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Ph.D. DISSERTATION
Design of Irregular SC-LDPC CodesBased on Various Optimization
Techniques
다양한최적화기법을통한비균일 SC-LDPC부호의설계
BY
KWAK HEE-YOUL
FEBRUARY 2019
DEPARTMENT OF ELECTRICAL ENGINEERING ANDCOMPUTER SCIENCE
COLLEGE OF ENGINEERINGSEOUL NATIONAL UNIVERSITY
Ph.D. DISSERTATION
Design of Irregular SC-LDPC CodesBased on Various Optimization
Techniques
다양한최적화기법을통한비균일 SC-LDPC부호의설계
BY
KWAK HEE-YOUL
FEBRUARY 2019
DEPARTMENT OF ELECTRICAL ENGINEERING ANDCOMPUTER SCIENCE
COLLEGE OF ENGINEERINGSEOUL NATIONAL UNIVERSITY
Design of Irregular SC-LDPC CodesBased on Various Optimization
Techniques
다양한최적화기법을통한비균일 SC-LDPC부호의설계
지도교수노종선
이논문을공학박사학위논문으로제출함
2019년 2월
서울대학교대학원
전기컴퓨터공학부
곽희열
곽희열의공학박사학위논문을인준함
2019년 2월
위 원 장:부위원장:위 원:위 원:위 원:
Abstract
In this dissertation, two main contributions are given as i) design methods of irreg-
ular spatially-coupled low-density parity-check (SC-LDPC) codes with non-uniform
degree distributions by linear programming (LP) and ii) rate-loss mitigation methods
of SC-LDPC codes without degradation of the finite-length code performance using
differential evolution algorithms.
First, new design algorithms of irregular SC-LDPC codes with non-uniform de-
gree distributions are proposed using LP optimization techniques. In general, irregular
SC-LDPC codes with non-uniform degree distributions are difficult to design with
low complexity because their density evolution equations are multi-dimensional. To
overcome the problem, proposed design algorithms are based on three main ideas; a
local design of degree distributions, pre-computation of the input/output message re-
lationship, and selection of a proper objective function. These ideas make it possible
to design degree distributions of irregular SC-LDPC codes by solving low complexity
LP problems over the binary erasure channel (BEC). It is shown that the proposed ir-
regular SC-LDPC codes designed by the proposed algorithms are superior to regular
SC-LDPC codes in terms of both asymptotic and finite-length performances over the
BEC. It is also confirmed that the proposed irregular SC-LDPC code achieves better
performance compared to an optimized irregular block LDPC code in the same block-
length, which implies that the proposed design algorithms also provide a new way to
construct capacity-approaching block LDPC codes.
Second, new optimization methods are provided to mitigate the rate-loss of SC-
LDPC codes without finite-length performance degradation over the binary erasure
channel. In the SC-LDPC codes, the rate-loss of SC-LDPC codes is one of the main
issues to be addressed. One way to mitigate the rate-loss is attaching additional vari-
able nodes with an irregular degree distribution to the boundary check nodes, where
i
the degree distribution of additional variable nodes is optimized with a constraint that
the BP threshold should not be degraded by attaching them. However, it is observed
that the degree distribution obtained with the BP threshold constraint degrades the
finite-length performance. In order to maintain the finite-length performance, a proper
design constraint is given using the expected graph evolution and then the constraint
is imposed on the proposed optimization method, which is based on differential evolu-
tion algorithms. From the well-designed degree distribution, the rate-loss of SC-LDPC
codes is mitigated by 54% without sacrificing the finite-length performance. It is also
shown that the rate-loss mitigation can be translated into a performance improvement
if the conventional SC-LDPC codes and the proposed SC-LDPC codes are compared
for the same code-rate.
keywords: Density evolution, error correcting codes, linear programming (LP)
problem, low-density parity-check codes, non-uniform degree distribution, rate-loss,
spatially-coupled low-density parity-check codes (SC-LDPC).
student number: 2013-20743
ii
Contents
Abstract i
Contents iii
List of Tables vi
List of Figures vii
1 INTRODUCTION 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Overview of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Overview of LDPC Codes 6
2.1 LDPC Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Decoding of LDPC Codes in the BEC . . . . . . . . . . . . . . . . . 9
2.2.1 MAP Decoding of LDPC Codes . . . . . . . . . . . . . . . . 9
2.2.2 BP Decoding of LDPC Codes . . . . . . . . . . . . . . . . . 9
2.3 Analysis of LDPC Codes . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.1 Density Evolution . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.2 Optimizing Degree Distribution . . . . . . . . . . . . . . . . 11
3 Design of Irregular SC-LDPC Codes with Non-Uniform Degree Distribu-
tions by Linear Programming 13
iii
3.1 Code Structure of SC-LDPC Ensembles . . . . . . . . . . . . . . . . 16
3.1.1 Construction of SC-LDPC Ensembles . . . . . . . . . . . . . 16
3.1.2 DE Equations of SC-LDPC Ensembles . . . . . . . . . . . . 19
3.1.3 Expected Graph Evolution of SC-LDPC Ensembles . . . . . . 20
3.2 Proposed Design Algorithms of SC-LDPC Ensembles . . . . . . . . . 23
3.2.1 Pre-Computation of δu(z) . . . . . . . . . . . . . . . . . . . 23
3.2.2 Objective Function 1: Maximizing Design Rate . . . . . . . . 24
3.2.3 Objective Function 2: Minimizing the Number of Required It-
erations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.4 Improving the Performance by Permitting Non-Uniform Check
Node Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.1 Performance Improvement Using Multi-Edge Type Check Nodes 37
3.3.2 Windowed Decoding of Proposed SC-LDPC Codes for Mod-
erate and Large L . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.3 Comparing the BP Decoding Performance of SC-LDPC Codes
for Small L . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3.4 Applying the Proposed Algorithms to SC-RA Codes . . . . . 50
3.3.5 Comparing Finite-Length Performances of SC-LDPC and SC-
RA Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 Rate-Loss Mitigation of SC-LDPC Codes without Degradation of Finite-
Length Performances 58
4.1 Code Structure of SC-LDPC Codes . . . . . . . . . . . . . . . . . . 60
4.1.1 (l, r, L) SC-LDPC Ensemble . . . . . . . . . . . . . . . . . . 60
4.1.2 (l, r, L, λ(x)) SC-LDPC Ensemble . . . . . . . . . . . . . . 62
4.1.3 DE Equations of the (3, 6, L, λ(x)) Ensemble . . . . . . . . . 63
4.2 Optimizing Methods of Degree Distribution λ(x) . . . . . . . . . . . 64
4.2.1 Minimizing the Rate-Loss While Maintaining the BP Threshold 64
iv
4.2.2 Minimizing the Rate-Loss with Target Local Threshold . . . . 66
4.2.3 Minimizing the Rate-Loss with Target Local Threshold and
without Local Minimum of r1 . . . . . . . . . . . . . . . . . 67
4.2.4 Optimizing the (3, 6, L,B, αA) Ensemble . . . . . . . . . . . 71
4.3 Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3.1 Independence of L, z, and Decoding Algorithms for Opti-
mized Results . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.2 Comparison in the Same Design Rate . . . . . . . . . . . . . 77
5 Conclusions 81
Abstract (In Korean) 92
v
List of Tables
3.1 Parameters of the Tanner graphs in Figure 3.1 . . . . . . . . . . . . . 18
3.2 λu(x) for the proposed irregular SC-LDPC ensemble from Algorithm
3.3 for L = 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Comparison of the BP threshold of the regular SC-LDPC and the pro-
posed irregular SC-LDPC ensembles . . . . . . . . . . . . . . . . . . 33
3.4 Comparison of the BP thresholds between the regular SC-LDPC en-
sembles and proposed irregular SC-LDPC ensembles with the MET
structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5 BP and windowed thresholds of the regular and proposed irregular SC-
LDPC ensembles along with their folded versions for L = 40 . . . . . 43
3.6 BP thresholds of the regular SC-RA and the SC-RA ensembles de-
signed by Algorithm 3.3 . . . . . . . . . . . . . . . . . . . . . . . . 51
3.7 Coefficients of λu(x) for the SC-RA ensemble designed by Algorithm
3.3 for L = 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.8 Comparing the BP thresholds of the regular SC-RA ensembles with
the ensembles obtained by the proposed algorithm . . . . . . . . . . . 53
