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0, 1*

ORçBe�/�ë(E8GH�yp/+��F3a��(c��

0 ∪ 0 = 0, 0 ∪ 1 = 1, 1 ∪ 0 = 1, 1 ∪ 1 = 1�`\(FJa

2.1 � �"! #(t, n)-VSSS ��« 9 � �.¬�­H®¯£�`° ��% F 2

<5*%$ Î'& ®�¯+*)(Z73h�95F(G5� �(��!� 1� *�����4I6�>$� n � * �H�.� 7 ? �)@(/Ð�BF5G5�`p!¾Ë� %)� 9�F3ak²x³ 2 µ 4I6k7����(46i�yE�F(t, n)-VSSS

*&¬­�®�¯+*�� �g7+*0VHFJan � *J���Pb*&� � 7 P = {1, 2, . . . , n}

�Ax�t7

2 ≤ t ≤ n7�á ' E',�æ���E0F!a 1 � *����(4�6b� n � * �H�.���3? �)@ #3% � i �

* ����� ��� � \ � Î£È �� ��g#3%�� ���P i �-� � % F3a (t, n)-VSSS�Iv5w�/ � � #3%F`¬�­�®�¯873h59 � ��ì #3% F!a.%/

2.1 ([1])v5w�*

2<H*�0�1g7�á ' E n

®q¯b*�$ Î2& ®¯

X0, X1

7&�(t, n)-VSSS

*3¬­�®�¯i��°�3Ba1. S

7|S| = t

7�á ' EI���b* P*�4��� � �yE0F+������1HF��æ

α > 0�

0 < d ≤ qp

5 � A �h[OR(X0[S])] ≤ d− αq (2.1)

h[OR(X1[S])] ≥ d (2.2)p � 276 <�a ' ¿+A�� Xi[S](i = 0, 1)����� %0��% Xi(i = 0, 1)

7S/�Ä�� #�% F�® �8:9+A � /0��F�®¯;�xE(FJa�M ' � h[OR(Xi[S])](i = 0, 1)

���Xi[S](i = 0, 1)

*�;�®Z7� �=<(� � OR

ç�eiA ' q�)q®?>A@ ÈB& *2CED Ç�F l �/H1)F!a

2. S7

|S| < t7`á ' E���Ð* P

*-4)��� � ��E�Fb�I�&�X0[S]

�X1[S]

���� �.\�¯b*G ¼'H V �0� 2 �JI ®¯ � \(FJa

4

��2.1*α�5Æ ÇkÈ^É�Ê È �y° �I% F�Ì É�Í.Î)Ï /�1 2 � ��I� 7 t � lHm ' �I��*3�q546+*0²!4I·£�=³04I·b* � F # *3Ó0/H1HFJa�,B× � Æ ÇkȪÉIÊÑÈ α p�Ø$�J9HÙЧx�.���� � ��(4�6(p Ú+�)� ¾)� #3% F!aZG�G&/���Ì ÉÍBÎ5Ï α ���Bä (2.1), (2.2) � 2

α = minS:|S|=t

h[OR(X1[S])] − h[OR(X0[S])]

q(2.3)��u��+�F�G�� ��� �HE5F�a q ���b*��.æ(\*I/I�ªä (2.3) � 2 h[OR(X1[S])]

�h[OR(X1[S])]*3Ó0p�Ø��I� %5� α

* µ p�Ø��b:Ò\ 2 �B���(4I6�p Ú+��� ¾�� #�% F3a h[OR(X1[S])] ≤ q><

h[OR(X0[S])] ≥ 0/�1 2 ��M ' h[OR(X1[S])] > h[OR(X0[S])]

\�*./.�α*Ð� 2  bF µ *�H���

0 < α ≤ 1/51)F!a+M ' � q �J¬­)®�¯ X0, X1

*J¯æ�/51HF!a ; ����� �Iq0*����4�6Ð*q � *3Ø$� #.� \�F.*B/B� q � � # 9�ÙZ§ �+9Z� #�% FJa

� � �2.17`á ' E (t, n)-VSSS

*3¬�­�®¯0*3¢]�BA � � [1] �I« ��F (2, 4)-VSSS*&¬­�®�¯

X0, X1

73v5w �3¥ EIaX0 =

0 0 0 1

0 0 0 1

0 0 0 1

0 0 0 1

, X1 =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

(2.4)

ä(2.4)

*&¬.­�®B¯X0, X1

*J� ���0*2®+7

ORç.e�A ' 4

��q�> @ È & *�C D Ç2F l �H7)� �' u3*B�J�J� %H�J% 1, 2�`\)F`a �J�x� � d = 2, α = 1

4 , q = 4/�� �

2.1*-0 1

17yá ' EIa �c!�

|S| = 27´á ' E.E ¼B� * S ⊂ {1, 2, 3, 4} �3¸ A � � h[OR(X0[S])] = 1, h[OR(X1[S])] = 2��\ 2 C D Ç F l � � Ó0pH15F�*�/���ä (2.3)

*α*����)ä0>$�ª�

α = 2−14 = 1

4

�&\(F(G(�p ���5/+�BF3a(M ' �I¬B­)®¯ X0, X1

*!�5� %(�I% ���Ð* 1®Ð7 »�¼ F+�ª��¯�*�� % >�V ��B����� I ®�¯i��\5F(GH�`pBnH>(F ' �B��� � 2.1

*�0�127`á ' E�a �

� �2.1/ �H� ' n×q *`¬�­�®.¯ X0, X1

7´h�9�F�G�� �)���Ò� � 1 � *��I��436)>Ð� n � * �J��B��? ��@HE5F)G)�xp/���F&a n � * ���!����? �@5E(F�_B�87&v)wB/ �(� E(F3a Xi(i = 0, 1)*�¯b*2G ¼ H V07�E+F(G(� �0���!�  �� % F q! � *�®�¯Ð*J� � 7�� %+��% Ci(i = 0, 1)�`EF!a

1 � *I����4�6k7 n � * �������J? ��@�E+F+�.�.�I����+4�6b*�4�· <(� � v5w)*���D7�� 2�� Aª®�r¡a1.���04I6b*�²!4�·�p ? �)@ #&% Fb�.��� C0

*IqX̃0

73,�� É(Ç�� �"! 3Ba X̃0

*i®

!�I« 9 � � 07)²!4I· � 1

7�³(4.· �"# ÕiA�� # Õ]A ' q � *.4.·87 i � * �H���* ¸¹ E�F�4I·£�xE+F z ' ¿bAy� i = 1, 2, . . . , n~éa

2.���04I6b*J³04�·�p ? �)@ #&% Fb�.��� C1

*IqX̃1

73,�� É(Ç�� �"! 3Ba X̃1

*i®

!�I« 9 � � 07)²!4I· � 1

7�³(4.· �"# ÕiA�� # Õ]A ' q � *.4.·87 i � * �H���* ¸¹ E�F�4I·£�xE+F z ' ¿bAy� i = 1, 2, . . . , n~éa

���04�6b*14�·+���+� %0�I% * �H����.« 9 � q � *�4I· �3? �)@ #3% F ' �.����04I6�&¸ A ������� � q � *!Øg� #�� \(FJa

5

���2.1*)0�1

1���B���Z*

t � * �H�.� 7 l0m FÐ�´��+4�6+pI5q #&% F(G��I7����;A� 9�F!a���� 2.1*�0%1

1 � 2 � OR(X0[S])*2C D Ç F l �H�

d � 2 � # : � OR(X1[S])*

C D �F l �H�dv5� � \+F*�/�� h[OR(X0[S])] < h[OR(X1[S])]

/�1(FIaOR(X0[S])

�OR(X1[S]) � 2 u���C D �F l �p � # 9H*B/ 1

z ³�4.·�~0* � �Ð*3æ+p�[0\]: � ¸ � � ²�BA � ¾�� #&% F3a�� � � OR(X1[S])�

OR(X0[S]) � 2 u 1z|³(4�·�~(* � ��æ0p�â�9 ' �� ¸ � � ³]��A � ¾�� #&% F!a h[OR(X0[S])], h[OR(X1[S])]� %���% *)C D Ç'F l �)*JÓ�pØg��9�ÙZ§ª� ����� 7 t � l5m ' �.� � �B���(4�6(p Ú+�)� ¾)� #3% F5G5� � \(FJaM ' � � � 2.1

*�0�12���

t− 1 � vHw)* ����� >$�`�����04�6 ��R A � ��� � S�T�p. � % \9+GH�!7����A � 95F&a � � 2.1*-0�1

2 � 2 � X0[S]�X1[S]

��� �I\�¯0*�G ¼'H V�0� 2 � I ®)¯£�`\0F ' �B� C0

>$� !� ¿ X̃0[S]u

C1

>$� !� ¿ X̃1[S]u��B¯Ð*2G ¼ >HV�5� 2 �JI ®¯ � \5F&a t− 1 � v)w�* ����� >��`�I� X̃0

7�h59 ' *B> X̃1

7�hH9 ' *.>.�� �5p<�>�\�9�aIs � � h[OR(X0[S])] = h[OR(X1[S])]p � 2 6 < ' �I� ���.� t− 1 � v)w(7l5m � u��$§��b��p+²!/k§��b��p�³+\�*�>2�b*� ���p<�>¦I�B���04�60p�Hq #3% \�9�a

2.2 � �"! #�� � �����

2.2.1 ��� �"!�#%$'&%(� �

2.17yá ' E (t, n)-VSSS

*3¬B­)®¯��I�!â.ã�äç�e �(���3�  £� % F [2]a!â�ã�ä�>k�

(t, n)-VSSS*!¬�­�®�¯87& ÐF ' � � �8G!**)�/5����®�¯i� � ãä���* ¸¹8R�++� <�9 � �0�F3a

17i � � 0

7n− i � �0<�¯�>:@ È & v = [0, 0, . . . , 0, 1, 1, . . . , 1]T

7 �)V5F!a¯%> @ È &v*�E ¼�� * � �0*�G ¼ >�V57�¯ � u�< n

®n!¯(*3®�¯87

Mn,i

�´E(F&a�¢;�.A� � M4,0

�[0, 0, 0, 0]T

74!¯Zu�<J®B¯�/�1 2 �

M4,1,M4,2

�J��� %)�J% [0, 0, 0, 1]T , [0, 0, 1, 1]T*IE ¼�� * � �+*�G ¼ >V57�¯ � u`< 4× 4!®�¯H/�1)F3aJ®�¯

Mn,i

7 � ã�ä0/ an−izi�¡ë�EG)� � E(F&a(GG&/�� a � 0

z ²J4�·~ � � z � 1z ³54�·�~ �3¸�¹ A � 9HF&a M4,0,M4,1,M4,2��� %(��%� ãä+/ a4, a3z, a2z2

�^ë(EIa� � ��¯%> @ È & v

*��-,�\5F Á ·�*�G ¼'H V�7�¯ � u`< n®(n

i

)¯�*!®�¯87

Nn,i

�xE(FJa¢]�A � �B®�¯ N4,0, N4,1, N4,2

�)� %0��%N4,0 =

0

0

0

0

, N4,1 =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

, N4,2 =

0 0 0 1 1 1

0 1 1 0 0 1

1 0 1 0 1 0

1 1 0 1 0 0

(2.5)

��\�F3ai = 0, 1, . . . , n �&¸ A � �3®�¯ Mn,i

�3®�¯Nn,i

7(n− i)!i! �/./0 A ' ®�¯]�y\HF&a Mn,i

�� ã�ä(/ an−izi

� ë;A �B« 2 � (n− i)!i! · an−izi

(n−i)!i! = an−izi\)*�/��

Nn,i

� � ãä(/ an−izi

(n−i)!i!

�ë5E8G)� � E(F3a`ä (2.5)*!®�¯

N4,0, N4,1, N4,2

����� %5��%�� ã�ä0/ a4

4! ,a3z3!1! ,

a2z2

2!2!

�¡ë)o0F3a

6

M ' �®¯b* .0 7��B� �/IëiA��Bâãä8*���� ¸�¹g# o+F3a�¢]�A � � N4,0 �N4,1

N4,0 �N4,1 =

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

��\ 2 ��â�ãäa4

4! + a3z3!1! �3¸�¹�# obFJa

Mn,i

* � ã�äÐ�.� Nn,i

* � ã)ä�7 (n − i)!i! � A ' äb\5*B/�B®�¯ Mn,i

>;�Ò®�¯Nn,i

Q* # �0� Nn,i

>i�Mn,i

Q0* # �8�B�â�ãHäk*.d�eb/���� � /Z��FIa��+c���®)¯ M4,0

��®¯N4,0

74! �".0 A ' ®�¯0/(1 2 (a4 = 4! · a4

4! )�M4,1

��®�¯N4,1

73!1! �".�0 A ' ®�¯

(a3z = 3!1! · a3z3!1! )

/51)F�*./��Bâãä)ç�e0/��� � # �0/b�BFJa� � ����� ������ ÎHÏ ψ = ∂∂a

+ ∂∂z

� � �����Ja��B��� � ã�ä�� k � ψ���� �! �"$#&%

'&(*)+��,�-�����.�/�02143�5��k.�687 ! .�/��:9<;����:=�>$?*@&%�'&(�)

a2z2

2!2!

�ψ�

1

� � �BAC% az2

1!2! + a2z2!1!

AED���=az2

1!2! + a2z2!1!

��,�-��8��.�/$#&%

0 0 1 0 1 1

0 1 0 1 0 1

1 0 0 1 1 0

(2.6)

A�D���=F)(2.5)

��.�/N4,2 G8H %�) (2.6)

#a2z2

2!2!

