11

,

0 14

1 4

11 13

,

0 5

8 10

13 15

,

0 2

4 1

7 15

,

0 13

1 15

2 12

,

0 13

1 9

4 6

,

0 6

9 7

14 15

"

@o(X,G,∅,A)§Ù¥A = ∪B∈B{B + 2i mod 16 : 0 ≤ i < 8}§Ò´��©�(3, 3× 2)-CS(82 : 0)"

y3-S = {x, y}§¦�S ∩X = ∅§¿-

C =

x y

2i 2i+ 2

2j + 1 2j + 3

: i, j ∈ {0, 2, 4, 6}

"@o(X ∪ {x, y}, S,G,A ∪ C)Ò´��©�(3, 3× 2)-CS(82 : 2)"

·�ïá©�3-�O�35��#�Ã¡a"

½n 10.19. �3��©�3-(v, 3× 2, 1)�O��=�v ≡ 2 mod 8"

y². 7�^�v ≡ 2 mod 8d·K10.5��"

Huber3©[92]¥�Ñ�3��©�3-(10, 3 × 2, 1)�O§¤±·�ò�

Äv > 10��¹"Pv = 8m+ 2§é,��êm ≥ 2"-X´��m+ 1 �:�

8Ü§�¹:∞"éN´�y(X, {∞}, {{x} : x ∈ X \ {∞}},(X3

))´��(3, 3)-

CS(1m : 1)"A^½n10.16§Ñ\��©�(3, 3 × 2)-CS(82 : 2)£Ún10.18¤

156 Aa|Ü?è¯KïÄ

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�3-(8m+ 2, 3× 2, 1)�O"

10.6 (((ØØØ

�k§cÚt��§(½kk� G�§(t − 1)-�S��Ð¢��`c-©�

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¯K"3�Ù¥§·�ÏL�E©��O§��üa#�©�@yè"

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[1] H. Zhang and G. Ge, Optimal ternary constant-weight codes of weight four

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(SCI)

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176 Aa|Ü?è¯KïÄ

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position [2, 2] and distance six, in manuscript.

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