4.1 Comparing the (3, 6, 20, λ1(x)) and (3, 6, 20,B1, 0.05) SC-LDPC en-
sembles with respect to γ and the ratio of BLERs at ε = 0.45. . . . . . 75
vi
List of Figures
1.1 Block diagram of a digital communication system. . . . . . . . . . . 2
2.1 Tanner graph of LDPC code. . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Binary erasure channel with parameter ε. . . . . . . . . . . . . . . . . 9
3.1 Examples of Tanner graphs of SC-LDPC codes for L = 4, w = 2, and
M = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Design procedure of the proposed algorithms with L = 4, w = 2, and
M = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Average variable node degree of each position for the SC-LDPC en-
semble designed by Algorithm 3.2. . . . . . . . . . . . . . . . . . . . 27
3.4 Evolution of average convergence speed as the for-loop in Algorithm
3.3 proceeds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5 Evolution of r1(τ) for the SC-LDPC ensembles designed by Algo-
rithm 3.2 as the inner for-loop proceeds. . . . . . . . . . . . . . . . . 31
3.6 Evolution of r1(τ) for the SC-LDPC ensembles designed by Algo-
rithm 3.3 as the inner for-loop proceeds. . . . . . . . . . . . . . . . . 32
3.7 Correlation between two numbers of required iterations. . . . . . . . 34
3.8 Design rate and BP threshold of the regular SC-LDPC and proposed
irregular SC-LDPC ensembles for various value L. . . . . . . . . . . 41
vii
3.9 Threshold εWINu of the regular SC-LDPC and the proposed irregular
SC-LDPC ensembles along with their folded versions for L = 40. . . 42
3.10 Tanner graphs of an original SC-LDPC code for L = 6 and w = 3 and
the corresponding folded SC-LDPC code. . . . . . . . . . . . . . . . 44
3.11 Block and bit erasure probabilities of the folded regular SC-LDPC
code and the folded irregular SC-LDPC codes for L = 40 under win-
dowed decoding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.12 Block and bit erasure probabilities of the folded regular SC-LDPC
code and the folded irregular SC-LDPC codes for L = 100 under
windowed decoding. . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.13 BP decoding performances of the regular SC-LDPC code, the pro-
posed irregular SC-LDPC codes, and the block irregular LDPC code
with the same design rate and blocklength 9,900. . . . . . . . . . . . 48
3.14 BP decoding performances of the regular SC-LDPC code, the pro-
posed irregular SC-LDPC codes, and the block irregular LDPC code
with the same design rate and blocklength 1,500. . . . . . . . . . . . 49
3.15 Tanner graph of the SC-RA ensemble for q = 3, L = 2, and M = 2. . 51
3.16 Two types of stopping sets produced by low degree variable nodes. . . 53
3.17 Block erasure probability of the SC-RA codes for L = 20. . . . . . . 55
3.18 Block erasure probability of the SC-RA codes for L = 30. . . . . . . 56
4.1 BLER of the (3, 6, 20) conventional SC-LDPC code and the (3, 6, 20, λ1(x))
SC-LDPC code in [51]. . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 BLERs of the (3, 6, 20) code, the (3, 6, 20, λ1(x)) code, and the pro-
posed (3, 6, 20,B1, 0.05) code along with local BLERs of the (3, 6, 20, λ1(x)),
(3, 6, 20, λ2(x)), (3, 6, 20, λ3(x)), (3, 6, 20,B1, 0.05) codes. . . . . . 68
4.3 Evolution of r(`)1 (ε) of the (3, 6, 20) and (3, 6, 20, λ(x)) SC-LDPC en-
sembles with λ1(x), λ2(x), and λ3(x) at ε = 0.4681. . . . . . . . . . 69
viii
4.4 Evolution of r(`)1 (ε) of the (3, 6, 20) and (3, 6, 20, λ(x)) SC-LDPC en-
sembles with λ1(x), λ2(x), and λ3(x) at ε = 0.4781. . . . . . . . . . 70
4.5 Plots of r(`)1 (ε) and Pr(failure at x(`)1 ) against x(`)1 at ε = 0.4781 for
the (3, 6, 20, λ1(x)) SC-LDPC ensemble. . . . . . . . . . . . . . . . 71
4.6 Design rates of the (3, 6) regular LDPC ensemble and (3, 6, L), (3, 6, L, λ1(x)),
and (3, 6, L,B1, 0.05) SC-LDPC ensembles for various values of L. . 74
4.7 Comparison of BLERs of C(L) and P(L) for various values of z to
show independence of z. . . . . . . . . . . . . . . . . . . . . . . . . 75
4.8 Comparison of BLERs of C(L) and P(L) for various values of L to
show independence of L and the decoding algorithm. . . . . . . . . . 76
4.9 Comparison of BLERs of C(L) and P(L) for different values of L
with target design rate 0.5. . . . . . . . . . . . . . . . . . . . . . . . 78
4.10 BP thresholds of C(L) and P(L) to achieve target rate 0.5 by puncturing. 79
4.11 Comparison of BLERs of punctured C(L) andP(L) with target design
rate 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
ix
Chapter 1
INTRODUCTION
1.1 Background
The fundamental goal of digital communication is reliable transmission of information
from source to sink via noisy channel. In 1948, Claude Shannon proved in his landmark
paper [1] that errors induced by noisy channel can be removed to any desired level by
error-correcting codes as long as the code rate is less than the channel capacity. Since
then, error correcting codes plays a pivotal role in the digital communication systems.
The digital communication systems can be categorized as three main functional units;
source coding, channel coding, and (de)modulation, as illustrated in Figure 1.1 [2], [3].
In the transmitter, information to be transmitted is converted into a sequence of binary
information bits by the source encoder. Next, the channel encoder adds redundant bits
to the information bits by an error correcting code to ensure a reliable transmission
over noisy channel.
Unfortunately, Shannon used a random coding technique, which is efficient to
prove the theoretical result but impractical to implement, and did not report how to
find practical error-correcting codes. Ever since, coding theory has been concentrated
on finding good error correcting codes with practically implementable encoding and
decoding schemes [2]–[7]. In the first 50 years, coding theory mainly focuses on block
1
Source
EncoderSource
Channel
EncoderModulator
Channel
DemodulatorChannel
Decoder
Source
DecoderSink
Noise
Figure 1.1: Block diagram of a digital communication system.
codes based on algebraic background including Hamming codes [8], Golay codes [9],
Reed-Muller codes [10], [11], Bose-Chaudhuri-Hocquenghem (BCH) codes [12], [13],
and Reed-Solomon (RS) codes [14]. Another important class of error correcting codes
is probabilistic coding such as convolutional codes [15], turbo codes [16], and low-
density parity-check (LDPC) codes [17].
In 1963, LDPC codes were originally invented by Gallager [17]. However, due
to the limits of implementation of the probabilistic iterative decoding scheme in the
1960s, the excellent potential of LDPC codes is undiscovered at the beginning. As
a result, LDPC codes have been forgotten and neglected over 30 years until turbo
codes were introduced by Berrou, Glavieus, and Thitimajshima in 1993. The origi-
nal turbo code is just a parallel concatenation of two convolutional codes but attain
near-Shannon limit performance with low complexity iterative decoding. Shortly after
achieving the remarkable improvement, probabilistic coding based on iterative decod-
ing is vigorously investigated and, in 1996, LDPC codes were rediscovered by MacKay
and Neal [18], [19].
Since the rediscovery of LDPC codes, LDPC codes have been the main research
topic in the channel coding area. As a major breakthrough, Luby et al. introduced
irregular LDPC codes which have very good error-correcting performance closely
approaching the theoretical limit over the binary erasure channel (BEC) [20]. These
ideas are developed further by Richardson and Urbanke to design capacity approach-
2
ing LDPC codes over large class of binary input channels including the additive white
Gaussian noise (AWGN) channel [21]. They also establish the belief propagation (BP)
decoding algorithm, which is the optimal iterative decoding algorithm over the AWGN
channel, and propose an important analysis tool, called density evolution, to analyze
the performance of codes under BP decoding. Along with the advent of code design
techniques, LDPC codes have been adopted in many communication systems such as
wireless metropolitan area network (WMAN), wireless local area network (WLAN),
10GBase-T Ethernet, and digital video broadcasting [22]-[25].