��,�-����I.�/�0B11.<687 ! .�/$�JDLK

M 7���N�A�O�P*08��=Q")�R&S��OJ%�T�U2.1�WV$! �

(t, n)-VSSS�:X�Y�.�/$#&%�.�/

Mn,i Z ! #:.�/ Nn,i

�./0 �! .�/[A %:\�]����LNA�O�R8^J�F=8N�N�R�% Mn,i, Nn,i

� ./0 F! .�/���,��8�W_J()��F`*a �Jb�c�d�c

H+n =

{

n∑

i=0

γian−izi : γi ∈ Z, γi ≥ 0,∀i = 0, . . . , n

}

,

K+n =

{

n∑

i=0

γian−izi

(n− i!)i!: γi ∈ Z, γi ≥ 0,∀i = 0, . . . , n

}

Ae�L��A4%e.J/Mn,i

� .�0 R:fg1 c �E.</L�I,<-*�L��_�(<)�# H+n

��%e.</Nn,i

�IhJiLR:f21c ��.�/8��,�-L����_J(�)�#

K+n

��j�����=H+

n ,K+n

��T�U�0B1k%Fl�mγi ≥ 0 , i = 0, 1, . . . , n�En� �! `*a �Jb�c�d�c %

Hn,Kn

Akoqpr=

2.2.2 s+tvu�wyx�z�{g|~}�{v�$�+������N&���8R#&%�_�(�)����80B14T�U2.1�WV$! �

(t, n)-VSSS�:X�Y*.�/ �E�8� ����� �W�8�

���:=JT�U2.1#

(t, n)-VSSS�FX�Y.�/���,�-��8��_�(�)+#&%*���� G�� D&��� ���� =

7

���2.1 [2] (t, n)-VSSS

�IXJY<.</ �Y0, Y1

A %Y0, Y1

# b�cdFcMn,i

��h<i A M T � �W=f0

#�.�/Y0

�W,<-����W_�(�)%f1

#�.�/Y1

�I,�-��L�I_J(<)�A ���W=��*�����C0, C1(C0 >

C1)O�� M

ψn−tf0|z=0 = C0at, ψn−tf1|z=0 = C1a

t (2.7)

ψn−t+1f0 = ψn−t+1f1 (2.8)O:] H� � = Z ! % Nn,i � h�i8R�� F! X�Y*.�/�� X0, X1

A %g0#�.�/

X0 � ,�-����W_(�)�%g1#�.�/

X1 � ,�-�����_<(*)�A�����A %ψn−tg0|z=0 = C0

at

t!, ψn−tg1|z=0 = C1

at

t!(2.9)

ψn−t+1g0 = ψn−t+1g1 (2.10)O�] H� � =)(2.7) ��� 7 M % ψn−tf0, ψ

n−tf1

#�% b�c�d�c X�Y�.�/Y0, Y1 � 3�5 � n− t

.��E687 ! ./ � 3�5 � t

.��� "! H$# F! .�/�% � ,*-��&��_�(*) � D��F= C0, C1

#&% bJc�d�c X�Y�./Y0, Y1 � 3�5 � t

. �'� 7 M'( Z c �W]*)�O*��+ M 0AED��

Mt,0 ��, �8R-�-�:= C0 > C1� V+! �vNLAIR<%<X�Y�.*/Y0, Y1

b&c8d&c � 3�5 � t..�

OR��� :!

qQ0/-1�24365 �'78�9;:=<&> �"? O'@4A�� � R<%<) (2.7)

#:T�U2.1 � lm 1 � ,*- M 7��:=J) (2.9) ��� 7M �CBCD � % C0 > C1

� V�! ��N�A�R<%&X�Y.�/X0, X1

b&c�d�c � 3�5 � t.E�

OR���

!qQ-/�1�2F3�5 �'7 8*9�:�<&> �"? O@GA'� � R<%<T�U 2.1 � l*m 1 � ,*- M 7��F=)(2.8)

#:X�Y.�/Y0, Y1 � 3�5 � t− 1

. � .�/ � ,�-0����_<(�)8OH 7vAI7 � N�AIR���F=:3�5 � t− 1. �$IEJ D�/ �CK +�L�?�� �! .�/ � ,�-0���W_<(�)�#MH 78N�A�# � 10D � R�%�) (2.8)#:T�U

2.1 � lm 2�WS M 7��:= B�D � ) (2.10)

#J%JX*Y�.�/X0, X1

� 3�5 � t− 1. � .�/ � ,�-��N��_�(�)$OCH 7 � R�%JT�U 2.1 � l�m 2 � ,�- M 7���=

Z ! %�) (2.7), (2.8) ��� 7 M f = f0 − f1

A � p�AC% ψn−t, ψn−t+1 ��OQP �80B1 f # Ft,n

� j����:=vN*N �Ft,n =

{

f ∈ Hn : ψn−t+1f = 0, ψn−tf |z=0 = Cat (C > 0)}

(2.11)A4T�U :! =SR�1 � % Ft,n

O�Q ��P R�o� ��LNLA � SFR c M 7�� [2]=

Ft,n =

{

n−t∑

i=0

βie(i)t,n : β0, . . . , βn−t ∈ Z,

n−t∑

i=0

βi > 0

}

(2.12)

N�N �e(i)t,n = an−t−izi(a− z)t (i = 0, 1, . . . , n− t)A4T�U :! =

)(2.12) ��� 7 M % ∑n−t

i=0 βi > 0� Vy! �;T-�

βi(i = 0, 1, . . . , n − t)��35 �'U�V R�%

f =∑n−t

i=0 βian−t−izi(a− z)t

A � 7 M )�WJ������A %�_<(�) f#�)

(2.11) � l�m�)E� V�! �&=��� � % f ∈ Ft,n

R��-�f�$XL7 M

(t, n)-VSSS � X<Y*.�/��I\�]0���W���E��S0� ([2], [7])=

8

1.∑n−t

i=0 βi > 0� V�! �"T'�

βi(i = 0, 1, . . . , n−t)�e3&5 ��U-V R:% f =

∑n−ti=0 βia

n−t−izi(a−

z)tA � pr=

2._�(*)

f�f =

∑ni=0 γia

n−izi �MPE����� % f = f0 − f1

AID�� G��"� %����8O'� �( ��� O f0

%��;�$O � ( ��� O f1

A4T � �:=3. G =gcd{γi(n−i)!i! : i = 0, 1, . . . , n}

A T � %g =

∑ni=0

γi(n−i)!i!G

an−izi

(n−i)!i!

A � p = gcd{u0, u1

, . . . , ui}#u0, u1, . . . , ui ���� ���� �$R��0��=

4. g = g0 − g1AID�� G��"� % g � ( ��� R��Q�8O'� � ( ��� � g0

%��;�$O� � ( ��� �g1AkT � �F=

g0, g1 � ,�-��&��.*/�� X0, X1

A�����A %�.*/X0, X1

O(t, n)-VSSS �X�Y*.*/[AWD0�:=

t, n�

2 ≤ t ≤ n� V+! �J3�5 � T���A % βi(i = 0, 1, . . . , n− t)

�∑n−t

i=0 βi > 0� V+! �

TQ��A��&�8AJ^I%f ∈ Ft,n � ,*-0����X�Y�.�/ ��� 9 3��� 3 α #

α =

2n−t∑

i=0βi

(

n

t

)

‖f ‖

(2.13)

R���?21 c �[2]=vN*N � ‖f ‖

#�%f =

∑n−ti=0 βie

(i)t,n

�f =

∑ni=0 γia

n−izi ��PE����� :! A^ � �Q� ��� ,�� ��� %���D�P! ‖ f ‖=∑n

i=0 |γi|A�" M T�U R c �:=�)

(2.13)#&%

α � TU�)(2.3)

�W_�(�)f ∈ Ft,n

�MX�7 M og^F0*?�# � � R-�-�F=>�A�" M %(2, 4)-VSSS � X*Y./S��_�() f ∈ F2,4

0g1e\�]$" M-> �:=β0 = 3, β1 = 2,

β2 = 1A�����A %

f = 3a2(a− z)2 + 2az(a− z)2 + z2(a− z)2

= 3a4 − 4a3z + z4 (2.14)

= 4!

[

3a4

4!−a3z

3!1!+z4

4!

]

(2.15)

ACD-��=�)(2.14) G�H % f0 = 3a4 + z4, f1 = 4a3z

A�D���= Z #�%I) (2.15) G�H g0 = 3a4

4! + z4

4! ,

g1 = a3z3!1!

A � p�A % g0, g1 � ,�--����.�/�# b:cLd�c ) (2.4) � X<Y*.�/J) X0, X1

AED0��= �9 3��� 3α#�)

(2.13) G8H % α = 2·(3+2+1)

(4

2)(3+4+1)= 1

4

AED���=

2.3 %'& (*),+.-0/2143 5$687:9<;=1?>=@=ACBED4FEG (PBVSSS)Q�H ��I�J AK" M %�N � ��R#:%�L�M �ON�P�Q O�R*/�R c �TS�ULD�V�W�RKX�YKZQ)K[�� � �7 M ��� ����=�L\M ��N�P�Q OR�/�R c �8AI#&% t ] ��^�_\` � <�a #+A&^ ��NbPQ O\L\M �R�X$R$^���N�A�A B 58R-�-�F= NbP�Q ��L\M � R�X�R+^�� (t, n)-VSSS�dc�' � # � % (t, n)-

PBVSSS (PB:Perfect reconstruction of Black Pixels)Aeo p =

(t, n)-PBVSSS#J% N!P�Q ��L�M

9

� R�/�����# � %TR�/ P�� O���W "��*�*7�A�7 ����� O�� H R Z���Z D� ��LO&.*P c M 70�[7, 8, 9, 10]

=Blundo

1[8, 9]

R�� n ≤ 10 ��� a � α� �� �� �N�+AW7 � 5���R ����I����� >�M 70�F=

[7]R�� _�(�)*��� ����� 0B1����:" M �[H 2 ≤ t ≤ n

R � (t, n)-PBVSSS

� X�Y*.*/��=@�]�����_<(*)S�IS:" M 7���=(t, n)-PBVSSS

� ^�_�` � t ] < a �$A N PKQ OKL�M � R0/.R c � S�U$D (t, n)-VSSSR��0��=PBVSSS � X�Y*.�/ � T�U � (t, n)-VSSS � T�U 2.1

l*m1 � d = q

AID��8AJ^�R0���=-�*D�Pb (t, n)-PBVSSS � T�U��FT�U 2.1

lm1 � ) (2.1), (2.2)

� b�c�d�ch[OR(X0[S])] < q

h[OR(X1[S])] = q (2.16)

A"! c 0�?�# P R-���F=F) (2.16)� X�Y.�/ X1 �'� 7 M 3�5 � t

.��OR���:"�#

qQ*/

1N2 3 5 ��7 8�9':�<�> O X<Y�.�/ � /�� qA H "�pkD-�NAC�C5#� " M 70��=

OR(X1[S])

�'7 8*9;:�<�> O�X�Y.*/ � /Q� A H$"�7+NLA$� 3�5 � t..�

OR�*�$"�#

qQ0/�1�243

5 � ��+ M � ]*)�O 1� N�P�Q %*R��-� � R 3<5 � t ] ��^�_�` � <�a #+A�^ �� Y�Z P��

��N P�Q ��L�M � R�/.R c ��=%(t, n)-PBVSSS � X�Y�.�/ � >.���L� � S��&= (2, 4)-PBVSSS � X�Y.�/ X0, X1

X0 =

0 1 1 1

0 1 1 1

0 1 1 1

0 1 1 1

, X1 =

0 1 1 1

1 0 1 1

1 1 0 1

1 1 1 0

(2.17)

A�D���=�X�Y�.�/X1 �'� 7 M 3�5 � 2

.��OR��� "�#

4Q�/�1&2 365 � ]�) �'��+ M 1

D� R 7 8*9�:=<&> � 4

AWD H X*Y�.�/ � /Q� 4A H*"$p D��:= ^�_K` � 2 ] <�a #�AJ^OYZ P&�&��N�P\Q OKL\M � R-/ER c �:= '

Z # Mn,i � h*i8R�(�FR c � (t, n)-PBVSSS � X�Y.�/ Y0, Y1 � ,�-�����_<(*)S� b�cd�cf0, f1

A�"�#�A�^ �-�=�Q� C0

O�Q�*" Mψn−tf0|z=0 = C0a

t, ψn−tf1|z=0 = 0 (2.18)

ψn−t+1f0 = ψn−t+1f1 (2.19)

O�] H � =WN�N�R ) (2.18)�WT&)

2.1 � ) (2.7) � C1 = 0A�" # P�� DJK M 7*�E= ψn−tf1|z=0

�$*,+�� at � (�O.-vK M 7��8A 3�5 � t.E�

OR���*"�#

qQ�/-1&243 5 � �*+ M � ]�)O

1 � D21�D*7�# � �N�P\Q OKL\M � R�/.R c D�7<=)(2.18), (2.19) G8H/ f = f0 − f1

A � pJA �� ��G�� D&T�)$O�] H� � =���2.2 ([7]) f = f0 − f1

A � p�A ψn−t, ψn−t+1 ��OQP ��021f ∈ Pt,n =

{

Hn : ψn−tf |z=0 = Cat (C > 0), ψn−t+1f |z=0 = 0}

10

A4T�U R c �:=Pt,n

��Q ��GL�"� oE ���N�AIO�� 1 c M 7���=

Pt,n =

{

n−t∑

k=0

δk p(k)t,n : δk ≥ 0,

n−t∑

k=0

δk > 0

}

N�N �p(k)t,n =

n−t∑

i=k

(

t− 1 + i− k

t− 1

)

an−t−izi(a− z)t (0 ≤ k ≤ n− t) (2.20)

A4T � �:= Z # ) (2.20) ����� ).� p(k)t,n =

∑ni=0 γia

n−iziA�����A

γi =

1 (i = k)

0 (0 ≤ i ≤ n− t, i 6= k)

(−1)i−n+t

(

i− 1 − k

n− t− k

)(

n− k

i− k

)

(n− t+ 1 ≤ i ≤ n)

A oE ��:=_<(�)S� ��� "�#�)�� an−kzk � �;�8O 1

R an−kzk � (L� n ��� 1(8021

� n− t( Z R�;�$O

0 � D��8AW7 � S����<D�) � D�K M 70�:=>�A\" M (2, 4)-PBVSSS � X�Y�.�/���@�] " MQ> ��= k = 0A\" M )

(2.20) ��� !-����A p(0)t,n =

2∑

i=0

(

i+ 1

1

)

a2−izi(a− z)2

= a4 − 4az3 + 3z4

= 4!

[

a4

4!−az3

1!3!+ 3

z4

4!

]

A�D H g0 = a4

4! + 3 z4

4! , g1 = az3

1!3!

A�D��&=g0, g1 � ,-&�&��XYL./ X0, X1

� b<c$dJc )(2.17) � X0, X1

AED���=

11

�3 � � � � � � � � � � � � � ���� � � � � � � � � � � �

3.1 � "!4(^�_�`$# t ]&% a('*) ^,+.-0/21436571 α #98;:(�(<>=@?BA.CBDFEHGBIBJ +$KMLON QP6R�6RTS (�(U S>POV NBW = [1, 2, 3, 4, 5, 6] X LOY"L n U6Z,[ � : ^HW (n ≥ 5 S*\ ) �$]>E(t, n)-VSSS ^ � 'F)`_ -$/Q1a3b571 α #c8B:(�.<d=@?&A$CeD"#Of W 'g) L@NFh9i(j;k6l #t me%.n '") ^b+Hoqp V>=brtsFu&vxwaSxy iHz;{|L 'H}&~ r�s Ut�;�xLH+ w W ) WQ���&�(U;�= X0�d�O^ i %(n = jBk,l E me� # t+ 1 m.�.� +@�0�2L9NTibz;{|L '�}&~ rbs Ub�>�B+@�B�P@VF= (t, n)-VSSS #b� _ = XRb� i�j6kHl # t+1 m.�B�O%$n 'F) ^ E zB{ rHsdE -;/g1�3H5�1 #��,�.<0= X t # 2 ≤ t ≤ n#c� ' <b� � ) Lci s # t ≤ s ≤ n #`� ' <�� � ) <d= X �e� 2.1 #c� ' L Nn,i E@�e� ^H� PVM= (t, n)-VSSS Et?&A.CeD"# X0, X1

) Lci6� ] S # S ⊂ PYB�

|S| = s #9� ' < � ] ) L'g) ^9iFj;k&l # s mB%0n 'g) ^ E -$/21a365�1 αs #αs = min

S=|S|=s

h[OR(X1[S])] − h[OR(X0[S])]

q(3.1)) �>�d= X h[OR(X1[S])] ≤ q

Y;�h[OR(X0[S])] ≥ 0 S0E ^ i>j$kTl # s mB%(n '*) ^ E -/"143b5�1

αs EH�T�e  (3.1) ¡ y i αs E ) y o =@¢MEO£>¤;u i αs ≤ 1 ^ � = X -;/*143b5¥1U : ^,¦ Vd§ j.k,l # %dn '") ^6+�z${ P�V>= }&~ r�s U6�F�;+O�$� P�VF= ' � i*-$/21435�1¨U1+@©Mª wc«g\a¬ W ) P�VM= X­*+�iMj$kTl # s mB%dn '*) ^ E -./214365®1 αs #Of WeN&i %Mn = j$kTl E me� # t + 1m$�.� +9�.� < ) t m&%.n 'g) ^ ¡ y �M�e+ }T~ r�s UtzB{ P9Vd=`¯;] U;� = � ) #�°(< XH±) LON,i[1] ^b² _ p V ' (2, 4)-VSSS Et?&A.CeD

X0 =

0 0 0 1

0 0 0 1

0 0 0 1

0 0 0 1

, X1 =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

(3.2)