As a convolutional counterpart of block LDPC codes, Felstrom and Zigangirov
introduced LDPC convolutional codes, which is later called spatially-coupled LDPC
(SC-LDPC) codes [26]. There have been lots of research results on construction and
analysis of LDPC convolutional codes [27]–[35]. Especially, Sridharan et al. [30] de-
rived the BP threshold as a theoretical performance limit of the iterative decoding and
Lentmaier et al. [33] empirically showed that the BP threshold of convolutional LDPC
codes approaches to the MAP threshold of underlying regular block LDPC codes. This
remarkable improvement of the BP threshold is named as the threshold saturation ef-
fect, which is analytically proved in [36] over the BEC and general binary memoryless
symmetric (BMS) channels [37]. Proving the threshold saturation effect also has been
done through many different approaches such as a potential function [38], [39] and an
one-dimensional continuous coupled system [40]. These approaches utilize the unique
decoding process of SC-LDPC codes, where reliable messages are propagated from
the both ends to the center of the graph in a wave-like manner.
SC-LDPC codes are constructed by coupling L disjoint LDPC codes with a bound-
ary condition. From the boundary condition, the wave-like decoding is triggered and
propagated into the center of the graph. However, at the cost of the wave-like propaga-
tion, the rate-loss is induced, where the design rate of SC-LDPC codes is reduced from
that of underlying uncoupled codes. Since the rate-loss converges to zero at a speed
1/L, an asymptotically good code ensemble is obtained for sufficiently large L while
3
the wave-like decoding process is still successful even for extremely large L. The long
block length induced by large L values can be addressed by the windowed decoding
scheme [53] whose decoding complexity and latency are independent of L. However,
an in-depth analysis regarding a scaling law of SC-LDPC codes shows that large L is
not good for the finite-length performance [41]; the block erasure probability is lin-
early increasing with L. Thus, it is practically important to adjust L considering the
trade-off between the asymptotic and finite-length performances of SC-LDPC codes.
Furthermore, the enhancement of SC-LDPC codes for a finite value of L by varying
code structures can be a good research topic.
1.2 Overview of Dissertation
This dissertation is organized as follows. In Chapter 2, the notation and review of
LDPC codes are introduced. Section 2.1 presents basic descriptions of LDPC codes
such as Tanner graph representation of LDPC codes and irregular LDPC ensembles.
In Section 2.2, decoding algorithms of LDPC codes are illustrated and their decoding
complexities are discussed. In Section 2.3, density evolution of LDPC codes and an
optimization method of irregular LDPC ensembles are briefly described.
In Chapter 3, new design methods for irregular SC-LDPC codes are proposed.
Section 3.1 introduces the construction method of irregular SC-LDPC ensembles and
presents analysis tools including the DE equations and expected graph evolution. In
Section 3.2, new design methods of irregular SC-LDPC codes are proposed and the
performance of proposed SC-LDPC codes is evaluated in terms of the BP threshold.
The performance improvement of proposed SC-LDPC codes is verified by comparing
the finite-length performance in Section 3.3.
In Chapter 4, an optimization algorithm of degree distributions of additional vari-
able nodes is proposed to reduce the rate-loss. In Section 4.1, the code structure of
SC-LDPC codes to be optimized and their DE equations are presented. The optimiza-
4
tion algorithm of degree distributions is proposed in Section 4.2 and the validity of
the designed codes is confirmed via numerical analysis. Finally, the conventional and
proposed SC-LDPC codes are compared in the same design rate in Section 4.3.
5
Chapter 2
Overview of LDPC Codes
In this chapter, some preliminaries of LDPC codes are introduced. First, the basic
concepts of LDPC codes are described and the backgrounds including decoding algo-
rithms and density evolution are provided. Finally, protograph-based LDPC codes are
introduced.
2.1 LDPC Codes
In this section, review some preliminaries of LDPC codes are given. An LDPC code is
a linear block code defined as a collection of binary codeword c of length n satisfying
HcT = 0T, where H is a parity-check matrix of size m×n. The matrix H is sparse in
the sense that the number of non-zero entries is increases linearly with n, rather than
with n2. The sparseness of H guarantees that the decoding complexity increases only
linearly with n under an iterative decoding. An LDPC code can be represented by a
bipartite graph, generally called Tanner graph, with n variable nodes, m check nodes,
and their connecting edges. Each variable node and check node correspond to a column
and a row of H, respectively. Edges are defined as the connections between variable
nodes and check nodes. Figure. 2.1 shows an example of LDPC codes, where circles
stand for the variable nodes and squares stand for the check nodes. Lines between
6
Figure 2.1: Tanner graph of LDPC code.
circles and squares represent edges of the code. The corresponding parity check matrix
H of the Tanner graph in Figure 2.1 is given as
H =
1 1 1 0 0 0
1 0 0 1 1 0
0 1 0 1 0 1
0 0 1 0 1 1
.
The number of edges connected to a node is referred as the degree of the node. An
LDPC code is called (l, r) regular LDPC codes if each variable node has degree l and
each check node has degree r. For example, the code in Figure 2.1 is a (2, 3) regular
LDPC code. On the contrary, if the degrees of variable (check) nodes are different
from each other, the code is said to be irregular.
Irregular LDPC codes have a pair of degree distributions (L,R) as
L(x) =∑i
Lixi, R(x) =
∑i
Rixi (2.1)
where Li is the fraction of degree-i variable nodes and Ri is the fraction of degree-i
check nodes. The numbers of variable nodes and check nodes of degree i are nLi and
mRi, respectively. From an edge perspective, the degree distributions are represented
by λ(x) =∑
i λixi−1, ρ(x) =
∑i ρix
i−1, where λi (ρi) is the fraction of edges
connected with variable (check) nodes of degree i. The relationship between (L,R)
7
and (λ, ρ) is given by
L(x) =
∫ x0 λ(z)dz∫ 10 λ(z)dz
(2.2)
R(x) =
∫ x0 ρ(z)dz∫ 10 ρ(z)dz
. (2.3)
Also, the average variable and check degrees, denoted by lavg and ravg, are expressed
as
lavg =1∫ 1
0 λ(z)dz(2.4)
ravg =1∫ 1
0 ρ(z)dz. (2.5)
Then total number of sockets from which edges emanate on variable (or check) nodes
is lavgn, which is equal to ravgm.
Given a degree distribution pair (λ, ρ), define (λ, ρ) LDPC ensemble as a collec-
tion of LDPC codes following the degree distributions. To be specific, an code instance
of the (λ, ρ) ensemble with blocklength n consist of nLi variable nodes of degree i
and mRi check nodes of degree i. Then the ith socket on variable nodes is connected
to the π(i)th socket on check nodes, where π is a permutation of size lavgn. In other
words, the permutation π determines edge connectivity of LDPC codes and there is
one-to-one mapping between a code instance of the (λ, ρ) ensemble and selection of
π. Then, the set of LDPC codes generated by all possible permutations of size lavgn is
defined as the (λ, ρ) ensemble. The design rate of the (λ, ρ) ensemble is given by
R =n−mn
= 1−∫ 10 ρ(z)dz∫ 10 λ(z)dz
. (2.6)
If all parity equations of a code instance from the ensemble are linearly independent,
the code rate is the same with the design rate.
8
𝑋 𝑌?
0
1
0
1
𝜖
𝜖
1 − 𝜖
1 − 𝜖
Figure 2.2: Binary erasure channel with parameter ε.
2.2 Decoding of LDPC Codes in the BEC
In this section, the decoding method of LDPC codes is described. In this disserta-
tion, the BEC is mainly considered, where each transmitted bit is received correctly or
known to be erased. Denote the channel input at time t as xt ∈ {0, 1} and the corre-
sponding channel output as yt ∈ {0, 1, ?}, where ? indicates an erasure. For erasure
probability ε, erasure occur independently with probability ε, i.e., Pr{yi =?} = ε.
Figure 2.2 shows the BEC with ε. The decoding problem over the BEC is to find the
values of the erased bits. Note that the capacity of the BEC with ε is known as 1− ε.
2.2.1 MAP Decoding of LDPC Codes
For a given yi, the MAP decoder finds xMAPi that maximizing a posterior probability,
i.e., the MAP decoding rule is
xMAPi = argmax
α∈{0,1}PXi|Y (α|y). (2.7)
For the BEC, the MAP decoding rule is equivalent to solving a linear system of equa-
tions, which is accomplished in complexity at most O(n3). In other words, the MAP
decoding is optimal decoding but the decoding complexity is not feasible for large n.
2.2.2 BP Decoding of LDPC Codes
In general, the BP decoding algorithm is used for LDPC codes, where its operation can
be represented by passing messages along edges in a Tanner graph. The BP decoding
9
algorithm is an iterative decoding algorithm in the sense that the decoding algorithm
proceeds iteratively. The decoder stops until it finds a valid codeword satisfying parity
check equations or the number of iterations reaches the predetermined number.
For a given received sequence y = {y1, . . . , yn}, let Mi,j ∈ {0, 1, ?} and Ej,i ∈
{0, 1, ?} be outgoing messages from the ith variable node to the jth check node and
from the jth check node to the ith variable node, respectively. At initialization, the
outgoing messages of the ith variable node are equal to its received value yi, i.e.,
Mi,j = yi if there is an edge between the ith variable node and the jth check node.
Then, the jth check node receives dj incoming messages, where dj is the degree of the
jth check node. From the incoming messages, the jth check node calculate message
Ej,i by adding all incoming messages except Mi,j , where Ej,i becomes an erasure
if there is an erasure in the adding messages. If an erased variable node receives a
non-erased incoming message, the variable node is recovered by the value of the non-
erased incoming message. In other words, check nodes can recover an erased variable
node when only one of the variable nodes connected to the check node is erased. This
process of message exchanging is repeated until all of erased variable nodes are recov-
ered, or until the number of iterations reaches the predetermined maximum number of
iterations.
The decoding complexity of BP decoding linearly increases with a function of the
number of edges. For a given average variable node degree lavg, the number of edges
lavgn is a linear function of n, which implies that the decoding complexity of BP
decoding grows linearly with codelength n. Note that when the BP decoding algorithm
is performed sequentially, where only one variable node can be recovered at a iteration,
it is called peeling decoding. It is known that the performances of BP and peeling
decoding are the same in the BEC.
10
2.3 Analysis of LDPC Codes
2.3.1 Density Evolution
In [21], Richardson et al. analyzed decoding algorithms of LDPC codes using DE
which is one of important tools to analyze LDPC codes. This technique is commonly
used for optimizing the degree distributions of capacity-approaching LDPC ensem-
bles. Over a general binary memoryless symmetric channels, DE tracks the evolution
of probability density functions of an ensemble with assumption that infinity block
length. However, for the BEC, the DE approach is much simplified since it only tracks
the erasure probability of messages emitted from variable nodes.
Let x(`) be the erasure probability of messages emitted from variable nodes at
iteration `. For the (λ, ρ) irregular LDPC ensemble, the evolution of x(`) is expressed
as
x(`) = ελ(1− ρ(1− x(`−1))), (2.8)
with the initial condition x(0) = ε. If x(`) goes to zero as ` increases for a given ε, the
channel erasure is likely to be corrected by the ensemble using BP decoding with high
probability. The BP threshold εBP is the limit of channel erasure probability such that
the decoding is guaranteed to be successful in asymptotic settings of the DE analysis,
i.e., εBP is defined as the maximum ε for which x(`) goes to zero as ` increases.
2.3.2 Optimizing Degree Distribution
Using the DE equation, the (λ, ρ) irregular LDPC ensemble can be optimized to in-
crease the BP threshold close to the Shannon limit. From the DE equation in (2.8), a
sufficient condition on the parameters of the DE equation for successful decoding is
represented by
ελ(1− ρ(1− x)) < x for 0 < x ≤ ε. (2.9)
11
Also, recall that the design rate of the (λ, ρ) ensemble is given by
R = 1−∫ 10 ρ(z)dz∫ 10 λ(z)dz
. (2.10)
Since the condition of successful decoding and the design rate are linear functions of
λ(x), the optimization of λ(x) is performed by solving the following LP problem as
minimize1∫ 1
0 λ(x)dx
subject to ελ(1− ρ(1− x)) < x for 1 ≤ x ≤ ε (2.11)
where the objective function is to maximize the design rate for a given channel param-
eter ε
12
Chapter 3
Design of Irregular SC-LDPC Codes with Non-Uniform
Degree Distributions by Linear Programming
Most studies of SC-LDPC codes focus on regular SC-LDPC codes composed of reg-
ular LDPC codes. However, irregular LDPC codes can be coupled to construct ir-
regular SC-LDPC codes, where the threshold saturation effect also occurs [37]–[39].
Thus, there have been researches on constructing SC-LDPC codes from block irreg-
ular LDPC codes that outperform block regular LDPC codes. The first approach to
construct irregular SC-LDPC codes is coupling protograph-based codes such as repeat-
accumulate (RA) [42], accumulate-repeat-jagged-accumulate (ARJA) [35], [43], and
MacKay-Neal (MN) [45] codes. Those irregular SC-LDPC codes show better asymp-
totic or finite-length performances than regular SC-LDPC codes.
Similar to protograph-based irregular SC-LDPC codes, randomly constructed ir-
regular SC-LDPC codes can be constructed by coupling block irregular LDPC codes
with variable and check node degree distributions λ(x) and ρ(x). However, it is dif-
ficult to globally optimize degree distributions λ(x) and ρ(x) of irregular SC-LDPC
codes with low-complexity because their DE equations [36] are multi-dimensional
for the number of positions L. Thus, many studies assume a nearly regular degree
distribution with two distinct degrees because its optimization is possible by exhaus-
13
tive searches. For example, the SC-LDPC codes with two distinct degrees on check
nodes are designed to improve the BP threshold over wide range of code rate [46].
In [47], they show that slightly imposing irregularity on λ(x) can improve the con-
vergence speed of SC-LDPC codes. Also, it has been reported that this improvement
in the convergence speed also results in an improvement of the finite-length perfor-
mance [48]. Recently, an optimization method for irregular SC-LDPC codes without
any constraints on degree distributions is proposed [49]. Further, in [49], the SC-LDPC
ensembles with so called non-uniform degree distributions are designed, where degree
distributions can differ for each position. By optimizing degree distributions using a
genetic algorithm, it is shown that the optimized SC-LDPC codes exhibit improved
decoding performance compared to regular SC-LDPC codes.
In this chapter, motivated by the ideas in [49], a systematic algorithm with low-
complexity is proposed to design irregular SC-LDPC codes with non-uniform degree
distributions. To reduce the design complexity, the proposed design algorithm itera-
tively solves LP problems. The LP problem is widely used when optimizing block
irregular LDPC codes over the BEC [7], [50]. However, it is hard to directly applythe
LP optimization technique to the designing of degree distributions of SC-LDPC codes
because the DE equations are multi-dimensional. Thus, three important ideas are in-
troduced to establish an LP model for optimizing irregular SC-LDPC codes. First, in-
stead of designing degree distributions at once, a local design is iteratively performed,
where the variable node degree distributions of target positions are obtained one at a
time while keeping the degree distributions of the other positions constant. Second,
the input/output message relationship between variable nodes at target positions and
the remaining graph is pre-computed. These two methods allow us to obtain the one-
dimensional DE equation and represent the subjective function of the established LP
model as a linear function. Finally, an objective function suitable for the designed LP
problem is given.