+,�$WBN � _ = X ?BA.C;D X0, X1

+T³.WeN&i   (3.1) ¡ y αs(s = 2, 3, 4) #�´&µd<>= ) � V¶ V α2 = 14 , α3 = 1

2 , α4 = 34

) S(= Xg� E ) ^ αs = s−14 (s = 2, 3, 4)

) Sd· NeW = ' � i %n = j;k&l E me�M^ � = s EH¢Q#9: ^ w¸<&Vd§ i s m&%(n '") ^ E -0/*1¹3b571 αs

U : ^ wSxy iTz${xL '*) ^,+ }B~ r6s U6�>�$+@�$� P�VM= X12

  (3.2) E (2, 4)-VSSS E�?&ABCeD X0, X1 #9f WeN�� I L ' j&k�le+b³$WTN,i %.n = jTk�lE mB� #�� _ N6z;{ L '") ^ E,r6s2#�� 3.1+ °M< X � 3.1 E (a) ¡ y i>j$kTl 1 md^ u }&~rbs +$KMLON���� �>U�� V N$W S W � ) U�.Y = X R ' i � 3.1 E (b), (c), (d) ¡ y 2 m.�� j.k&l # %(n = ) ^ u iFj$kTl # %(n = mB� U��dW «*\ z.{ r�s Ub�g�;+��$� P�VM= � )U�.Y = Xj.k&l # %(n '") ^ E z.{ r6s U��g�;+��B� P�V>= ' � + u iFj$kTl # s m&%dn '") ^ E-$/Q1a3�5�1

αs Et¢*# ^ ^ =� ¦ 1+�©(ª0¦ =���� U$� = X0�d�O^ i(j;k,l # s m&%(n '")^ E -$/*1a3b5�1 αs E�¢ U���� : ^ w¸S0= ^ ���F� i %(n = jBk,l E me� U n m E ) ^ E z{ r6s>E -$/21a3�571 αn E�¢ U ^ ^ =� ¦ 1

+O©dª w ¡ � S (t, n)-VSSS+����TU.� = X$��Ti

t Et¢*#��t� L '") ^ E j.k,l # n mB%0n 'g) ^ E -0/*143�571 αn

U&in #! #"b: +FL'F) ^�+

1+9©dª w ¡ � S (t, n)-VSSS EO?eA;C&D*#t� _ = XT��$ i.jek,l # n mT%0n 'g) ^E -0/2143�571 αn

U,in #! #"b: +>L '*) ^6+ 1

+O©Mª w (t, n)-VSSS Et?&A.CBD*# i&%&©8B¬�'*# h@� (t, n)-VSSS EH?eA$CBD ^ � = ))( � � ) + <>= XR6� i (2, n)-VSSS+6³$W&N&iMj.kTl # n me%(n '") ^ E -$/21a365�1 αn

U�%B© 8$¬ + S=O?BA.C$D +,�0WeN � _ = X j;k&l # n m;%dn 'Q) ^ E -0/Q1�36571 αn

U�%e© 8.¬ + SM=?TA$CeDMu i+* u [1] ^t² _ p V ' (2, n)-VSSS EO?eABCeD � E h E ^ � = X [1] ^H² _ p V '(2, n)-VSSS E�?BA;C;D X0, X1 u

X0 =

1 0 . . . 0

1 0 . . . 0...

.... . .

...

1 0 . . . 0

, X1 =

1 0 . . . 0

0 1 . . . 0...

.... . . 0

0 0 . . . 1

(3.3)

)-, ¦ = X   (3.1) ¡ y   (3.3) EO?eABCeD X0, X1

+b³$WTN,in mT%0n '") ^ E -$/"143b5¥1u αn = n−1

n^ � y iM�"p`Y.+ n→ ∞ ^ αn → 1

) S(= X$��$ + 2.2.2 . E�  (2.12) E+/ �0 µ Yxpai   (3.3) E (2, n)-VSSS EH?eA$C;D U,oqp VF= � ) #9°d< X n # n ≥ 2 #@� ' <b�� ) L�i f∗2,n ∈ F2,n

) L@NTiβi = n− i− 1 (i = 0, 1, . . . , n− t) #�132 )

f∗2,n =n−2∑

i=0

(n− i− 1)an−2−izi(a− z)2 (3.4)

= (n− 1)an − nan−1z + zn

= n!

[

(n− 1)an

n!−

an−1z

(n− 1)!1!+zn

n!

]

) S(= X   (3.3) E�?&A;CBD X0, X1

+54&6 <d= �7   g0, g1 u � V ¶ V g0 = (n− 1)an

n! + zn

n! ,

g1 = an−1z(n−1)!1!

) S0= � ) U�.Y = X3.2 8:9<;>= s ?A@CBED FHGJILKCM NPORQ Nt ≥ 3 E@¯$]"#b� _ =�S +bi � E .M^ u i(j;k6l # s mT%(n '")UT E z;{ rbs>E -$/Q1a3b51αs #9�&  V(<d= X αs u i   (3.1) E ¡ �O+ �&�xP@V>= UTi���7  2#�W*· N �X$ E ¡ �@+ �

13

(a) (b)

(c) (d)

� 3.1 (2, 4)-VSSS E j.k,l # %dn '*)�T E&r�s X (a)j$kTl

1 m E )�T X (b) ��� E j.kTl 2m&%dn '*)UT X (c) ��� E j$kTl 3 me%(n '")�T X (d) ��� E j$kTl 4 me%(n '")�T X

14

� <>= � ) U ^ T = X���3.1 t, n # 2 ≤ t ≤ n #�� ' < � � Eb� � ) LOi s # t ≤ s ≤ n #H� ' < � � E6�� ) <g= X R ' i βi (i = 0, 1, . . . , n − t) u � V ¶ V i ∑n−t

i=0 βi > 0 #t� ' <T� � ) <g= Xf =

∑n−ti=0 βia

n−t−izi(a− z)t) LON,i

f ∈ Ft,n

+�4�6 <>= (t, n)-VSSS EH?eA$C;D X0, X1

Yp � I L ' jek6l E ���Hi ��� E s(t ≤ s ≤ n) m E jBk6l # %0n '")�T E -$/"143b571 αs uαs =

2n−s∑

i=0

(

n− t− i

s− t

)

βi

(

n

s

)

‖ f ‖

(3.5)

) , w � ) U ^ T = X��� jek�l # s m&%.n '")�T E z;{ r�sME -;/Q1¸3b5�1 αs u�  (3.1)+ ¡ y �T�|P9V NBW = X  (3.1) E��gu i (t, n)-VSSS EH?&A;CBD X0, X1 E ��� E s C"# OR0 µ L9N�o�p V ' q

­{ ���x1�� E���� /�� %�� E�� ^ � = X   (3.1) E���FEt¢>u i f ∈ Ft,n

) LON,iψn−sf |z=0#��d� '>)UT +Oo p V(= as E � � +"!xL@W X RH� i ψn−sf |z=0 #��d�$= ' � +�i ψn−se

(i)t,n (i =

0, 1, . . . , n− t) ##�d�0= � ) + <M= X ψn−se(i)t,n #$�M�.=�% +ti ��$ E 3�&�')(+*-, E/.T  U �� ) S0= X u1, u2 # ��� E �7   i k # k ≥ 1 #9� ' < ��� E�� � ) L '")UT +�id3�&�'�(+*, E.&  ¡ y

ψk(u1u2) =k∑

j=0

(

k

j

)

(ψk−ju1)(ψju2) (3.6)

)  ���0 ^ T = X$��$ i   (3.6) #�f WeN ψn−se(i)t,n ##�>�d= X i # 0 ≤ i ≤ n− t E ��� E��� ) <d= ) i   (3.6) #Of WeNTi

ψn−se(i)t,n =

n−s∑

k=0

(

n− s

k

)

{

ψn−s−k(an−t−izi)}{

ψk(a− z)t}

(3.7)) , ¦ = X"�;�O^ i k ≥ 1 E )UT ψk(a− z)t = 0) S0=&E ^ i   (3.7) E�1�2Fu k = 0 E 7 �¦6U43 y

ψn−se(i)t,n =

{

ψn−s(an−t−izi)}

(a− z)t (3.8)) Sd= X   (3.6) #Of WeNTi   (3.8) #��50d<M= ) iψn−se

(i)t,n =

n−s∑

k=0

(

n− s

k

)

(ψn−s−kan−t−i)(ψkzi)(a− z)t (3.9)) Sd= X   (3.9) u i n− s− k > n− t− i E )�T u ψn−s−kan−t−i = 0) Sxy i k > i E )�Tu ψkzi = 0

) Sd=,E ^ i   (3.9) uψn−se

(i)t,n =

min{n−s,i}∑

k=max{0,t−s+i}

(

n− s

k

)

(ψn−s−kan−t−i)(ψkzi)(a− z)t (3.10)

) , w � ) h ^ T = X$��$ i 4 6 y + ¯;]� ¦ <d= X15

1. i+ t− s ≥ 0Ye�

n− s ≥ i E )�T  (3.10) ¡ y i�� # ) =O¢>EH£M¤.u i+ t− s ≤ k ≤ i) Sxy i

ψn−se(i)t,n =

i∑

k=i+t−s

(

n− s

k

)

(n− t− i)!

(k + s− t− i)!as+k−t−i i!

(i− k)!zi−k(a− z)t

=i∑

k=i+t−s

(n− s)!(n− t− i)!i!

k!(n− s− k)!(k + s− t− i)!(i− k)!as+k−t−izi−k(a− z)t

=i∑

k=i+t−s

(n− s)!i!

k!(i − k)!·

(n− t− i)!

(n− s− k)!(k + s− t− i)!as+k−t−izi−k(a− z)t

=i∑

k=i+t−s

(n− s)!

(

n− t− i

k + s− t− i

)(

i

k

)

as+k−t−izi−k(a− z)t (3.11)

) S0= X   (3.11)+z = 0 #!/��d<d= ) i k 6= i E 7 uT<�� N 0

) S0=&E ^ iψn−se

(i)t,n|z=0 = (n− s)!

(

n− t− i

s− t

)

as (3.12)

U I�y�� � X2. i+ t− s ≤ 0

Ye�n− s ≥ i E )�T  (3.10) ¡ y i�� # ) =O¢>EH£M¤.u 0 ≤ k ≤ i

) S y iψn−se

(i)t,n =

i∑

k=0

(

n− s

k

)

(n− t− i)!

(k + s− t− i)!as+k−t−i i!

(i− k)!zi−k(a− z)t (3.13)

) S0= X   (3.11))�� +bi   (3.13) #c ���0M<d= )

ψn−se(i)t,n =

i∑

k=0

(n− s)!

(

n− t− i

k + s− t− i

)(

i

k

)

as+k−t−izi−k(a− z)t (3.14)

) S0= X   (3.14)+z = 0 #!/��d<d= ) i k 6= i E 7 uT<�� N 0

+ S0= ' � iψn−se

(i)t,n|z=0 = (n− s)!

(

n− t− i

s− t

)

as (3.15)

U I�y�� � X3. i+ t− s > 0

Ye�n− s < i E )�T  (3.10) ¡ y i�� # ) =O¢>EH£M¤.u i+ t− s ≤ k ≤ n− s

) Sxy iψn−se

(i)t,n =

n−s∑

k=i+t−s

(

n− s

k

)

(n− t− i)!

(k + s− t− i)!as+k−t−i i!

(i− k)!zi−k(a− z)t (3.16)

16

) S0= X   (3.11))�� +bi   (3.16) #c ���0M<d= )

ψn−se(i)t,n =

n−s∑

k=i+t−s

(n− s)!

(

n− t− i

k + s− t− i

)(

i

k

)

as+k−t−izi−k(a− z)t (3.17)

) , ¦ = X�� � ¡ y i− k ≥ i− (n− s) > 0 S.E ^ i   (3.17)+z = 0 #!/��d<d= ) i

ψn−se(i)t,n|z=0 = 0 (3.18)U I�y�� � X

4. i+ t− s ≤ 0Ye�

n− s < i E )�T  (3.10) ¡ y iψn−se

(i)t,n =

n−s∑

k=0

(

n− s

k

)

(n− t− i)!

(k + s− t− i)!as+k−t−i i!

(i− k)!zi−k(a− z)t (3.19)

) , ¦ = X   (3.11))�� +bi   (3.19) #9  ��0(<>= )

ψn−se(i)t,n =

n−s∑

k=0

(n− s)!

(

n− t− i

k + s− t− i

)(

i

k

)

as+k−t−izi−k(a− z)t (3.20)

U,oqp VM= X�� � ¡ y i− k ≥ i− (n− s) > 0 S$E ^ i   (3.20)+z = 0 #�/��(<M= )

ψn−se(i)t,n|z=0 = 0 (3.21)) S0= X

¡ · N6i ψn−sf |z=0 =∑n−t

i=0 βi(ψn−se

(i)t,n)|z=0 S;E ^ i ¯B]5 ¦ # L ' 4

� E�  (3.12), (3.15),

(3.18), (3.21) # � V ¶ Vg#�/ �0<>= ) iψn−sf |z=0 =

n−s∑

i=0

βi(ψn−se

(i)t,n)|z=0 +

n−t∑

i=n−s+1

βi(ψn−se

(i)t,n)|z=0

=n−s∑

i=0

βi(n− s)!

(

n− t− i

s− t

)

as (3.22)

) S(= XF�e�@^ i h[OR(X1[S])]− h[OR(X0[S])] E����>u�  (3.22) E as E/� � + s! #��� 'h E +�!qL@W X R ' i ?&A.CeD>EHD � # �7  |#@f W&N � � <d= ) i [2] ¡ y ‖f‖2 n! ^ � = �) U&�;Y · N;W = X jek,l # s mT%(n 'g)�T E -./*143b5�1 E��e�&  (3.1)

+ti � V p E���#�/ � LON,i

αs =

n−s∑

i=0

βi(n− s)!

(

n− t− i

s− t

)

s!

‖ f ‖

2n!