The concept of the one-dimensional DE equation is firstly used in [51] to optimize
14
the degree distribution of additional variable nodes to mitigate the rate-loss of SC-
LDPC codes, where the objective function for the optimization is to maximize the de-
sign rate. However, it is observed that the objective function of maximizing the design
rate is not proper for optimizing irregular SC-LDPC codes. Thus, another objective
function that aims to minimize the number of required iterations for successful decod-
ing [50], [52] is applied. The proposed design algorithm finds the degree distributions
of target positions one at a time to minimize the number of required iterations in the
one-dimensional DE equation. Although the proposed algorithm aims to minimize the
number of required iterations in the one-dimensional DE equation, it is observed that
the overall number of required iterations in the multi-dimensional DE equations is also
properly decreased by the proposed design algorithm. In addition, the BP threshold is
improved by the proposed design algorithm if all of the degree distributions are lo-
cally designed, which is confirmed graphically from an analysis of the expected graph
evolution [41]. Numerical results show that proposed irregular SC-LDPC codes exhibit
superior BP threshold compared to regular SC-LDPC codes. Specifically, the improve-
ment of the BP threshold is noticeable for small and moderate values of L. While the
BP threshold of regular SC-LDPC codes approaches the Shannon limit for a large L,
the proposed design algorithms provide capacity-approaching codes even for small
and moderate values of L. It is practically important to design SC-LDPC codes with
high performance for small and moderate values of L since a large L can worsen the
finite-length performance [41] and minimum distance properties [35]. The proposed
design algorithms are validated on the finite-length performance under windowed de-
coding [53], [54] for moderate and large values of L and under BP decoding for a
small L. The performance improvement is shown for all values of L, especially, the
proposed irregular SC-LDPC code with a small L achieves better performance than an
optimized block irregular LDPC code with the same code rate and blocklength under
BP decoding. Since the proposed design algorithms avoid degree-two variable nodes,
which are essential to optimized block irregular LDPC codes, the proposed algorithms
15
give a new way to design capacity-approaching block LDPC codes without degree-two
variable nodes.
This chapter is organized as follows. Section 3.1 introduces the construction meth-
ods of SC-LDPC ensembles and the tools for analyzing the ensembles. In Section 3.2,
the methods to design the degree distributions of SC-LDPC ensembles are proposed.
In Section 3.3, the finite-length performance of the proposed SC-LDPC codes is eval-
uated and several methods for improving the performance further are presented.
3.1 Code Structure of SC-LDPC Ensembles
3.1.1 Construction of SC-LDPC Ensembles
Irregular SC-LDPC ensembles considered in this chapter assume that variable node
degree distributions of each position can be irregular while check node degrees at each
position are regular but can differ at each position. Let l and r denote the variable and
check node degrees of regular LDPC ensembles, respectively, and L and w denote the
chain length and coupling width, respectively, where w ≥ 0 and L ≥ w. An irregular
SC-LDPC ensemble consists of M variable nodes located at each of L positions with
Mc check nodes located at each of L + w − 1 positions, where Mc , M lr ∈ N.
Variable nodes at position u follow the edge perspective degree distribution λu(x) =∑d λu,dx
d−1 [7] while check nodes at position v have degree rv ∈ N. The number of
edges between positions in the graph is represented by a (L+w− 1)×L connectivity
matrix T , where the entry Tv,u is the number of edges between check nodes at position
v and variable nodes at position u. For example, a regular SC-LDPC ensemble [36]
with (l, r) has code parameters of λu(x) = xl−1 for 1 ≤ u ≤ L, rv = r for 1 ≤ v ≤
L+ w − 1, and Tv,u = Ml/w for u ≤ v ≤ u+ w − 1.
In Figure 3.1, Tanner graphs of three different SC-LDPC codes with L = 4, w =
2, and M = 2 are shown as an example. Note that there are multiple edges in the
Tanner graphs to represent various degree distributions with a small number of nodes.
16
Position 1 Position 4
(a) Graph 1
Position 1 Position 4
(b) Graph 2
Position 1 Position 4
(c) Graph 3
Figure 3.1: Examples of Tanner graphs of SC-LDPC codes for L = 4, w = 2, and
M = 2.
First, Graph 1 represents a regular SC-LDPC code, where the degree of all of the
variable nodes is equal to 4. In contrast, the degree distributions of variable nodes
at positions 1 and 4 in Graph 2 are irregular. Graph 2 represents a SC-LDPC code
with non-uniform degree distributions in that degree distributions of variable nodes
at each position can differ from each other. In addition, Graph 2 becomes Graph 3
after permitting a non-uniform check node degrees, or equivalently, a different number
of connected edges between positions, which gives more freedom in the design of
degree distributions. Note that a symmetric structure is assumed, that is, λu(x) =
λL+1−u(x), rv = rL+w−v, Tv,u = TL+w−v,L+1−u for all codes in this chapter.
For the given code parameters λu(x), rv, and T , the detailed code construction
method from the SC-LDPC ensemble is described as follows.
1. Let [m,n] , {m,m + 1, . . . , n} for m < n. First, place M variable nodes
with degree distribution λu(x) at position u for u ∈ [1, L]. Then, there are
M/∫ 10 λu(x)dx variable node sockets at position u, where 1/
∫ 10 λu(x)dx is the
average variable node degree of variable nodes at position u. To connect all
of the sockets, the number of edges connected to variable nodes at position u
should be equal to the number of sockets, that is,M/∫ 10 λu(x)dx =
∑i Ti,u. Let
πu be a random permutation on [1,∑
i Ti,u] for position u. Divide πu into w dis-
joint subsets denoted by π1u, . . . , πwu such that the size of πtu becomes Tu+t−1,u
17
Table 3.1: Parameters of the Tanner graphs in Figure 3.1
λi(x) T
Graph 1
λ1(x) = x3
4
4 4
4 4
4 4
4
λ2(x) = x3
λ3(x) = x3
λ4(x) = x3
Graph 2
λ1(x) = 38x
2 + 58x
4
4
4 4
4 4
4 4
4
λ2(x) = x3
λ3(x) = x3
λ4(x) = 38x
2 + 58x
4
Graph 3
λ1(x) = 38x
2 + 58x
4
5
3 4
4 4
4 3
5
λ2(x) = x3
λ3(x) = x3
λ4(x) = 38x
2 + 58x
4
for t ∈ [1, w].
2. Similarly, placeMc check nodes of degree rv at position v for v ∈ [1, L+w−1]
in the graph. Accordingly, there are Mcrv check node sockets at position v but
some sockets cannot be filled if∑
j Tv,j < Mcrv.
3. Divide the randomly selected∑
j Tv,j check node sockets among the Mcrv
check node sockets at position v into w groups randomly such that the size of
group t becomes Tv,v−t+1 for t ∈ [1, w], where Tv,u = 0 for u < 0. Then, the
jth check node socket in group t at position u+ t−1 is connected to the πtu(j)th
variable node socket at position u, where πtu(j) denotes the jth element of πtu.
The total numbers of variable nodes V and check nodes C placed in the graph are
V = LM and C = (L + w − 1)Mc, respectively. However, some check nodes at
position v for v ∈ [1, w] or v ∈ [L+1, L+w−1] cannot be connected to any variable
18
nodes in the graph because the number∑
j Tv,j of edges connected to check nodes is
lower than the number of socketsMcrv. To be specific, a check node socket at position
v for v ∈ [1, w] or v ∈ [L+ 1, L+w− 1] cannot be connected to any variable node in
the graph with probability 1−∑
j Tv,j/(Mcrv). Then, the expected number of check
nodes C with at least one connection to the variable nodes in the graph is given as
C =
(L+ w − 1− 2
w−1∑v=1
(1−
∑j Tv,j
Mcrv
)rv)Mc,
which is less than C = (L + w − 1)Mc. Therefore, the design rate RSC = 1 − C/V
of the SC-LDPC ensemble is expressed as
RSC =
(1− l
r
)− l
r
w − 1
L+l
r
2w−1∑v=1
(1−
∑j Tv,jMcrv
)rvL
. (3.1)
Because the last term of (3.1) is generally much smaller than the other terms, the last
term is ignored when calculating the design rate for convenience.
3.1.2 DE Equations of SC-LDPC Ensembles
Let x(`)u and y(`)v denote the average erasure probabilities of messages at iteration `
emitted from variable nodes at position u and check nodes at position v, respectively.
Set the initial conditions as x(0)u = ε for all u. Then, the evolution of x(`)u can be
expressed as
y(`)v = 1−
(1−
∑j Tv,jx
(`)j
Mcrv
)rv−1, x(`+1)
u = ελu
(∑i Ti,uy
(`)i∑
i Ti,u
). (3.2)
With the DE equations in (3.2), the BP threshold, which is defined as the supremum
of ε for which x(`)u goes to zero for all u, is obtained. Additionally, the overall number
of required iterations Ir for successful decoding is defined as the minimum ` such that
x(`)u < δ for sufficiently small δ such as 10−10 and define the average convergence
speed as L/Ir [47].
19
3.1.3 Expected Graph Evolution of SC-LDPC Ensembles
In [41], the scaling law of SC-LDPC codes is derived to predict the finite-length perfor-
mance of SC-LDPC codes. The scaling law of SC-LDPC codes depends on the scaling
parameters derived by analyzing the statistical behavior of the number of degree-one
check nodes in the remaining graph under peeling decoding. In [41], a system of cou-
pled differential equations to compute the expected number of degree-one check nodes
is derived for regular SC-LDPC ensembles. Because the SC-LDPC ensemble consid-
ered in this chapter has the non-uniform degree distributions, the differential equations
to compute the expected number of degree-one check nodes should be modified as
follows.
Consider degree distributions of the original graph before the peeling decoder
is initialized. Define the type of a variable node at position u using the vector x =
(x1, . . . , xw), where xt represents the number of edges connected to check nodes at
position u + t − 1. Although the profile of degrees of each variable node is sufficient
to represent the DE equations in (3.2), the type of each variable node should be con-
sidered when calculating the expected graph evolution. Let Pu(x) be the probability
that a variable node chosen at random from position u in the original graph is of type
x. Considering the construction method of SC-LDPC ensembles in Section 3.1.1, it is
given that vector x = (x1, . . . , xw) for variable nodes at position u follows a multino-
mial distribution with probabilities p = (p1, . . . , pw), where pt = Tu+t−1,u/∑
i Ti,u.
Thus, probability Pu(x) is given as
Pu(x) =|x|!
x1! · · ·xw!
w∏t=1
(Tu+t−1,u∑
i Ti,u
)xtwhere |x| =
∑i xi. Next, consider check nodes at position v. Let ρm,v be the proba-
bility that a check node chosen at random from position v in the original graph is of
20
degree m, which is given as
ρm,v =
(rvm
)(∑j Tv,jMcrv
)m(1−
∑j Tv,jMcrv
)rv−m, if v ∈ [1, w − 1]
1, if m = rv, v ∈ [w,L]
0, if m < rv, v ∈ [w,L]
ρm,L+w−v, if v ∈ [L+ 1, L+ w − 1].
For iteration ` of the peeling decoder, let τ be the number of iterations normalized
by M , i.e., τ = `/M . At time τ , let Rj,v(τ) be the number of edges connected to the
check nodes of degree j, j = 1, . . . , rv, at position v, v ∈ [1, L + w − 1]. Likewise,
let Ux,u(τ) be the number of edges that are connected to variable nodes of type x at
position u, u ∈ [1, L]. After the initialization of the peeling decoder, the expected
value of Rj,v(0) is expressed as
E[Rj,v(0)] = jMc
rv∑m≥j
ρm,v
(m
j
)εj(1− ε)m−j .
In addition, the initial values of the expected value of Ux,u(`) can be computed as
E[Ux,u(0)] =
ελu,|x|
M∫ 10 λu(z)dz
Pu(x), u ∈ [1, L]
0, otherwise.
Let E[∆Rj,v(τ)] = E[Rj,v(τ + 1/M)−Rj,v(τ)] and E[∆Ux,u(τ)] = E[Ux,u(τ +
1/M) − Ux,u(τ)], where the expectation is determined given the degree distributions
in the remaining graph at time τ . In order to compute the expectation of Rj,v(τ) and
Ux,u(τ) from the initial values, the following system of differential equations is solved.
∂Rj,v(τ)
∂τ=
E [∆Rj,v(τ)]
1/M,∂Ux,u(τ)
∂τ=
E[∆Ux,u(τ)
]1/M
.
The procedure used to obtain E[∆Rj,v(τ)] and E[∆Ux,u(τ)] is described as fol-
lows. Let φm,x,u(τ) be the probability that a variable node of type x connected to a
21
degree-one check node at position m belongs to position u. Then,
φm,x,u(τ) =
xm−u+1|x| Ux,u(τ)∑
i∈S(m)
(∑x′
x′m−i+1|x′| Ux′,i(τ)
) , if u ∈ S(m)
0, otherwise
where S(m) = {j|min(m− (w − 1), 1) ≤ j ≤ m}. When a degree-one check node
from positionm and the variable node connected to it are removed, ξm,v,t(τ) is defined
as the probability that t edges of the removed variable node are connected to the check
nodes other than the removed check node at position v. Then,
ξm,v,t(τ) =
∑
i∈S(v)
( ∑x:xv−i+1=t
φm,x,i(τ)
), if m 6= v
∑i∈S(v)
( ∑x:xv−i+1=t+1
φm,x,i(τ)
), if m = v
for t ≤ lmax − 1, where lmax denotes the maximum degree of variable nodes. The
average number of degree-j check nodes losing one edge when t edges are randomly
removed from check nodes at position v is given as
Fj,v,t(τ) =t∑
k=1
k
(t
k
)δkj,v(τ) (1− δj,v(τ))t−k
where δj,v(τ) = Rj,v(τ)/rv∑q=1
Rq,v(τ) for j ≤ rv and δrv+1,v(τ) = 0. Then,
E[∆Ux,u(τ)|pos(τ) = m] = −|x|φm,x,u(τ)
E[∆Rj,v(τ)|pos(τ) = m] =
jlmax−1∑t=1
ξm,v,t(τ)
(Fj+1,v,t(τ)− Fj,v,t(τ)
)− 1, if v = m, j = 1
jlmax−1∑t=1
ξm,v,t(τ)
(Fj+1,v,t(τ)− Fj,v,t(τ)
)otherwise
where pos(τ) is the position at which a degree-one check node is removed at time τ .
Finally, the expectations of ∆Rj,v(τ) and ∆Ux,u(τ) are described as
E[∆Rj,v(τ)] =
L+w−1∑m=1
E[∆Rj,v(τ)|pos(τ) = m]pm(τ)
E[∆Ux,u(τ)] =L+w−1∑m=1
E[∆Ux,u(τ)|pos(τ) = m]pm(τ)
22
where pm(τ) = R1,m(τ)/L+w−1∑v=1
R1,v(τ). Then, the expectation of the total number of
degree-one check nodes E[R1(τ)] in the remaining graph at τ becomes∑
v E[R1,v(τ)]
and the normalized number of degree-one check nodes r1(τ) is computed as r1(τ) =
E[R1(τ)]/M . In the next section, the expected graph evolution is utilized to show the
decoding behavior of irregular SC-LDPC codes to be designed.
3.2 Proposed Design Algorithms of SC-LDPC Ensembles
In this section, new design algorithms for code parameters λu(x), ρv, and Tv,u of
irregular SC-LDPC ensembles are provided. In the following subsections, the methods
to establish the optimization problem of irregular SC-LDPC codes are introduced.
3.2.1 Pre-Computation of δu(z)
Since the DE equations in (3.2) are multi-dimensional for L, it is difficult to obtain
a simple sufficient condition similar to (2.9) for irregular SC-LDPC codes. However,
this problem can be addressed by a procedure called pre-computation of δu(z) which
is described it in Algorithm 3.1. This procedure, firstly used in [51], is essential for the
proposed design algorithms.
In Algorithm 3.1, the graph is considered as two parts; variable nodes at target
positions u and L − u + 1 and the remaining graph. Then the input/output message
relationship between two parts is obtained. First, the input message z to the remain-
ing graph is passed from variable nodes at the target positions to the remaining graph.
Second, the messages in the remaining graph are updated using the DE equations (3.2)
until all of the messages are saturated while the messages from variable nodes at the
target positions to the remaining graph are fixed at z. Finally, let the incoming mes-
sage to variable nodes at the target positions be δu(z). In other words, Algorithm 3.1
calculates the output message δu(z) corresponding to the input message z from the
perspective of the remaining graph.
23
Algorithm 3.1 Pre-computation of δu(z) [51]Input: {λ1(x), . . . , λL(x)}, {r1, . . . , rL+w−1},T , Q, ε, u
1: for q = 1 : Q do
2: Set zq = ε qQ .
3: Update x(`)i for all i except u and L − u + 1 by DE equations (3.4) until the
messages are saturated while fixing x(`)u = zq, x(`)L−u+1 = zq for all `.
4: Let δu(zq) be the incoming message to variable nodes at position u, that is,
δu(zq) =
∑i Ti,uy
(`)i∑
i Ti,u.