17

=

2n−s∑

i=0

βi

(

n− t− i

s− t

)

(

n

s

)

‖ f ‖

) Sd= X2¡ · N,i���� 3.1U °xPOV ' X �

  (3.5) E jBk6l # s m&%(n 'g)�T E -./"1a3�5�1 αs E�.&  +�³;W&N,i s = t E@¯;] ^ � =αt u α

) � � ^ � = X * % i   (3.5) E αs E/.e  + s = t #�/ �(<>= ) i αt =2∑n−t

i=0βi

(n

t)‖f‖)

S0= X R ' i   (3.5) E αs E.&  +6³.WBNTi s = n EO¯.] ^ � = i>j.k,l # n me%dn '")�TE -0/*1¹3b5®1 αn uαn =

2β0

‖ f ‖(3.23)

) S(= X ��� 3.1 E ± ) L@N,i β0 = 3, β1 = 2, β2 = 1) L '")UT E (2, 4)-VSSS E�?&A$CeD>E-$/21¹3b5®1 #9  (2.14), (2.15) ¡ y��F�0= ) ‖ f ‖= 8) Sxy i α2 = 1

4 , α3 = 12 , α4 = 3

4

) S= � ) U���� ^ T = X3.3

� ����� =������������ I����3.3.1 PBVSSS �! #"%$'&)(+*�,�-/.1032%$

3 ≤ t ≤ n E (t, n)-VSSS+T³0WBN&iFj.k,l # n me%Mn '")T E z0{ r6s"E -0/|1�365®1

αn

U�%&© 8B¬ + S0=`?eA;C&D +b�;WeN � _ = )UT + u it­ E �&7   h(k)t,n

) � E+4#5t  U76M+�� X���

3.2 t, n # 2 ≤ t ≤ n #9� ' < ��� EO� � ) L�i k # 0 ≤ k ≤ n− t #`� ' <b� � ) <d= XW R i��7   h(k)t,n ∈ Hn # i

h(k)t,n =

n−t−k∑

i=0

(

n− i− k − 1

t− 1

)

an−t−izi(a− z)t (3.24)

) �B�(<>= X <d= ) i h(k)t,n E8495b  h

(k)t,n =

∑ni=0 γia

n−izi uγi =

(−1)i

(

n− i− k − 1

t− i− 1

)(

n− k

i

)

(0 ≤ i < t)

0 (t ≤ i ≤ n, i 6= n− k)

(−1)t (i = n− k)) , ¦ = X18

� 3.1���

3.2 E�  h(0)t,n E 4#56  h

(0)t,n =

∑ni=0 γia

n−izi E/� �.Xi 0 1 . . . t− 1 t t+ 1 . . . n− 1 n

an−izi E/� � γ0 γ1 . . . γt−1 0 0 . . . 0 (−1)t

� 3.2   (2.20) E p(0)t,n E 4 5b  p

(0)t,n =

∑ni=0 γia

n−izi E/� �.Xi 0 1 2 . . . n− t n− t+ 1 n− t+ 2 . . . n

an−izi E� � 1 0 0 . . . 0 γn−t+1 γn−t+2 . . . γn

��� � � 2.2 E@  (2.20)+ ² _$' (t, n)-PBVSSS EO?eA;C&D*#`GBI0<d= �7   p

(k)t,n #9f W = X

p(k)t,n u�� !  dS$E ^ i a, z � V ¶ V + ��� E����Q#�/�� L`NFh 152FEO¢ )� 2FE�¢Mu ! L w¸Sy i a ) z # �.V Y _.'

n−t∑

i=0

(

t− 1 + i− k

t− 1

)

aizn−t−i(z − a)t

= akzn−k +n∑

i=n−t+1

(−1)i−n+t

(

i− 1 − k

n− t− k

)(

n− k

i− k

)

aizn−i (3.25)

U I7y � � X 1�2>u � 2Q# ∑ni=0 γia

izn−i E�0 + 415 L '   ) S.· NBW = X   (3.25) #�� � LN,i� 2|# (−1)t ^ � = )n−t−k∑

j=0

(

n− j − k − 1

t− 1

)

an−t−jzj(a− z)t

=t−1∑

j=0

(−1)j

(

n− j − k − 1

t− j − 1

)(

n− k

j

)

an−jzj + (−1)takzn−kz

) S y i � �dU °|P�V ' X ����3.2 E h

(k)t,n E 4#56 FE� � γi(i = 0, 1, . . . , n) u i   (2.20) E (t, n)-PBVSSS

+6³(¦ =O?ABC&D"#cG&I.<(= ��7   p(k)t,n # 415 L '  >E�� � + (−1)t # Y$¦ ' (−1)tγn−i(i = 0, 1, . . . , n)) � � ^ � = X h(0)

t,n, p(0)t,n E � V ¶ V0E84 5 L '  FE� � # � 3.3.1

) � 3.3.1+ � V ¶ V6°0< X± ) LON,i (3, 8)-VSSS E k = 0 E )�T i

h(0)3,8 = 21a8 − 48a7z + 28a6z2 − z8

p(0)3,8 = a8 − 28a2z6 + 48az7 − 21z8

) S y i h(0)3,8, p

(0)3,8 u�� W$+ a

)z #�0V Y _ N (−1) � L '   + S>· N;W = � ) U&�$Y = X

19

3.3.2 t = 3 �����n # n ≥ 4 #9� ' < ��� EO� � ) L 'g)�T i(jek6l # n m&%0n '")�T E -$/"143b571 αn

U%T© 8B¬ ) S0= (3, n)-VSSS E�?eABCeD U GBI ^ T = � ) #c°d< X R�� i n U�� � E )�T0)��� E )�T + ¯B]� ¦ #�<d= X m # m ≥ 2 #c� ' <b� � ) L`N6i n U�� � E )�T u n = 2m

i �� E )UT u n = 2m+1

) L9NTi � V ¶ V(E@¯;] +b³;WTNU%T© 8;¬&'*# h@� ?eA;C&D>E��"#°(< X1. n = 2m E )�Tn = 2m (m ≥ 2)

) L 'g)�T idj;k,l # n m&%0n 'g)�T E -$/*1¹3b571 αn

U�%e©� 8B¬ )S0= (3, 2m)-VSSS Et?eA$CBD U G;I ^ T = � ) #9°(< X ω0, ω1 #`� � ) L�i���� 3.2 E h(k)t,n#@f W&N&i f∗3,2m # h(0)

3,2m

)h

(m−1)3,2m E��50 �B] ) LONf∗3,2m = ω0h

(0)3,2m + ω1h

(m−1)3,2m (3.26)) <>= X ��� 3.2 ¡ y h(k)

3,2m u e(i)3,2m (i = 0, 1, . . . , 2m − 3) E���0 �$] ^ � = Yxp¹i f∗3,2m

he(i)3,2m (i = 0, 1, . . . , 2m − 3) E���0 �;] ^ � = � ) +�� � <>= X   (3.26) E ω0, ω1 # i � �Y&�

a2m−2z2 E 7 E� � U 0+ Sd= ¡ �@+ 132 XQ�B��^ ��� 3.2 ¡ y i h(0)

3,2m, h(m−1)3,2m u � V¶ V

h(0)3,2m = (m− 1)(2m − 1)a2m − 4m(m− 1)a2m−1z +m(2m− 1)a2m−2z2 − z2m

h(m−1)3,2m =

(m− 1)m

2a2m − (m+ 1)(m− 1)a2m−1z +

(m+ 1)m

2a2m−2z2 − am−1zm+1

)-, ¦ =,E ^ i a2m−2z2 E 7 # 0+ <d= ' � + u i ω0m(2m− 1) + 1

2ω1(m+ 1)m = 0U I�y

�� ¡ � S ω0, ω1 #�132+� � U$� = X���� u ω0, ω1 # ω0 = m+ 1, ω1 = −2(2m− 1)

) 1��� X2� V p #9  (3.26)

+ / �0<M= )f∗3,2m = (m+ 1)h

(0)3,2m − 2(2m− 1)h

(m−1)3,2m (3.27)U�o�p VF= X  (3.27) ^ oxp V ' �U7   f∗3,2m

U(3, n)-VSSS E@?6A&CTD>#¹G6I;<0= ' �&E ��7   ^ � = '� + u i f∗3,2m ∈ F3,2m ^ � =)�U� UT� = X F3,2m

+�� <$= ' � + u i e(i)3,2m (i = 0, 1, . . . , 2m−3)E)� � E �>U�� ) S p S ¦ V(§BS p S W X e(i)3,2m (i = 0, 1, . . . , 2m − 3) E)� � E � #H´eµd<= ) iω0

2m−3∑

i=0

(

2m− 1 − i

2

)

+ ω1

m−2∑

i=0

(

m− i

2

)

=1

3(m+ 1)m(m− 1)(2m − 1)

) Sd= X m u m ≥ 2 #`� ' <�� � ) L ' E ^ i e(i)3,2m (i = 0, 1, . . . , 2m − 3) E)� � E � u �) Sd= � ) U��;Y = X20

­"+tif∗3,2m ∈ F3,2m ^ � = f∗3,2m

Yxp G;I2P�VM=@?eA$CBD +6³$WeNTiMj.k,l # n mB%0n ')UT E z.{ r6s>E -$/21a365�1 α2m #-�F�0= X   (3.27) ^ o p V ' �&7   f∗3,2m # 495 LON� � <d= ) if∗3,2m = (m− 1)(2m − 1)a2m − 2(m2 − 1)a2m−1z

+2(2m− 1)am−1zm+1 − (m+ 1)z2m (3.28)) , ¦ = X�� · N,i   (3.28)) j$k&l��

n m&%dn '*)UT E -0/*1¹3b5®1 αn EH  (3.23) ¡ yα2m =

2m2 − 3m+ 1

2m2 +m− 1→ 1 (m→ ∞)) S(= X(¡ · N6i n = 2m E )�T n ��� &: T w <(= ) αn

U1+���� <M= ¡ � S (3, n)-VSSSE�?BA;C;D U ��d<d= � ) U/�MY � p V ' X

2. n = 2m+ 1 E )�Tn = 2m+ 1 (m ≥ 2)

) L 'g)�T iFj;k&l��n m(n��� T E������������ αn

U�%&© ��¬ S�� (3, 2m+1)-VSSS E����� U�!" ^ T � �� � °�#%$ ω0, ω1

� � �& �' i ��� 3.2(h

(k)t,n

� f�)�* i f∗3,2m+1

�h

(0)3,2m+1 h(m)

3,2m+1

( �50 �;] +' *f∗3,2m+1 = ω0h

(0)3,2m+1 + ω1h

(m)3,2m+1 (3.29)

#��,$�/�

3.2 ¡ y h(k)3,2m+1 u e(i)3,2m+1 (i = 0, 1, . . . , 2m−2)

( �40��T] ^ � �.- p�i f∗3,2m+1he(i)3,2m+1 (i = 0, 1, . . . , 2m−2)

( ��0 �T] ^ � � �� 0/ � � #��1$   (3.29) / ³ )%* i n = 2m( T � / ω0, ω1

�a2m−1z2 ( 7 ( � �32 0 /�4 � ¡ � / 1 2�$ ��� u ω0 = m + 1,

ω1 = −2(2m+ 1) 1 �X�%$ �,5 p6�   (3.29) / / � ' � � #7� f∗3,2m+1 = (m+ 1)h

(0)3,2m+1 − 2(2m+ 1)h

(m)3,2m+1

=2m2a2m+1−(m+1)(2m+1)a2mz+2(2m+1)amzm+1−(m+1)z2m+1 (3.30)

04 �.$  (3.30) ^ o¥p 57� � 7   f∗3,2m+1 2 (3, n)-VSSS( �8����

��!�"#9� � � ( � 7   ^�

� � � / u i f∗3,2m+1 ∈ F3,2m+1 ^ � ����� 2 � ��$ f∗3,2m+1 2 F3,2m+1 /�#3� �7 u i

e(i)3,2m (i = 0, 1, . . . , 2m− 2)

( � � ( � 2ω0

2m−2∑

i=0

(

2m− i

2

)

+ ω1

m−2∑

i=0

(

m− i

2

)

=1

3m2(m+ 1)(2m+ 1) > 0

04 � �� - p@�*p - ^ � �.$ ­ / i f∗3,2m+1 ∈ F3,2m+1 ^ � � f∗3,2m+1 -p:!�"<; 5 �1���

�� /³)+*i,j,kbl��

n m=.n>�3 T ( z;{ rts ( �8�?�@���A� α2m+1

� �d����$c  (3.30)

j.k&l�� n m(n��� T ( �>�B�C�%�D� αn(   (3.23) ¡ y

α2m+1 =m2

m2 + 2m+ 1→ 1 (m→ ∞)

21

� 3.3 (3, n)-VSSS / ³d¦ � i f∗3,n =∑n−3

i=0 γian−izi 495 '6�� T ( an−izi ( � � γi(i =

0, 1, . . . , n)( � � ' i 0 ≤ n ≤ 10) $n γ0 γ1 γ2 γ3 γ4 γ5 γ6 γ7 γ8 γ9 γ10

4 1 −2 0 2 −1

5 8 −15 0 10 0 −3

6 5 −8 0 0 5 0 −2

7 9 −14 0 0 7 0 0 −2

8 21 −30 0 0 0 14 0 0 −5

9 32 −45 0 0 0 18 0 0 0 −5

10 6 −8 0 0 0 0 3 0 0 0 −1

C4 �1$ ¡ ·C* i n = 2m+1( T n � � � T w #�� αn 2 1 / � � #>� ¡ � 4 (3, n)-VSSS( ���8�8 2 ��7#7� �> 12 � -B� p 57� $  (3.28), (3.30) /;� 5 ¶ 5 m = 2, 3, 4, 5

� / � '6�� T ( an−izi(i = 0, 1, . . . , n)( � �( ¢ � � 3.3 / °�#%$ � 3.3 u i ( )�� _ � i 4 ≤ n ≤ 10 ^ ( f∗3,n =

∑ni=0 γia

n−izi 4#5'��� T ( γi(i = 0, 1, . . . , n)

( ¢ � ° ' *8)���$ � 3.3 -pai

4 ≤ n ≤ 10( £M¤ ^ u i γ0 u� ( � i γ1, γn u�� ( � i γ2 = 0 ,4 ·�*�)7�.$

3.3.3 t = 4 �����n�n ≥ 6

� � � # ��� ( � � �'.�? T i(jek6l>� n m+0n��� T ( �����C���D� αn 2%&©8� ¬ 14 � (4, n)-VSSS( ���8�8 2

!" ^ T � �� � ° #%$ t = 3( ¯.]>u i n ( ¢ �� � i � � ( 2 6 y / ¯$]5 ¦>� '���2 i ��� ( t = 4

( ¯$] / u i m � m ≥ 2� � � #b�� ' * i n = 3m,n = 3m+ 1, n = 3m+ 2

(3 6 y / n

( ¢ ^ ¯;]� ¦�� #3�.$T¯;]� ¦'.�$��5 ¶ 5 ( n

( ¢ ^ %e©�� ¬ ' �.hO� ���� ( � � °�#%$1. n = 3m

( Tn = 3m (m ≥ 2) '.�? T idj;k,l�� n m+0n7�? T ( ���������D� αn 2

%e©� �� ¬ 4 � (4, 3m)-VSSS

( �+�8� 2!+" ^ T � �> � °�#�$ ωi (i = 0, 1, 2)

� � � =' i � � 3.2(h

(k)t,n ¡ y i f∗4,3m

�h

(0)4,3m, h

(m−1)4,3m , h(2m−2)4,3m

( �50 �;] +' *f∗4,3m = ω0h

(0)4,3m + ω1h

(m−1)4,3m + ω2h

(2m−2)4,3m (3.31)

#?�%$� �

3.2 ¡ y h(k)4,3m u e

(i)4,3m (i = 0, 1, . . . , 3m − 4)

( � 0 �0] ^ � ��- pci f∗4,3mhe(i)4,3m (i = 0, 1, . . . , 3m − 4)

( ��0 �$] ^ � � �7 �/ � � # �6$6  (3.31) / ³ )�* i � �ωi (i = 0, 1, 2)

�a3m−2z2, a3m−3z3 ( ��5 ¶ 5 ( 7 ( � �?2 0 /4 � ¡ � / 1 2$ ��� u

ω0 = (m+ 1)(m+ 2)(2m+ 1), ω1 = −3(3m− 1)(m+ 1)(m+ 2), ω2 = 3m(3m− 1)(2m+ 1)

22

1�� i   (3.31) / / ��#3� f∗4,3m = (m+ 1)(m+ 2)(2m+ 1)h

(0)4,3m − 3(3m − 1)(m+ 1)(m+ 2)h

(m−1)4,3m

+ 3m(3m − 1)(2m + 1)h(2m−2)4,3m (3.32)

, ¦ �.$�� � i   (3.32)( �7   f∗4,3m 2 (4, n)-VSSS

( ���8�� �0!�"

#�� � � ( �7  ^ � � � � / u i f∗4,3m ∈ F4,3m ^ � ���&� 2 � ��$ e(i)4,3m (i = 0, 1, . . . , 3m− 4)( � � ( � u

ω0

3m−4∑

i=0

(

3m− i− 1

3

)

+ ω1

2m−3∑

i=0

(

2m− i

3

)

+ ω2

m−2∑

i=0

(

m+ 1 − i

3

)

=1

4(m− 1)2m(m+ 1)(m+ 2)(2m + 1)(3m − 1) > 0

04 y � 4 ( ^ i f∗4,3m ∈ F4,3m ^ � � �� 02 � -;� p 5 �6$­ / i f∗4,3m ∈ F4,3m ^ � � f∗4,3m -p !8" ; 5 �,����� / ³ )* iMj.k,l>� n m�0n7�

T ( z${ r�s ( �>���C��� � α3m

� �M���.$b  (3.32)� 4 5 ' * oqp 5 �9  i(j$k,l>�

n m+(n��� T ( �>�B�C�%�D� αn(   (3.23) ¡ y

α3m =(m− 1)2(m+ 1)(3m − 1)

3m(m+ 2)(m2 + 1)→ 1 (m→ ∞)

�4 �1$ ¡ ·�* i n = 3m '1�3 T / i n ��� � T w #>� i αn 2 1 / �%� #>� (4, n)-VSSS( ���8�8 2 ��7#7� �> 12 � -B� p 57� $2. n = 3m+ 1

( Tn = 3m+ 1 (m ≥ 2) �'6�? T iFj;k&l�� n m(n��� T ( ����������� αn 2

%&© ��¬ ,4 � (4, 3m + 1)-VSSS

( �+�>� 2!8" ^ T � �> � ° #6$ ωi (i = 0, 1, 2)

� � � �' i���3.2(h

(k)t,n ¡ y i f∗4,3m+1

�h

(0)4,3m+1, h

(m)4,3m+1, h

(2m−1)4,3m+1( �50 �;] +' *

f∗4,3m+1 = ω0h(0)4,3m+1 + ω1h

(m)4,3m+1 + ω2h

(2m−1)4,3m+1 (3.33)

#��,$�/�

3.2 ¡ y h(k)4,3m+1 u e(i)4,3m+1 (i = 0, 1, . . . , 3m−3)

( �40��T] ^ � �.- p�i f∗4,3m+1he(i)4,3m+1 (i = 0, 1, . . . , 3m− 3)

( �50��e] ^ � � �> �/ � � #��.$ n = 3m � / ωi (i =

0, 1, 2)�a3m−1z2, a3m−2z3 ( �65 ¶ 5 ( 7 ( � ��2 0 04 � ¡ � / 1 2%$ � � u ω0 = 2(m−

1)(m+1)(m+2)(2m+1), ω1 = −3(m+1)(m+2)(2m−1)(3m+1), ω2 = 6m2(2m+1)(3m+1)

1P2%$ � ( T i   (3.33) / ωi(i = 0, 1, 2)� / � '�� ��7   f∗4,3m+1 2 (4, n)-VSSS

( �=��+

��!+"#7� � � ( ��7   ^ � � � � / u i f∗4,3m+1 ∈ F4,3m+1 ^ � ���� 2 � �.$ e(i)4,3m+1

(i = 0, 1, . . . , 3m− 3)( � � ( � u

ω0

3m−3∑

i=0

(

3m− i

3

)

+ ω1

2m−3∑

i=0

(

2m− i

3

)

+ ω2

m−2∑

i=0

(

m+ 1 − i

3

)

=1

4(2m− 1)(m− 1)m2(m+ 1)(m+ 2)(2m + 1)(3m+ 1) > 0

23

,4 y � 4 ( ^ i f∗4,3m+1 ∈ F4,3m+1 ^ � � �� 12 � -.� p 5 ��$,  (3.33) / ωi(i = 0, 1, 2)� / � '6� ��7   f∗4,3m+1 -p@!8"<; 5 �,�����8 / ³ )* i>j;k&l>� n m�(n �� T ( z{ r6s ( ���B��� �D� α3m+1

� �>�7��$T  (3.33) / ωi(i = 0, 1, 2)� / � ' 495 ' * o�p 5

�`  iFj;k&l�� n m(n��� T ( �>�B�C�%�D� αn(   (3.23) ¡ y

α3m+1 =3m2(m− 1)(m+ 1)(2m− 1)

(m+ 2)(2m− 1)(3m + 1)(m2 +m+ 1)→ 1

04 � �� �2 � -7�.$ ¡ ·�* n = 3m+ 1( T i n �0� � T w:# � αn → 1

� � � # ¡� 4 (4, n)-VSSS( ���8�� 2 ��# � �� 02 � -;� p 5�� $

3.n = 3m+ 2( T

n = 3m+ 2 (m ≥ 2) �'6�? T iFj;k&l�� n m(n��� T ( ����������� αn 2%&© ��

¬ ,4 � (4, 3m)-VSSS( ��>� 2

!�" ^ T � �� � ° #6$ ωi (i = 0, 1, 2)� � � +' i���

3.2(h

(k)t,n ¡ y i f∗4,3m+2

�h

(0)4,3m+2, h

(m)4,3m+2, h

(2m)4,3m+2( �50 �;] +' *

f∗4,3m+2 = ω0h(0)4,3m+2 + ω1h

(m)4,3m+2 + ω2h

(2m)4,3m+2 (3.34)

#��,$�/�

3.2 ¡ y h(k)4,3m+2 u e(i)4,3m+2 (i = 0, 1, . . . , 3m−2)

( �40��T] ^ � �.- p�i f∗4,3m+2he(i)4,3m+2 (i = 0, 1, . . . , 3m− 2)

( �50 �B] ^ � � �� ./ � � #3��$ n = 3m, 3m + 1 � / ωi (i = 0, 1, 2)

�a3m−1z2, a3m−2z3 ( ��5 ¶ 5 ( 7 ( � � 2 0 14 � ¡ � / 1#2�$ � � u

ω0 = (m+1)(m+2)(2m+1), ω1 = −(m+2)(3m+1)(3m+2), ω2 = (3m+1)(3m+2)(2m+1)

1�� ��$ � ( T i   (3.34) / ωi(i = 0, 1, 2)� / � '6� �&7   f∗4,3m+2 2 (4, n)-VSSS

(������

�.!>"#?� � � ( ��7   ^ � � � � / u i f∗4,3m+2 ∈ F4,3m+2 ^ � ����� 2 � �%$

e(i)4,3m+2 (i = 0, 1, . . . , 3m− 2)

( � � ( � uω0

3m−2∑

i=0

(

3m+ 1 − i

3

)

+ ω1

2m−2∑

i=0

(

2m+ 1 − i

3

)

+ ω2

m−2∑

i=0

(

m+ 1 − i

3

)

=1

4(2m− 1)(m− 1)m2(m+ 1)(m+ 2)(2m + 1)(3m+ 1) > 0

,4 y � 4 ( ^ i f∗4,3m+2 ∈ F4,3m+2 ^ � � �� 12 � -.� p 5 �6$ ­ / i   (3.34) / ωi(i =

0, 1, 2)� / � '�� ��7   f∗4,3m+2 -

p:!�" ; 5 ������� / ³ )�* i>j$k&l�� n m�0n7�� T ( z.{ r6s ( ���B���%�D� α3m+2

� �F�7��$T  (3.34) / ωi(i = 0, 1, 2)� / � ' 4 5 ' *oqp 5 �@  iMj.k&l�� n m(n��� T ( �>�B� � � � αn

(   (3.23) ¡ yα3m+2 =

m2(2m+ 1)(3m + 1)

(m+ 1)(m+ 2)(2m + 1)(3m+ 2)→ 1

2 � � ^ T ��$ ¡ ·6* n = 3m + 2( T i n ��� � T w #9� αn → 1

� � � # ¡ � 4(4, n)-VSSS

( ���8�8 2 ��# � �� 02 � -;� p 5�� $6 ≤ n ≤ 14

( £>¤ / ³ )+* i f∗4,n

�f∗4,n =

∑n−4i=0 γia

n−izi 4#5 '.�� T ( an−izi ( � �γi(i = 0, 1, . . . , n)

( ¢ � � 3.4 / °�#%$ � 3.4 / ³ )* i�� u 0 � � ( � ��2 �*·�*�)>��$ �24

� 3.4 (4, n)-VSSS / ³d¦ � i f∗4,n =∑n−4

i=0 γian−izi 495 '6�� T ( an−izi ( � � γi(i =

0, 1, . . . , n)( � � ' i 6 ≤ n ≤ 14) $� u 0 � � ( * � $

n γ0 γ1 γ2 γ3 γ4 γ5 γ6 γ7 γ8 γ9 γ10 γ11 γ12 γ13 γ14 γ15

6� �

0 0� � �

7� �

0 0� �

0�

8� �

0 0�

0�

0�

9� �

0 0 0�

0�

0�

10� �

0 0 0�

0�

0 0�

11� �

0 0 0�

0 0�

0 0�

12� �

0 0 0 0�

0 0�

0 0�

13� �

0 0 0 0�

0 0�

0 0 0�

14� �

0 0 0 0�

0 0 0�

0 0 0�

14� �

0 0 0 0 0�

0 0 0�

0 0 0�

3.4���

� iγ0, γ1 u γ0 6= 0, γ1 6= 0

( � �F^ � y i γ2 = 0, γ3 = 0 ,4 ·6*8) � �� 12 �-��6$

(3, n)-VSSS( ¯;]Mu i n ( ¢ � � � i � � ( 2 6 y / ¯;]5 ¦�� ' * %&©8� ¬ ' ���d� �

���� ( �� ° '���2 i � � ( (4, n)-VSSS

( ¯;] / u i m � m ≥ 2� � � #�� � +' * i

n = 3m,n = 3m+1, n = 3m+2(

3 6 y ¯e] ¦8� ' * %,©+� ¬�' ���.� ���8�+ ( � �° '�� $ � � i (3, n)-VSSS ^ u i � � ωi(i = 0, 1) u an−2z2 ( 7 ( � �72 0 /�4 � ¡ � / 1��� 2 i (4, n)-VSSS ^ u i � � ωi(i = 0, 1, 2) u an−2z2, an−3z3 ( 2

� ( 7 ( � �72 0 /=4� ¡ � / 1P2+�&� 2 � ��$ � ( ¡ � / i n ( ¢ 2 � _ � / � 5 * i�%T©� ¬�' �;h9� ������ ( �

� °�# � � / i ¯B]5 ¦8� #�� 6 y � ha� _ i ��5 ¶ 5 ( ¯B] ( ωi(i = 0, 1, . . . , n−2)� �F��� � � (����   h4� _ *�) w $3.3.4 t = 5 �����n�n ≥ 8

� � � # ��� ( � � �'.�? T i(jek6l>� n m+0n��� T ( �����C���D� αn 2%&©8� ¬ ,4 � (5, n)-VSSS( ���8�8 2

!�" ^ T � �� � °7#�$ m � m ≥ 2� � � #6� �

�' *in = 4m,n = 4m + 1, n = 4m + 2, n = 4m + 3

(4 6 y / ¯��� ¦�� ' * i �%5 ¶

5 %B©� ¬�' �$h�� ���8�8 2 ��7#7� �> � °7#%$n = 4m (m ≥ 2) '.�? T idj;k,l�� n m+0n7�? T ( ���������D� αn 2

%e©� �� ¬ 4 � (5, 4m)-VSSS

( �+�>�� 2!" T � �� � ° #�$ ωi (i = 0, 1, 2, 3)

� � � ' i����3.2(h

(k)t,n ¡ y i f∗5,4m

�h

(0)5,4m, h

(m−1)5,4m , h(2m−2)5,4m , h(3m−3)5,4m

( ��0 ��� ' *f∗5,4m = ω0h

(0)5,4m + ω1h

(m−1)5,4m + ω2h

(2m−2)5,4m + ω3h

(3m−3)5,4m (3.35)

#?�%$� �

3.2 ¡ y h(k)5,4m u e

(i)5,4m (i = 0, 1, . . . , 4m − 5)

( � 0 ��� d� ��- pci f∗5,4m

25

he(i)5,4m (i = 0, 1, . . . , 4m − 5)

( ��0 ��� .� � �7 �/ � � # �6$6  (3.35) / ³ )�* i � �ωi(i = 0, 1, 2, 3)

�a4m−2z2, a4m−3z3, a4m−4z4 ( �%5 ¶ 5 ( 7 ( � �72 0 /�4 � ¡ � / 1#2=$

ωi(i = 0, 1, 2, 3)� 1P2 � � / u i 3

{ ( � � �,­����   ��� w �&� 2 � �,$ ωi(i = 0, 1, 2, 3)( ´&µFu���$´Tµ u 4 w Windows XP 2� � # ��´Tµ�� � �  �� ����� ������� Mathe-

matica 5.1���)�*,´&µ '�� $ * %��&´&µgu���� /+4 � 2 � ωi(i = 0, 1, 2, 3) u6�8 e  � ´&µ

#7� �> ��! / �M�7� �� �2 T �.$ ��� uω0 = (m+ 1)(m+ 2)(m+ 3)(2m+ 1)(3m + 1)

ω1 = −4(m+ 1)(m+ 2)(m+ 3)(2m + 1)(4m− 1)

ω2 = 6m(m+ 2)(m+ 3)(3m + 1)(4m− 1)

ω3 = −4m(m+ 1)(2m+ 1)(3m + 1)(4m− 1)

1�����$ � ( T �`  (3.35) / ωi(i = 0, 1, 2, 3)� /�� '�� ��7   f∗5,4m 2 (5, n)-VSSS

( �����

�0!+"#7� � � ( ��7   $� � � � / u�� f∗4,3m+2 ∈ F4,3m+2

$������ 2 � ��$ e(i)5,4m

(i = 0, 1, . . . , 4m− 5)( � � ( � u

ω0

4m−5∑

i=0

(

4m− 1 − i

4

)

+ ω1

3m−4∑

i=0

(

3m− i

4

)

+ ω2

2m−3∑

i=0

(

2m+ 1 − i

4

)

+ ω3

m−2∑

i=0

(

m+ 2 − i

4

)

=1

5(m− 1)3m(m+ 1)(m+ 2)(m+ 3)(2m + 1)(3m+ 1)(4m − 1) > 0

04 y � 4 ( � f∗5,4m ∈ F5,4m

.�� �� 02 � -;� p 5 �6$­ / �   (3.35) / ωi(i = 0, 1, 2, 3)� /�� '1� �U7   f∗5,4m -

p ! "�; 5 � �=�+� / ³ ) *"�# �$� � n % �&8�3 T (('�)�*�+7( ���9�:�.�A� α4m

� �-,��,$/. (3.35) / ωi(i = 0, 1, 2, 3)� / � ' 4 5 ' *021 5 �3. � # ��� � n %4& �� T ( ���B��� �D� αn( . (3.23) 56 � α4m 7

α4m =3(m− 1)2(m+ 2)

(m+ 3)(3m + 1)(m2 +m+ 6)→ 1 (m→ ∞)

04 �6$ 5 ·6*�� n = 4m( T � n �,� � T98 #3� αn → 1

��: � # (5, n)-VSSS( �

�8�8 2 ��7#7�-; 12 � - ,<1 57� $� � � n = 4m+ 1, 4m+ 2, 4m+ 3

( T 7�� n = 4m( T �� ( � � � �2=�; �/

4 � ( ��>�?@7�A B A /�� #C; #7�.$ n = 4m+ 1, 4m+ 2, 4m+ 3

( T ( ����� �� �D7 E 5-F 5

α4m+1 =3(m− 1)2m2(m+ 2)(3m − 2)

(m+ 3)(3m+ 1)(3m4 + 4m3 + 16m2 − 4m− 4→ 1 (m→ ∞)

α4m+2 =(m− 1)m2(m+ 2)(3m − 1)(3m+ 1)

(m+ 3)(3m+ 2)(3m4 + 8m3 + 17m2 − 4)→ 1 (m→ ∞)

α4m+3 =m3(m+ 2)(3m + 2)

(m+ 1)(m+ 3)(3m3 + 12m2 + 23m+ 12)→ 1 (m→ ∞)

26

04 �.$ 5�� *�� n �,� � T98 #7� αn → 1�(: � # 5 = 4 (5, n)-VSSS

( ��8�8 2 ��#3�!; �2 � -�, 1 5�� $

3.4 KCM NPORQ N I ���; (�� 7(� t = 2, 3, 4

( ��� ��� 9����� (t, n)-VSSS( �%�+�= /���� � s(t ≤ s ≤ n)