5: end for
Consider a message update scheduling such that the message xu is updated only
after the other messages are saturated to their fixed values. Under this message update
scheduling, xu is updated from z to ελu(δu(z)) because the incoming message to
variable nodes at position u is δu(z). In other words, pre-computation of δu(z) gives
a one-dimensional DE equation z(`+1) = ελu(δu(z(`))) for position u with an initial
value of z(0) = ε under the message update scheduling described before. Using the
one-dimensional DE, a sufficient condition for successful decoding can be represented
as
ελu(δ(zq)) < zq for 1 ≤ q ≤ Q.
In the following subsections, two objective functions including maximizing the design
rate [7] and minimizing the number of required iterations [50], [52] are applied to the
design algorithm for irregular SC-LDPC ensembles.
3.2.2 Objective Function 1: Maximizing Design Rate
First, the objective function that maximizes the design rate, which is used for
block irregular LDPC codes and used in [51], is considered. The corresponding de-
sign algorithm is described in Algorithm 3.2. Maximizing the design rate can be
24
Algorithm 3.2 Design algorithm for maximizing design rateInput: l, r, L, w, lmin, lmax, Q, Imax
1: Initialization: Code parameters of the regular SC-LDPC ensemble
2: for Iter = 1 to Imax do
3: for u = L/2 to 1 do
4: Calculate the BP threshold εBP by the DE equations (3.4)
5: Pre-computation of δu(z) by Algorithm 3.1
6: Obtain λ∗(x) =lmax∑k=lmin
λ∗kxk−1 by solving the following LP problem
minimize1∫ 1
0 λ∗(x)dx
subject to εBPλ∗(δ(zq)) < zq for 1 ≤ q ≤ Q,λ∗(1) = 1
7: λu(x) = λ∗(x)
8: end for
9: end for
achieved by solving the LP problem with the objective function to minimize the av-
erage variable node degree 1/∫ 10 λu(x)dx under the constraint of successful decoding
εBPλu(δu(zq)) < zq for q ∈ [1, Q]. Unlike the codes defined in Section 3.1.1, where
the number of variable nodes at each position is fixed to M , there are Ml∫ 10 λu(x)dx
variable nodes at position u because Ml edges heading to variable nodes at each
position are distributed to the variable nodes with the average variable node degree
1/∫ 10 λu(x)dx. Thus, minimizing the average degree 1/
∫ 10 λu(x)dx corresponds to
maximizing the number of variable nodes Ml∫ 10 λu(x)dx and the design rate.
In Algorithm 3.2, the graph is initialized with the regular SC-LDPC ensemble.
Then, degree distribution λu(x) is designed starting from u = L/2 ∈ N to maximize
the design rate and λL−u+1(x) is also determined by the designed degree distribution
because of the symmetric code structure. To design λL/2(x), the value δL/2(z) is pre-
calculated; this is used to establish the LP problem of λL/2(x). The degree distribution
25
Precomputation of 2(z) Optimizing λ2(x) and λ3(x)
Init
iali
zati
on
Fo
r-lo
op 1
Fo
r-lo
op 2
Precomputation of 1(z) Optimizing λ1(x) and λ4(x)
Figure 3.2: Design procedure of the proposed algorithms with L = 4, w = 2, and
M = 2.
for the next target position λL/2−1(x) is designed in the same manner after designing
λL/2(x). This local design is conducted until the number of iterations, denoted by
Iter, reaches Imax or λu(x) does not change with Iter. For the other design algorithms
to be introduced, the same design procedure is applied, in which the local design of
the degree distributions is conducted for each pair of positions one at a time, as in
Figure 3.2.
For input values l = 4, r = 8, L = 10, w = 3, lmin = 3, lmax = 10, Q = 1000,
and Imax = 10, the design rate is increased from 0.4 to 0.4360 by Algorithm 3.2.
Likewise, the design rate is increased from 0.45 to 0.4670 for L = 20. Figure 3.3
26
2 4 6 8 10 12 14 16 18 20
2
2.5
3
3.5
4
4.5
5
Figure 3.3: Average variable node degree of each position for the SC-LDPC ensemble
designed by Algorithm 3.2.
shows the average variable node degree 1/∫ 10 λu(x)dx of each position for the de-
signed SC-LDPC ensemble by Algorithm 3.2 for L = 10 and 20. It shows that the
degree distributions of center positions 5, 6 for L = 10 and 10, 11 for L = 20 are
properly designed to increase the design rate by minimizing the corresponding aver-
age variable node degrees. However, the variable node degrees of positions far from
the center remain nearly identical to initial degree l = 4, which means the fraction
of the number of positions to be improved by Algorithm 3.2 becomes smaller as L
increases. To achieve an improvement in all positions, a different objective function
for the LP problem is required instead of maximizing the design rate.
3.2.3 Objective Function 2: Minimizing the Number of Required Itera-
tions
Another objective function described in Algorithm 3.3 is minimizing the number of
required iterations. Consider a non-increasing iterative function f(x). Given an initial
value a, the iterative process is represented by x(0) = a and x(`) = f(x(`−1)). The
27
number of required iterations I for the target value b, b < a, is defined as the minimum
` such that x(`) ≤ b. In [50], the number of required iterations I is approximated as
I ≈n−1∑i=2
∆xixi − f(xi)
where x0 = b, xn = a, xi = b+ ia−bn , and ∆xi = xi+1−xi−1
2 .
Algorithm 3.3 Design algorithm for minimizing the number of required iterationsInput: l, r, L, w, lmin, lmax, Q, Imax
1: Initialization: Code parameters of the regular SC-LDPC ensemble
2: for Iter = 1 to Imax do
3: for u = L/2 to 1 do
4: Calculate the BP threshold εBP by the DE equations (3.4)
5: Pre-computation of δu(z) by Algorithm 3.1
6: Obtain λ∗(x) =lmax∑k=lmin
λ∗kxk−1 by solving the following LP problem
minimize
Q−1∑q=2
∆zqzq − εBPλu(δu(zq))
subject to1∫ 1
0 λ∗(x)dx
= l, εBPλ∗(δu(zq)) < zq for 1 ≤ q ≤ Q,λ∗(1) = 1
7: λu(x) = λ∗(x)
8: end for
9: end for
In the case of the one-dimensional DE z(`+1) = ελu(δu(z(`))), the iterative func-
tion is given as f(x) = ελu(δu(x)). Thus, the number of required iterations from
initial value zQ = ε to target value z1 = ε/Q is approximated as
Q−1∑q=2
∆zqzq − ελu(δu(zq))
(3.3)
where ∆zq = 1/Q. Thus, the design algorithm with the objective function that min-
imizes the number of iterations can be summarized as Algorithm 3.3. In Algorithm
28
0.48 0.485 0.49 0.495 0.5 0.505 0.51
0
0.05
0.1
0.15
0.2
0.25
0.3
Figure 3.4: Evolution of average convergence speed as the for-loop in Algorithm 3.3
proceeds.
3.3, there is an additional constraint 1/∫ 10 λ∗(x)dx = l that the average variable node
degree is equal to initial variable node degree l, which maintains the design rate. As
an example, the resulting degree distributions λu(x) designed by Algorithm 3.3 with
input values l = 4, r = 8, L = 20, w = 3, lmin = 3, lmax = 10, Q = 1000, Imax = 10
are shown in Table 3.2. Note that, to maintain the advantage of the regular SC-LDPC
codes that there are no degree two variable nodes [55], the minimum variable node de-
gree lmin is set to 3. Also, the algorithm is initialized with initial degree l = 4. Instead,
it cannot be initialized with l = 3 because a non-zero fraction of degree-two variable
nodes is unavoidably required to introduce variable nodes with degree larger than 3.
In order to investigate the change of irregular SC-LDPC ensembles according to
local designs, the evolution of the average convergence speed obtained by the DE
equations (3.2) is shown in Figure 3.4 as the inner for-loop in Algorithm 3.3 proceeds.
Figure 3.4 shows that the average convergence speed increases steadily as the local
design of each position proceeds. In addition, it is noted that an increase in the BP
threshold at for u = 5, where the BP threshold corresponds to the erasure probability
29
Table 3.2: λu(x) for the proposed irregular SC-LDPC ensemble from Algorithm 3.3
for L = 20
u λu(x)
1 x3
2 0.1916x2 + 0.4889x3 + 0.3195x4
3 0.3852x2 + 0.5679x4 + 0.0362x5 + 0.0017x8
4 0.4587x2 + 0.2065x4 + 0.3348x5
5 0.5260x2 + 0.2920x5 + 0.1820x6
6 0.5185x2 + 0.1283x3 + 0.0523x6 + 0.3009x7
7 0.6128x2 + 0.2644x7 + 0.0357x8 + 0.088x9
8, 9, 10 0.6429x2 + 0.3571x9
that the average convergence speed becomes zero.