% ( # ��� � &��� T ( �����C�%�D� αs /�)�*���� #7��$ � � � s % ( # ��� � &7�

��=/ � � � /=4 � ) =���� 6��� 4 ���B�C�%�D� ! #" ��$�$"� s % ( # ��� � &��� �� / ��� /=4 �8�>��� �%�D� �&%(' ( 5 = /&) ,��.$ t, n �

2 ≤ t ≤ n��: � #*�� (,+.- +' � βi (i = 0, 1, . . . , n− t) 7�� ∑n−t

i=0 βi > 0��: � # +�-

#7�6$ f ∈ Ft,n /&/.0 # � (t, n)-VSSS( ���8�8 X0, X1

( # ��� � s(t ≤ s ≤ n) %+4&�� �� ( ����������� ( ���#1 α̃s 7

α̃s = max

2n−s∑

i=0

(

n− t− i

s− t

)

βi

(

n

s

)

‖ f ‖

�) ,��6$�$��2�� �#�. �#�&� (2, n)-VSSS

( ���8�8 /2�3� �>�>� �C�%�D� / � )*4�5�#3�6$(2, 9)-VSSS � � 5 � (2, 10)-VSSS /��6� �"���# �(�# 879�&� �8���� ( ���<���=� � �# ��� 7 s(t ≤ s ≤ n) %+ &>�� :� ( �.� 4 �����@�%� � α̃s

7 E 54F65�; 3.5 /=< #6$�� � �; 3.5

((2, 9)-VSSS,(2, 10)-VSSS /2�>� ����� � �%� � ( 1?7�@�A9B � '��&C�D 7 E 5-F%5E

3.2 � E 3.3 /&< #%$ E 3.2 � E 3.3(�F�G

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(�F�G(b) 7��2V.W /&/.0

#T % -#X # ��� 7 I & U?Y �=/[Z � 4 J#KHL!N�O\L 7 < #].R (S(T $��U�� E 3.2

(�F�G (S(T(s23) 7�� # �� 7 2 % I & U^Y � � 3 % I & U^Y � /&Z � 4 J

K?L_NO\L 7 Y T=`.a(b.c (&I & T # ��� - / 5 T J(KML_N�OPL ( 1H7�@#A6B L ' U&]�R S�T$�d:egf�� E 3.2 h F�G (s35),(s58) 7���E:i F i # ��� 7 3, 4, 5 % I & U^Y � f Z �6j J

K^L!N�OkL 7 Y T�`�a�b�c h I & T # ��� - f 5 T J�KHLlN�O\L � # ��� 7 5, 6, 7, 8 % I & U

; 3.5 (2, n)-VSSS f �3� T s(2 ≤ s ≤ n) % I & U?Y ��h J>KHL_N�OmL (a), (b) n (a) 7:�.Z �� ?7>�[�2`�a(b�c h J>K?L_N:OPL n (b) 7 s % I & UoY �:h�Z ��j J>K?L_N:OPL n

n s 2 3 4 5 6 7 8 9 10

(a) 19

29

13

49

59

23

79

89

9 (b) 518

55126

2342

1321

2536

34

79

89

(a) 110

210

310

25

12

35

710

45

910

10 (b) 518

1330

815

3760

23

1115

79

45

910

27

; 3.6 (3, n)-VSSS f �3� T s(3 ≤ s ≤ n) % I & U?Y ��h J>KHL_N�OmL (a), (b) n (a) 7:�.Z �� ?7>�[�2`�a(b�c h J>K?L_N:OPL n (b) 7 s % I & UoY �:h�Z ��j J>K?L_N:OPL n

n s 3 4 5 6 7 8 9 10

(a) 235

17

625

1750

1125

2750

1625

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1370

1970

1132

1125

2750

1625

(a) 118

17126

29

1445

25

2245

2645

23

10 (b) 112

528

1142

13

410

1145

1645

23

; 3.7 (4, n)-VSSS f �3� T s(4 ≤ s ≤ n) % I & U?Y ��h J>KHL_N�OmL (a), (b) n (a) 7:�.Z �� ?7>�[�2`�a(b�c h J>K?L_N:OPL n (b) 7 s % I & UoY �:h�Z ��j J>K?L_N:OPL n

n s 4 5 6 7 8 9 10

(a) 4135

52675

29225

122675

157675

64225

9 (b) 263

14165

533

67

2581

3281

(a) 3130

17260

38325

111650

73325

181650

108325

10 (b) 135

8105

62455

93455

2591

1749

37

Y � f Z ��j J(KHL_NO\L 7 Y T[`.a(b.c h I & T # �� - f 5 T J(KML NOPL h 1H7 < �]�R #S>T n E 3.2 h (b) 7�� (s23),(s35),(s58),(a) h�Z��oh 1M7 Y ��� @.AgB L�� �� T nE

3.2 5 6 � (a) 7 # �"� 7 2 % I & U�Y ��h J#K^LlN�OkL h 1 7� � ��� � # �"� h I & T% - 7���� ������� J�K?L�N�OkL h 1 ���?� 8 j(T n�� f � (s23),(s35),(s58) 7����� 7 2 % I& U^Y �:h J#K?LlN�OkL h 1 7 (a) 5 6 � �8����n ��!"� � (s23),(s35),(s58) 7 � I & T % - 7��� � � �$#"%'& J�K^L N�O L h 1 �)(�* � � ��+ � � � � ��! (a) h � = � (s23),(s35),(s58)

5 6 � J>K?L_N:OPL h 1 �'�M� 8 j3T n+�U � E 3.2

� E3.3 5 6 � (a) 7 I & T ���' - �), j � Y � 7(J#KHLlN:OkL h 1 � � �� � I & T ���� - 7-�.�0/21��3� �.�"� J(KHL NO L � ��4 � � 9, 10 % I & U^Y � 7 � Z�6j J#KML NO L &#S>T (b) Y=d65 1H7 < � ��� T n7 f � (3, n)-VSSS h �. Z �# H7>�&�2`.a(b.c f �9� T J(KML!N2OPLlf � �0� ��� � T n

(3, 9)-VSSS� � 5)8 (3, 10)-VSSS f �6� T � �# Z �# 879�&��`�a>b#c h J>K8L_N2OmL�Y �

�0�' 7 s(t ≤ s ≤ n) % I & U^Y �:h[Z �3j J#K?LlN�O L α̃s

7 E�i F i ; 3.7 f < � n +2U �; 3.7 h (3, 9)-VSSS, (3, 10)-VSSS f �>� T J�KHLlN�O\L h 1^7�@�A(B L9��U C�D 7 E:i F iE

3.4� E

3.5 f < � n E 3.4� E

3.5 h (a) 7 � �� Z �� ?7#�&��`.a�b.c h I & T ���� -f 5 T J(KML NO\L 7 < � Q.R &(S(T n +�U � E 3.4

� E3.5 h (b) 7 � V.Wof /.0 � T % -X ���� 7 I & U?Y � f Z �6j J#KML NO L 7 < � ].R &(S(T nE

3.4� E

3.5 5 6 � (2, n)-VSSS h E 3.2� E

3.3 Y=d�e�f � (a) 7 I & T ���� - �', j �Y � 73J(KML!NO L h 1 � :� ��� � I & T ���� - 7��.� � Y�J>KHL_NO L � ��4 � � Z�6j J#KML NO L &#S>T (b) Y=d65 1H7 < � ;9Y �0; ! T nZ=< f (4, n)-VSSS h �# Z �( 73����`#a9b(c f �g� T J3K L�N�O L_f � ��� �3> T n

28

(4, 9)-VSSS� � 5)8 (4, 10)-VSSS f �6� T � �# Z �# 879�&��`�a>b#c h J>K8L_N2OmL�Y �

��� 7 s(t ≤ s ≤ n) % I & U?Y �h,Z �6j J#KML!N�O\L α̃s

7 E�i F i ; (3.4) f < � n +U � ; 3.4 h (4, 9)-VSSS,(4, 10)-VSSS f �9� T J#K?L!N:OPL h 1?7�@�A9B L �&U C�D 7 E�iF i E 3.6

� E3.7 f < � n E 3.6

� E3.7 h (a) 7 � �� Z �. H7(�,�2`�a(b�c h I & T ��

- f 5 T J#KML!NOPL 7 < � Q.R &(S>T n +�U � E 3.6� E

3.7 h (b) 7 � V�W^f /.0 � T% -#X ���� 7 I & U?Y � f Z �6j J#KML N�O\L 7 < � ].R &(S(T nE

3.6� E

3.7& 7 � �� � f Z �6j�`�a(b�c f �9� T J>KML!NO\L h 1 (a) � � Z ��j J

K?L_N:O\L h 1 (b) Y�� � 8 j>T ;>Y$7 j �2n ;=i47 � n h 1 � n = 9, 10& 7�:� � U�,#f �

(a) h 1 Y (b) h 1 � � � 8 j(T ;�Y � j � Y���� �=i T n n h 1^7 � 1=f �M� 8 Y i 1 � t = 4

h Y � &�� t = 2, 3 h Y � Y[d�e^f � �. � f Z �3j�`�a#b.c f �9� T J(KML NO L h 1 (a)

� � Z �6j J(K?L N�O L h 1 (b) Y�� � 8 j(T ;>Y ��.) �=i T n n h 1^7 � h��� + & �M�8 Y�i 1 � (a) h 1 Y (b) h 1 � � � 8 j3T h !�7 �� h Y�;�������� &>S#T n

29

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

2 3 4 5 6 7 8 9

������

��� ���� ��

(2,9)-VSSS

(a)(b)

(s23)(s35)(s58)

E3.2 (2, 9)-VSSS f �9� T J>KML!NO L (a),(b) h� ��9n (a) � �. Z �� H7>�[�2`.a#b�c f

�>� T J(K?L_N:O L n (b) � I�� T ��� - f�� T Z �6j J(K?L!N:O L n (s23),(s35),(s58) ���i��i (b) f�� � U `�a#b.c h J(KHL!N�O\L n

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

2 3 4 5 6 7 8 9 10

������

��� ���� ��

(2,10)-VSSS

(a)(b)

E3.3 (2, 10)-VSSS f �9� T J#K?L!N:O\L (a),(b) h� ��>n (a) � �� Z �� H7#�,�:`�a#b�c f

�>� T J>K?L_N:OPL n (b) � I�� T ���� - f�� T Z �gj J(KHL!N�O\L n30

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

3 4 5 6 7 8 9

������

��� ���� ��

(3,9)-VSSS

(a)(b)

E3.4 (3, 9)-VSSS f �9� T J>KML!NO L (a),(b) h� ��9n (a) � �. Z �� H7>�[�2`.a#b�c f

�>� T J>K?L_N:OPL n (b) � I�� T ���� - f�� T Z �gj J(KHL!N�O\L n

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

3 4 5 6 7 8 9 10

������

��� ���� ��

(3,10)-VSSS

(a)(b)

E3.5 (3, 10)-VSSS f �9� T J#K?L!N:O\L (a),(b) h� ��>n (a) � �� Z �� H7#�,�:`�a#b�c f

�>� T J>K?L_N:OPL n (b) � I�� T ���� - f�� T Z �gj J(KHL!N�O\L n

31

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

4 5 6 7 8 9

������

��� ���� ��

(4,9)-VSSS

(a)(b)

E3.6 (4, 9)-VSSS f �9� T J>KML!NO L (a),(b) h� ��9n (a) � �. Z �� H7>�[�2`.a#b�c f

�>� T J>K?L_N:OPL n (b) � I�� T ���� - f�� T Z �gj J(KHL!N�O\L n

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

4 5 6 7 8 9 10

������

��� ���� ��

(4,10)-VSSS

(a)(b)

E3.7 (4, 10)-VSSS f �9� T J#K?L!N:O\L (a),(b) h� ��>n (a) � �� Z �� H7#�,�:`�a#b�c f

�>� T J>K?L_N:OPL n (b) � I�� T ���� - f�� T Z �gj J(KHL!N�O\L n

32

�4 � � � � � � � � � �

4.1 ������������ ����� ��! "$# %'&$(*),+�-."$/.0.13254$6572 ≤ t ≤ n Y0�&U?Y98 � (t, n)-VSSS

7;:=< � T[`�a#b�c h?>A@ 2.1& � � ��� t− 1 BDCE & �GFIHKJML fON.� �9P�QSR=T.�9U,Q i j �WV Y�Y � t B I � UoYM8f FAHKJ?L.�9XZY �=iT V Y �M[O\ �=i ��� T n ��!"� � �0�' ^] t+ 1 B^C`_ I � U�Y98f F=HDJ?L2�Ma=b � f XY �,i T V Y�+ & �c[O\ �=i ��� j ��nAC E f � ���' `] t+ 1 B^C`_Gd � U�Ye8�f F=HfJML

�MX`Y �&ihg (t, n)-VSSS h `.a�b.c X0, X1 �9iWj �9k ] >hl g n>=@ 2.1 ] iWj � (t, n)-VSSS hnm=oOp=q ] X0, X1 r � � X0, X1 � Nn,i h;sZt &Ou ghr �g n9mIo^p=q X0, X1 ! QSv < �=ihg0���' ^] x(t+ 1 ≤ x ≤ n) BId � j r98:f FAHDJ9L3�MXY �&ihg h � � S ] P hGwZxZyOz r�� � |S| = x ] iWj � �D{ j y^z r�� �h[OR(X1 [S ])] > h[OR(X0 [S ])] (4.1)

� <}|�~h� r�8 &fu g nk(4.1) � �?�I�W� x p ] OR �I���njK� �A�?���=�9� d��O�n�D� u g � & � ��� ^] x Bd � j re8 �caZb � �?F=HhJMLh�9X`Y��n� g V r �G�O���n�O� gG��+ j � C E �ck � <�|�~� 1 � ���� ^] x(t+ 1 ≤ x ≤ n) B=d � j re8 �GF=HKJ9LD�ea=b � �GXOY & 8M� � �

h[OR(X1[S])] = h[OR(X0[S])] (4.2)

x ] t + 1 ≤ x ≤ n ] i�jI�e O¡ r �?j rI8n¢ k (4.1), (4.2) ] � � g V r � �A£ � ¢h¤O¥¦ ] x (x > t) BOd � j rI8 �MFOHhJ�L���XfY§�c� � � (t, n)-VSSS� mOofp`q ] v < � gV r �O¨ 8AgG��¤^¥ ¦ x (x > t) BAd � gWr F=HKJ9LK�eXOY��;� � � (t, n)-VSSS

� mAo`p=qX0, X1 � ¢ C E � �K{ �G>A©h¨ 8IgG� >A© 2.1 ] iWje� (t, n)-VSSS

� m=o^pZq ] X0, X1 r� ¢ X0, X1 � Nn,i

� sOtK¨ u g�r � g?� j, x ] t+ 1 ≤ j, x ≤ n ] i�j��e =¡ r � ¢ S ] P� w^xAy`zf¨ ¢ |S| = j ] i�je� �f{ª� yOz r �n�h[OR(X1[S])] = h[OR(X0[S])] (j = x) (4.3)

h[OR(X1[S])] > h[OR(X0[S])] (j 6= x) (4.4)

� <}|�~h� �V^VG¨ ¢ k (4.3)� r�8 � ¢h¤^¥ ¦ ] x BZd � j re8 �D«f��¬®­e¯�¬ αx � αx = 0 rn�KgV r �G° � � g?� V;��� ¢ k (3.1)

� ¤`¥ ¦ ] s(t ≤ s ≤ n) B=d � j re8 �K«^��¬®­e¯�¬ αs� >Z©K±§Q ¢h² Q³±=¨ u g?�33

� j ¢ mZoOpZq � >A© k (4.3), (4.4) �eC E � �f{ ����� k f ∈ Ft,n ] � �A�����cV r �Z¨8�g?� t, n ] 2 ≤ t ≤ n ] i�jI� �Z���  O¡ r � ¢ j, x ] t+ 1 ≤ j, x ≤ n ] i�je� �=� �  ¡ r � gG��¤^¥ ¦ ] x(t < x ≤ n) B=d � gWr F=HhJ9Lh�eX^Y��n� � � (t, n)-VSSS� mZo`pq X0, X1 ] :=< � g �� k ] f r � ¢ f ∈ Ft,n r �?j r�8

ψn−jf |z=0 = 0 (j = x)

ψn−jf |z=0 = Cjat (j 6= x)

� <}|�~h� � V=VG¨ ¢ Cj (j = t, t+ 1, . . . , n) ��=¡ r � gc�k(3.22) � | ¢

ψn−jf |z=0 =n−j∑

i=0

βi(n− j)!

(

n− t− i

j − t

)

aj

r;�Kg V r ���f± £ �`� g?� j ≥ n � | (n − j)! > 0 j���� ¢ 0! = 1 � � � ¨ ¢W¤O¥ ¦ ]x (t+ 1 ≤ x ≤ n) BZd � j r�8 �?FAHKJ9LK�9X^Y �n� � � (t, n)-VSSS

� m=o^pZq ����� �n−j∑

i=0

βi

(

n− t− i

j − t

)

= 0 (j = x)

n−j∑

i=0

βi

(

n− t− i

j − t

)

> 0 (j 6= x)

r³�Kgc�� r �;� ¢D¤^¥ ¦ ] 2 BZd � j re8;¢ �O��� � 4 B=d � j re8 �?FAHKJMLh�MXOY �;� g � ¢¤O¥ ¦ ] 3 B=d � g j r98 � �MFAHhJMLh�9X^Y �n� � � (2, 4)-VSSS� m=o`p=q X0, X1 ] :< �n�^� gG� βi(i = 0, 1, 2) � β0 + β1 + β2 > 0, 2β0 + β1 = 0, β0 > 0 ] iWjA������� u g?���� � f ∈ Ft,n r � � ¢ β0 = 1, β1 = −2, β2 = 4 r����

f = a4 − 4a3z + 9a2z2 − 10a3z + 4z4

= 4 · 3

[

2a4

4!− 2

a3z

3!1!+ 3

a2z2

2!2!− 5

az3

1!3!+ 8

z4

4!

]

r³�Kgc� mAo`p=q X0, X1 �n�O�f� g �� k ] g0, g1 r � ghr ¢ g0 = 2a4

4! + 3a2z2

2!2! + 8 z4

4! , g1 =

2a3z3!1! + 5 az3

1!3! ¨ u gG� αs

�nk(3.5) � | ¢ α2 = 1

28 , α3 = 0, α4 = 114 r �fgc�f¤Z¥ ¦ 2, 4 BId �j r�8 � «f� ¬ ­e¯�¬ �� �! � � ¨eX^Y��n� g � ¢�¤^¥ ¦ 3 B=d � j re8 � «`��¬ ­e¯�¬�� � 0 r³� | ¢ F=HKJ9LK�9X^Y �n� � �WV r �"�O± gc�

4.2 #%$'&)( *���"$# % &��,+�-."$/50.13254$657V �,+ ¨�� ¢ d � gA¤Z¥ ¦ ¡ t B`C^_-�f� ��£ � � ¢�. J-/ r�0 J�/`��1"2��cjGFAHDJML�MX`Y �c� g (t, n)-VSSS � � �A��354K� gG� t BfC`_ �G�A� � BZ¡K¨ u g x B � ¤^¥ ¦ ] d

34

� gWr ¢%. J/ r 0 J-/K��1�2��n�`��F=H�JeLK�eX^Y��G� g (t, n)-VSSS� mZo`pZq X0, X1

� ¢ >A© 2.1�����

1� � � | �MC E ��� { �n>Z©K¨ 8IgG�>=© 2.1 ] iWj9� (t, n)-VSSS

� m=oOp=q ] X0, X1 r � ¢ X0, X1 � Nn,i

� sZtK¨ u ghr �gc� x ] t ≤ x ≤ n ] i�jI�9 O¡ ¢ s ] t ≤ s ≤ n r � ¢ S ] P� w^x=y`z r � ¢ |S| = s ]i�jI� � {ª� yOz r �n�

h[OR(X1[S])] < h[OR(X0[S])] (s = x) (4.5)

h[OR(X1[S])] > h[OR(X0[S])] (s 6= x) (4.6)

� <}|�~h� � k (4.5)� r�8��e¢ mAo^pZq X0

��� { � X1

� | � ¢ �=��� x pAd�Dj OR �� �9���O��� dW�O�� 8 � �Kg � ¨ ¢ X1 �G���n� X0

��� �� Z���^� 0'r �;���^b �G� gM�s 6= x

� r�8���. J�/ ] :Z< � g mZo^pZq � X0 ¨ u g � ¢ k (4.5)�s = x

� re8�� mZoOpq X0 �,0 J�/ ] :=< � g m=oOpZq rª�Dgc����� � ¢ s 6= x� re8���0 J�/ ] :A< � g � � moOpOq X1 ¨ u g � ¢ k (4.5)

�s = x

� re8�� . J-/ ] :Z< � g V r � �DgG�� + � ¨ �M¢W¤^¥ ¦ ] s(t ≤ s ≤ n) BAd��Dj r98 �K«`��¬®­e¯�¬ αs

� >A©�±§Q ¢ αs

� r| {?g W����� � 0 < αs ≤ 1 r �;��� £ � 8 j � �;± � ¢ d� gA¤Z¥ ¦ ¡ � � £ � . J�/ r0 J-/f�!1�2,�nj?FAHKJMLD�MXOY �n� g (t, n)-VSSS ] � { V �,+ ¨ �M¢D¤=¥ ¦ ] s B=d��fjr98 �f«f� ¬ ­M¯}¬ αs

�5 ������ � 0 < |αs| ≤ 1 r � gG� αs

� ��� ��V ��� { � �fg � �e¢0 ≤ h[OR(X0[S])] ≤ q ± � 0 ≤ h[OR(X1[S])] ≤ q ¨ u g V rZr ¢h¤^¥ ¦ ] s BOd��Dj r�8�f«f��¬ ­e¯�¬

αs

� >Z© k (3.1) ± Q"!# g?�h¤`¥ ¦ ] s B=d�Dj r�8 � X^YfJ9L �D«f� ¬­?¯�¬ ��$ r³�Dg αs < 0 �?¢�. J�/ r 0 J�/D�!1�2,�cj?FAHfJ9LD�9XOY �n� g V r �c�=�f�gc� � j ¢W¤^¥ ¦ ] s B=d�Kj r�8 �D«`� ¬ ­M¯}¬ ��% � |αs|�� � 1 �'&�( � ��) FAHJMLh��* ² � �Zb§�;� gc�� j ¢ mZoOpZq � >A© k (4.5), (4.6) � C E ��� { ����� k f ∈ Ft,n ]'+ �A�����cV r �Z¨89gn� t, n ] 2 ≤ t ≤ n ] iKj9� �I���  A¡ r � ¢ s, x ] t ≤ s, x ≤ n ] ihj9� �A���  A¡ r �g;�e¤I¥ ¦ ] x B�d,�`j rM8 � ¢ . J5/ r 0 J5/`��1�2��;jGF�H`J?L`�GXZY��³� g (t, n)-VSSS� mZoOpOq X0, X1 ] :=< � g �� k ] f r � ¢ f ∈ Ft,n r �?j r�8

ψn−sf |z=0 = Csas (s = t, t+ 1, . . . , n) (4.7)

r �.- gG� V=VG¨ ¢ Cs(s = x) � $ �  Z¡�¨ u§| ¢ Cs(s 6= x) � �  Z¡�¨ u gG� k (4.7)�

/�0�M¢ k (3.22)

� |ψn−sf |z=0 =

n−s∑

i=0

βi(n− s)!

(

n− t− i

s− t

)

as (s = t, t+ 1, . . . , n)

r;�Kg V r � �`± £ �O� g?� s ≤ n � � ¨ (n − s)! > 0 �^² Q ± j ��� ¢ 0! = 1 � � � ¨t ≤ x ≤ n ¨ u g x B � ¤=¥ ¦ ] d��Dj r98 � ¢�. J�/ r 0 J!/D�51"2,�cjGFAHDJ?LK�?XOY �� g (t, n)-VSSS

� m=o^pZqD�9iWj���1 8 ��� �n−s∑

i=0

βi

(

n− t− i

s− t

)

< 0 (s = x)

35

n−s∑

i=0

βi

(

n− t− i

s− t

)

> 0 (s 6= x)

r³�Kgc�� r �ª� ¢9¤9¥ ¦ ] 2 B ¢ �=��� �=¤9¥ ¦ ] 4 BId,�^j r98�� F�H`JGLf�GX=Y��³� g � ¢M¤e¥¦ ] 3 BId�`j re8 � ��. J!/ r 0 J-/`�!12��njGFAHDJMLD�MXZY � � g (2, 4)-VSSS� mIopAq ] :=< � gn� βi(i = 0, 1, 2) � β0 + β1 + β2 > 0, 2β0 + β1 < 0, β0 > 0 ] iWje� � { � �� � �K� u g?� ��� �e¢ β0 = 1, β1 = −4, β2 = 6 r���� � � f ∈ Ft,n r � g�r ¢

f = a4 − 6a3z + 15a2z2 − 16az3 + 6z4

= 2 · 3!

[

2a4

4!− 3

a3z

3!1!+ 5

a2z2

2!2!− 8

az3

1!3!+ 12

z4

4!

]

rn� | ¢ g0 = 2a4

4! + 5a2z2

2!2! + 12 z4

4! , g1 = 3 a3z3!1! + 8 az3

1!3! r�� � r ¢ mOofp`q X0, X1 � � � kg0, g1 �G�O�K� g p^q r;�hgG� V � rI8 �D«D��¬ ­e¯}¬ � α2 = 1

44 , α3 = − 144 , α4 = 1

22 rn�gc� α2, α4 > 0, α3 < 0 � � ¨ ¢K¤O¥ ¦ ] 3 B=d�Dj r98 � XOYfJ9L�� - ¢ FZHDJ9L��c���n�. J�/ r 0 J�/f�-1"2 � �^� g V r ��`± gG�

36

�5 � � � �

o����W¨ ��¢ Naor r Shamir [1] � � £ ���,�?�hj (t, n) a^b`X� ��fF^H`x���� C E ¢(t, n)-VSSS r�� � g �`� � �=�"354��Gj � (t, n)-VSSS r�� . 0 2

J9L��n�f� g F=HAx��������� ¨ u�| ¢ 1 B � F=HfJMLK±�Q n B ��� �� �;�`jIJML ( ¤=¥ ¦ ) �?v < �;� ¢ n B � ¤Z¥¦ � n � � ��!�" ��#M��$?� 1 B&% �(' Q � gc� (t, n)-VSSS �G¢ �I�W� t �&)Z_I�Gy � | ¤I¥¦ ] t * (2 ≤ t ≤ n r � g ) )f_MyWlZ�9d��O��+-,�.0/1��9aOb��O�GX^YK¨32 g � ¢ �Z� �t− 1 ��)�4 � ��!�" �?y � £ � �5,�.0/-1W��6e� g87�9��;: �=<?>®� � � r � {A@�BDC � £�Z� gc� (t, n)-VSSS

� ¤^¥ ¦ � m=o^pZq rFE +e� g pZq C�+ �Z�(G�H��c� g?�o���f¨ �G¢ t+1 *�)�I ¤9¥ ¦ C d,� g�JOr � � £ � ¢?¤e¥ ¦ C t *�d,� gDr 2 � | � ¢LK�M�³�0NO,P.�/�1`��* ² � ��Q � ��N5R=z`� u N J^r CM² > ±O�D�8S � � S ¢5T�U � (t, n)-VSSS¨ �9¢ t * ¤Z¥ ¦ C5V ��S r 29� K�M � �WNF,�.�/-1 �`«`� ¬ ­M¯�¬ α C8X�Y � �Z��SI� ¢ o����¨ �9¢ s(t ≤ s ≤ n) * ¤Z¥ ¦ C�V ��S r 29� K�M �;�ZN;,�.0/-1 �f«^� ¬ ­9¯�¬ αs C8[\ �,� X�Y � S ��@ � ¢ t = 2, 3, 4, 5�^] 2 ¢D¤Z¥ ¦ C n *`��1O� V ��S ] 2 � K�M /(1W� �

-�N «f��¬®­e¯ ¬ αn � ¢ n → ∞ ¨ 1 ��_�`h�WN8a &�b&c&d C � e (t, n)-VSSS f�g�h�i�jf;k�l Ce² > ±�mWn S�o(a�&�b�c�d C&p e (t, n)-VSSS f g�h�i�jWf «�q ¬ ­M¯�¬;] ¢f¤=¥ ¦C s(t ≤ s ≤ n) * V �0S ] 2m b�r ]Fs N «�q ¬ ­e¯�¬ C ¡ �&m�t�uvn�S ] Jw?¢W¤O¥ ¦C5V �ZNO*=¡ C5x�y�z ��) ¢ a &�b�c�d C�p e (t, n)-VSSS f;g�h�i�jWf «&q ¬ ­M¯ ¬ f {]s(t ≤ s ≤ n) * V ��S ] 2m;b�r ]Os N «�q ¬ ­M¯}¬ f f�|�}~��%� s N J ] CO�&� n�S(o� S ¢�T�U f (t, n)-VSSS m(e���� V �ZN ¤=¥ ¦ ¡ �S��Zn ¢ t+1 *&)&I�f ���Wf�*I¡���m� £ � � ,�.0/(10} K�M �8� s ��R&��}���N J ] ¢ � � �e¢�V �ZN ¤Z¥ ¦ ¡ �S���n ¢ t *&)I���m � £ � ¢ . /�/ ] 0 /�/0}�1 2vn�S�,�.Z/-1Z} K&M � ��NFR���}���N J ] CI² >³±�mn So V ��N ¤`¥ ¦ ¡^m � £ � �9¢ ,�.Z/-1Z} K&M �;� s �(R�� ¢ . /�/ ] 0 /-/�}�1 2vn� K�M � �ZN8R���f 2 e�f (t, n)-VSSS }(�3S z � � CF[ \ �{n�S-ot ≥ 6 m;�vnF�a &�b&c s (t, n)-VSSS f g�h�i�j0}-k�l z N9± )�� ± �9¢ � f ] JPw;���� ¨��&N o�h�����f�� 3 ������{n S ����� ��� t }��2�� C ] N � ) R���� -�f;�Z} x #�P�&� } x {n;�Dn(��� � n * V �0S ] 2-f0� qD ¢¡-£¤  C ��¥ m n f \ �-¦ z§J ] �(¨&©� s �o t ≥ 6 f;R���� p�� a�&�b�c s g�h�i�j�}(k�l z N ]«ª�¬ � �;�0N(} �(­ ����m ®Z¯z N J ] ��°�± f(²�³W�&��N o

37

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h���� C��� N�m���S� � � ��� m s 8�Wn�S������ y�7�9�­ �������3f�� q��� ��� �� m�����>! �"�#?n(� z o%$�&�')(&����N�*,+.-0/,1�'�2 �,3�4.5 /6'�2 �87�9�:�;�<)= m>%? ��� m s 8�0n�SoW��S �6@,A�C�BDCFE 2FGIHP� C�J �6KPS ��L,M�N61�'�2^m > �"{n� ? O� z o@ m � *8+61�'�2§m > �8O�PQSR ��>UT � N ] 3 V.W �,XYC f Q W C�Z ���\[]$�&Z��S J2�Wn�S-o8^6_ 2 ` [�aIb m%ced�f Q W Cgf H�����S J 2 �P7�9�­�h m ? H�N��P�ji� s �k f8lnmZ� o�p Crq mPejH�N J ] }��32�Wn�So�csd�f�tvu ] n8�%w�¦?n S,x�m >�y�z mg$&{n;���&S J 2 � S��n{�|S �" nF� ? F� z o3��> m �8h8}6~�� m ? H&N O��6h8} f�� 2�� ��e� �,���3qD  �&f��e�]c qr�����v�-q;sv� �,��� mS���� y�z�C���� N J ] f V�� d C '� �&��S J�� �Wn�S-o,�6� ­�h ���6�W�%��S J ] > XYC ��°�± f���� q f��^mWn;�&� � S��] � ��� z o��S � ���6�^f�� q��� m > �P��iP� s ¡ � ? ��� m s F�Zn�¢o8£�¤�¥�¦�§�¨�©���ª�«¬ ���D�F������¢r­�®n�Wf�¯,°,±���| �%².³jC ­�®n�Wfg´6¯8µ,¶��g| ��·�¸,¹�º �g|&m > � ���m�»Wn8�%¼6½ s 1j¾ ¬ ��¢ J�� �Wn0¢-o8�����I¿ º�À�Á ¿���Â�m���¢�Ã�Ä6Å 5 / ��| ��ÆÇ ´�È�É�|jÊ > � ����Ë�ÌIÊ�»%ÍÃ�1j¾ >ÎjÏnÐ �%Ñ�Ò Ë�Ì)Ó � O ����Ë\Ê ? H�Ã�É���i�8Ó�ÔÖÕ �)��£ ¬S× ¢ J)� �Ø�¢nÙW�6¢%Ún� Ï ²�QSa Ê0�����eÊ�Û�ÜÝÉgÞ�¢ ­�ß�O ����à 2

V6�\¿0áeârã�´�ä�Ú �P£6��å ��� ß�O ����à 1 V6�\¿�æ�ç 5 ��äIÊ > Ú ²0a Ï ×vèné Ï�êSë×0ì�ív¬ G\HgÚ%î�ï6µ�ðñØgò%���\Ê�ó)ô é Ï�õ�ê �nö Ø0¢SÙ,���6�I¿��)÷ �� ê�ø à �nù6�å ��� ß6O ����à 1 V��.¿0ú�ûSü�ä�Ú.ý k�þ ä�Ú ß6O �nù���å O�ÿ 4 V��I¿��.W�����ä�Ú Æ����� ä�Ú� � º�� ä.Ê > Ú�c\d ê ¿�� � ¿�����w�� ¬�� ³ òSm × o�p ¬�� � ò × ¢ J)�ö Ø0¢ Ù��� Ê Ú������ ¬ > ³! Ú"��Â��$#������e�j÷ �� Ú ?IÐ�% É ö i ö Ó ¡ ê � ¬�& � ò CJ É�Kn¢ �n� ¢�'�Êg �" × ¢Ø ö ÍSÙ ø)(]õeÏ è [�i × ö Ø�¢%Ù

38

� � � �

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LNCS 950, pp. 1–12, Springer-Verlag, 1995.

[2] H. Koga, “A general formula of the (t, n)-threshold visual secret sharing scheme,” Ad-

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White Images,” Proc. of the 3rd Asia-Europe Workshop on Information Theory, Kamogawa,

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[4] D. Viet and K. Kurosawa, “Almost ideal contrast visual cryptography with reversing,” CT-RSA

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2005.

39

� �A (5, n)-VSSS � � � � � � � � � �

3.3.4 ��� (5, n)-VSSS Ê������ n = 4m+ 1, 4m+ 2, 4m + 3 � Ï�� ����� � ë Ó�� � �"!��#%$"&('�) × ò+*-,+.-� /10 J3254 n 6 n ≥ 8 79830+.�:�;<�>="?A@CB1. n = 4m+ 1 �(D �n = 4m+ 1 (m ≥ 2) D 2 0ED � �AF�÷EGIH ù G αn

õ �"��J �%K D9LA� (5, 4m + 1)-VSSS�M��� ��!3N #%$<O � �QPQD�7SR<.MB ωi (i = 0, 1, 2, 3) 79=T?UD 294"V"W 3.2 � h(k)t,n X ( 4

f∗5,4m+1 7 h(0)5,4m+1, h

(m)5,4m+1, h

(2m−1)5,4m+1 , h(3m−2)5,4m+1 �>Y"Z%[%\]D 29^f∗5,4m+1 = ω0h

(0)5,4m+1 + ω1h

(m)5,4m+1 + ω2h

(2m−1)5,4m+1 + ω3h

(3m−2)5,4m+1 (A.1)

D_.A�9B VMW 3.2 X ( h(k)5,4m+1 6 e(i)5,4m+1 (i = 0, 1, . . . , 4m−4) �9Y`Z [+\�O�a���bEc 4 f∗5,4m+1d

e(i)5,4m+1 (i = 0, 1, . . . , 4m − 4) ��YTZA[�\3OQa-�3PQD�'MeT;<.(��B ωi(i = 0, 1, 2, 3) 6

a4m−1z2, a4m−2z3, a4m−3z4 �%f�g-h�g���i��kj ? N 0 '+L�� X�l 'Sm3n`B o-p�6ω0 = 3(m− 1)2(m+ 1)(m+ 2)(m+ 3)(2m + 1)(3m+ 1)

ω1 = −2(m+ 1)(m+ 2)(m + 3)(2m − 1)(2m + 1)(3m− 2)(4m + 1)

ω2 = 6m2(m+ 2)(m+ 3)(3m − 2)(3m + 1)(4m + 1)

ω3 = −6m2(m+ 1)(2m− 1)(2m + 1)(3m+ 1)(4m + 1)

Dqm3r J BP��ED � 4�s (A.1) ' ωi(i = 0, 1, 2, 3) 79t�u 2 0kv"i s f∗5,4m+1

N(5, n)-VSSS �k���Tw! 79#%$Q.Q� 0�x��Mv�i s O�aT� 0�xT'�6 4 f∗5,4m+1 ∈ F5,4m+1 O�aA�Sy�z N aA�kB e(i)5,4m+1

(i = 0, 1, . . . , 4m− 4) �kj ?��>{<6ω0

4m−4∑

i=0

(

4m− i

4

)

+ω1

3m−4∑

i=0

(

3m− i

4

)

+ω2

2m−3∑

i=0

(

2m+ 1 − i

4

)

+ω3

m−2∑

i=0

(

m+ 2 − i

4

)

=1

10(m−1)2m2(m+1)(m+2)(m+3)(2m−1)(2m+1)(3m−2)(3m+1)(4m+1) > 0

D5L}|�~<L%� O 4 f∗5,4m+1 ∈ F5,4m+1 O�aA�TPAD N�� b%x�cIg3�>B� ' 4+s (A.1) ' ωi(i = 0, 1, 2, 3) 7�t�u 2 0�v"i s f∗5,4m+1 b�c�#�$��kg������Aw ! '�A� ^"43���+� 7 n ���Q�30�D � �M�����+�(�3F�����H`��� α4m+1 7S��x3�MB s (A.1) '40

ωi(i = 0, 1, 2, 3) 7�t"u 2����(2 ^�� cIg-� s D 4%� ��� 7 n � �T��0<D � �AF%�E� HM� � αn� s (3.23) X |α4m+1 =

3(m− 1)2m2(m+ 2)(3m − 2)

(m+ 3)(3m + 1)(3m4 + 4m3 + 16m2 − 4m− 4)→ 1 (m→ ∞)N���� O � �MB X K ^ n = 4m + 1 ��D � 4 n 7���� �� .<�3D αn → 1 798(0`. X-l L

(5, n)-VSSS �����%w !AN���� .Q�APAD N�� b%x�cIg�0�B2. n = 4m+ 2 �(D �n = 4m+ 2 (m ≥ 2) D 2 0ED � �AFA�E�IHM� � αn

N �"��J�� K D9LA� (5, 4m + 2)-VSSS�M����w !3N #%$<O � �QPQD�7SR<.MB ωi (i = 0, 1, 2, 3) 79=T?UD 294"V"W 3.2 � h(k)t,n X | 4

f∗5,4m+2 7 h(0)5,4m+2, h

(m)5,4m+2, h

(2m)5,4m+2 , h(3m−1)5,4m+2 �>Y"Z%[%\]D 29^f∗5,4m+2 = ω0h

(0)5,4m+2 + ω1h

(m)5,4m+2 + ω2h

(2m)5,4m+2 + ω3h

(3m−1)5,4m+2 (A.2)

D_.A�9B VMW 3.2 X | h(k)5,4m+2 6 e(i)5,4m+2 (i = 0, 1, . . . , 4m−3) �9Y`Z [+\�O�a���bEc 4 f∗5,4m+2d

e(i)5,4m+2 (i = 0, 1, . . . , 4m − 3) ��Y�Z�[�\AOAa���PTD>'Se�;-.Q�>B n = 4m+ 1 �ED � D��� ' 4 ωi(i = 0, 1, 2, 3) 6 a4m−1z2, a4m−2z3, a4m−3z4 ��f�g-h�g��>i��Mj"? N 0 ' LA� X�l'9m<n�B"o�pT6

ω0 = (m− 1)(m+ 1)(m+ 2)(m+ 3)(2m − 1)(3m+ 1)(3m + 2)

ω1 = −4(m− 1)(m+ 1)(m+ 2)(m+ 3)(2m + 1)(3m− 1)(4m + 1)

ω2 = (m+ 2)(m+ 3)(2m − 1)(3m − 1)(3m + 1)(3m+ 2)(4m + 1)

ω3 = −4m2(m+ 1)(2m + 1)(3m+ 1)(3m + 2)(4m + 1)

Dqm3r J BP��ED � 4�s (A.2) ' ωi(i = 0, 1, 2, 3) 79t�u 2 0kv"i s f∗5,4m+2

N(5, n)-VSSS �k���Tw! 79#%$Q.Q� 0�x��Mv�i s O�aT� 0�xT'�6 4 f∗5,4m+2 ∈ F5,4m+2 O�aA�Sy�z N aA�kB e(i)5,4m+2

(i = 0, 1, . . . , 4m− 3) �kj ?��>{<6ω0

4m−3∑

i=0

(

4m+ 1 − i

4

)

+ω1

3m−3∑

i=0

(

3m+ 1 − i

4

)

+ω2

2m−3∑

i=0

(

2m+ 1 − i

4

)

+ω3

m−2∑

i=0

(

m+ 2 − i

4

)

=1

30(m− 1)m2(m+ 1)(m + 2)(m+ 3)(2m− 1)(2m + 1)(3m− 1)(3m + 1)

×(3m+ 2)(4m + 1) > 0

D9L}| ~�LA�"O 4 f∗5,4m+2 ∈ F5,4m+2 OTa��-P-D N`� bTx�c g��kB � ' 4�s (A.2) ' ωi(i =

0, 1, 2, 3) 7 t%u 2 0Mv+i s f∗5,4m+2 b�c #%$���g<�9���%w ! '`�T� ^ 43���+� 7 n � �-��0D � �S�%�����Q�AF��E� H�� � α4m+2 7 �QxT�>B s (A.2) ' ωi(i = 0, 1, 2, 3) 7�t"u 2�����2^�� c�g<� s D 4<�%� � 7 n �"����0(D � ��FT�E�IHM� � αn � s (3.23) X |α4m+2 =

(m− 1)m2(m+ 2)(3m− 1)(3m + 1)

(m+ 3)(3m + 2)(3m4 + 8m3 + 17m2 − 4)→ 1 (m→ ∞)

41

N���� O � �MB X K ^ n = 4m + 2 ��D � 4 n 7���� �� .<�3D αn → 1 798(0`. X-l L(5, n)-VSSS �����%w !AN���� .Q�APAD N�� b%x�cIg�0�B3. n = 4m+ 3 �(D �n = 4m+ 3 (m ≥ 2) D 2 0ED � �AFA�E�IHM� � αn

N �"��J�� K D9LA� (5, 4m + 3)-VSSS�M����w !3N #%$<O � �QPQD�7SR<.MB ωi (i = 0, 1, 2, 3) 79=T?UD 294"V"W 3.2 � h(k)t,n X | 4

f∗5,4m+3 7 h(0)5,4m+3, h

(m)5,4m+3, h

(2m)5,4m+3 , h(3m)5,4m+3 �kY Z�[�\ D 2S^f∗5,4m+3 = ω0h

(0)5,4m+3 + ω1h

(m)5,4m+3 + ω2h

(2m)5,4m+3 + ω3h

(3m)5,4m+3 (A.3)

D_.A�9B VMW 3.2 X | h(k)5,4m+3 6 e(i)5,4m+3 (i = 0, 1, . . . , 4m−2) �9Y`Z [+\�O�a���bEc 4 f∗5,4m+3d

e(i)5,4m+3 (i = 0, 1, . . . , 4m− 2) �>Y+Z�[%\AO�a��TPTD�'Se+;-.-�>B n = 4m+ 1, n = 4m+ 2�ED � D � � ' 4 ωi(i = 0, 1, 2, 3) 6 a4m−1z2, a4m−2z3, a4m−3z4 �Tf�g-h`g-��i(��j ? N 0'`L-� X-l '�mQn+B+oQp%6

ω0 = (m+ 1)(m+ 2)(m+ 3)(2m + 3)(3m + 2)

ω1 = −2(m+ 2)(m + 3)(2m + 1)(2m + 3)(4m+ 3)

ω2 = 3(m+ 2)(m+ 3)(2m+ 1)(3m + 2)(4m+ 3)

ω3 = −2(m+ 1)(2m + 1)(2m + 3)(3m + 2)(4m + 3)

D m3r J B�P���D � 4+s (A.3) ' ωi(i = 0, 1, 2, 3) 79t�u 2 0�v"i s f∗5,4m+3

N(5, n)-VSSS�k���Tw ! 79#�$Q.3�+0"xT�Mv"i s OAaT�"0"xT'�6 4 f∗5,4m+3 ∈ F5,4m+3 O�aA�9y�z N aA�kB

e(i)5,4m+3 (i = 0, 1, . . . , 4m− 2) �Mj+?(��{�6ω0

4m−2∑

i=0

(

4m+ 2 − i

4

)

+ ω1

3m−2∑

i=0

(

3m+ 2 − i

4

)

+ ω2

2m−2∑

i=0

(

2m+ 2 − i

4

)

+ ω3

m−2∑

i=0

(

m+ 2 − i

4

)

=1

10m3(m+ 1)(m + 2)(m + 3)(2m+ 1)(2m + 3)(3m+ 2)(4m + 3) > 0

D9L}| ~�LA�"O 4 f∗5,4m+3 ∈ F5,4m+3 OTa��-P-D N`� bTx�c g��kB � ' 4�s (A.3) ' ωi(i =

0, 1, 2, 3) 7 t%u 2 0Mv+i s f∗5,4m+3 b�c #%$���g<�9���%w ! '`�T� ^ 43���+� 7 n � �-��0D � �S�%�����Q�AF��E� H�� � α4m+3 7 �QxT�>B s (A.3) ' ωi(i = 0, 1, 2, 3) 7�t"u 2�����2^�� c�g<� s D 4<�%� � 7 n �"����0(D � ��FT�E�IHM� � αn � s (3.23) X |α4m+3 =

m3(m+ 2)(3m + 2)

(m+ 1)(m+ 3)(3m3 + 12m2 + 23m+ 12)→ 1 (m→ ∞)

N���� O � �MB X K ^ n = 4m + 3 ��D � 4 n 7���� �� .<�3D αn → 1 798(0`. X-l L(5, n)-VSSS �����%w !AN���� .Q�APAD N�� b%x�cIg�0�B

42