In order to observe the change graphically as the local design of Algorithms 3.2 and
3.3 is performed, the expected graph evolution is used. With the differential equations
represented in Section 3.1.3, I plot in Figures 3.5 and 3.6 the evolutions of r1(τ) at
ε = 0.48 for the designed irregular SC-LDPC ensembles as the inner for-loop of the
algorithms proceeds. The evolution of r1(τ) at the initialization corresponds to the
case for the regular SC-LDPC ensemble that is analyzed in [41]. According to the
analysis in [41], r1(τ) remains constant at the local minimum for a certain interval of
τ , referred to as the second phase. On the basis of the second phase, the evolution of
r1(τ) is divided into three phases; the initial phase, the second phase, and the third
phase. Most of variable nodes located in the boundary positions have already been
recovered in the initial phase. Then, in the second phase, two decoding waves emerge
and travel along inner positions while recovering variable nodes through which the
decoding waves pass. If the local performance of a position is defined as the number of
degree-one check nodes that exist when the decoding waves pass through the position,
the steady value of r1(τ) in the second phase implies that the local performances of
30
1.E-02
1.E-01
1.E+00
0 2 4 6 8 10 12
100
10-1
10-2
Initialization
for = 9
for = 1
Figure 3.5: Evolution of r1(τ) for the SC-LDPC ensembles designed by Algorithm 3.2
as the inner for-loop proceeds.
inner positions are all the same. Moreover, it can be said that the BP threshold of
the SC-LDPC ensemble is determined by the local performance of inner positions.
Finally, in the third phase, two decoding waves meet around the center and variable
nodes located at center positions are recovered, which corresponds to an increase in
r1(τ).
Figure 3.5 shows the evolution of r1(τ) as the inner for-loop in Algorithm 3.2
proceeds. After for u = 9 in Algorithm 3.2, the degree distributions of center positions
from 9 to 12 are designed with decreasing their average variable node degree. However,
at the expense of lowering the average variable node degrees, the value of r1(τ) is
decreased in the third phase as shown in Figure 3.5. In addition, it is observed that
the minimum value of r1(τ) in the third phase becomes nearly identical to the value
of r1(τ) in the second phase due to the constraint that maintains the BP threshold. If
the minimum value of r1(τ) in the third phase becomes smaller than the value in the
second phase, the BP threshold will be degraded. In other words, because the value of
r1(τ) in the third phase is larger than the second phase value for the initial regular SC-
31
1.E-02
1.E-01
1.E+00
0 2 4 6 8 10 12
Initialization100
10-1
10-2
for = 9
for = 7
for = 5
for = 3
for = 1
Figure 3.6: Evolution of r1(τ) for the SC-LDPC ensembles designed by Algorithm 3.3
as the inner for-loop proceeds.
LDPC ensemble, there is a room for an improvement in the design rate by optimizing
degree distributions of center positions. However, an improvement in the design rate
cannot be achieved by optimizing degree distributions of inner positions due to the
constraint of the BP threshold. Accordingly, the average variable node degree of the
positions except center positions does not change, as indicated in Figure 3.3, and the
evolution of r1(τ) is unchanged from for u = 9 to for u = 1.
Compared to Algorithm 3.2, Figure 3.6 shows the evolution of r1(τ) for Algorithm
3.3. After for u = 9, where the local design for positions 9–12 has been performed,
r1(τ) in the right part of the second phase is increased. This means that the local
performance of the center positions is improved and consequently the length of the
second phase is shortened, or equivalently, the overall number of required iterations
is decreased. Likewise, from for u = 9 to for u = 7, the length of the second phase
is steadily shortened. After for u = 5, the second phase observed in the initialization
almost disappears and only a single local minimum point remains. The local minimum
value is greater than the value of r1(τ) in the second phase at the initialization, which
means that the BP threshold is increased. Further, after for u = 3, the second phase
32
Table 3.3: Comparison of the BP threshold of the regular SC-LDPC and the proposed
irregular SC-LDPC ensembles
εBP
L = 10 L = 20 L = 30
SC-LDPC, Regular, (3, 6) 0.4962 0.4881 0.4881
SC-LDPC, Regular, (4, 8) 0.4981 0.4977 0.4977
SC-LDPC, Alg. 3.3, (4, 8) 0.5241 0.5069 0.5027
SC-LDPC, Alg. 3.4, (3, 6) 0.5679 0.5247 0.5087
SC-LDPC, Alg. 3.4, (4, 8) 0.5435 0.5172 0.5089
observed in the initialization completely disappears, which implies that the BP thresh-
old is increased significantly. In other words, the BP threshold is increased after for
u = 5 because the local performances of positions 7–14 have already been improved.
Conversely, the BP threshold does not increase up to for u = 7 because the local
performances of positions 1–6 and 15–20, which are not yet designed at this point,
become a bottleneck despite the fact that the local performances of positions 7–14 are
improved by the local design. On the other hand, the BP threshold is increased when
the bottleneck positions 5 and 16 are designed in for u = 5. In summary, the BP thresh-
old can be increased if the degree distribution of all positions is properly designed by
Algorithm 3.3. In Table 3.3, the BP thresholds of the regular SC-LDPC ensembles and
the proposed irregular SC-LDPC ensembles from Algorithm 3.3 are shown. Note that
Table 3.3 represents variable and check node degrees for the regular LDPC ensem-
bles and initial variable and check node degrees for the proposed irregular SC-LDPC
ensembles.
Remark 1. The analysis of the expected graph evolution suggests that the BP thresh-
old can be increased by increasing r1(τ) at each local optimization. According to [41],
the value of r1(τ) is related to the DE equations and a larger r1(τ) is obtained as
increasing the convergence speed, or equivalently, decreasing the overall number of
33
145
150
155
160
165
170
20 25 30 35 40 45 50 55 60 65
Over
all
nu
mb
er o
f re
qu
ired
ite
rati
on
s
Number of required iterations in one-dimensional DE
for = 5
for = 3
Figure 3.7: Correlation between two numbers of required iterations.
required iterations. Algorithm 3.3, however, aims to minimize the number of required
iterations in the one-dimensional DE equation, which does not guarantee minimization
of the overall number of required iterations because the one-dimensional DE assumes a
different decoding scheduling from the conventional parallel scheduling as mentioned
in Section 3.2.1. However, it is numerically observed that these two kinds of num-
bers of required iterations are generally proportional. Figure 3.7 shows the correlation
between the numbers of required iterations in the one-dimensional DE equation and
in DE equations (3.2) at for u = 5 and for u = 3 in Algorithm 3.3. Each point in
Figure 3.7 is the numbers of required iterations measured at ε = εBP − 0.01 for a
randomly generated degree distribution. According to Figure 3.7, it can be seen that
both numbers of required iterations are generally proportional and thus the objective
function of Algorithm 3.3 is proper to increase r1(τ) at each local optimization.
3.2.4 Improving the Performance by Permitting Non-Uniform Check Node
Degrees
In Algorithm 3.3, a uniform check node degree is considered, i.e., rv = r for all
v. An additional improvement can be achieved by permitting non-uniform check node
34
Algorithm 3.4 Design algorithm for minimizing the number of required iterations with
non-uniform check node degreesInput: l, r, L, w,M, lmin, lmax, rmin, rmax, Q, Imax
1: Initialization: Code parameters of the regular SC-LDPC ensemble
2: for Iter = 1 to Imax do
3: for u = L/2 to 1 do
4: Calculate the BP threshold εBP by the DE equations (3.2) and set s = 0
5: for v = u to u+ w − 1 do
6: for rv = rmin to rmax do
7: Tv,u = rvMlr −
∑j 6=u
Tv,j , s← s+ 1, vs = v, rs = rv,Ts = T
8: Pre-computation of δu(z) by Algorithm 3.1
9: Obtain λ∗s(x) =lmax∑
k=lmin
λ∗s,kxk−1 by solving the following LP problem
minimize I(s) =
Q−1∑q=2
∆zqzq − εBPλ∗s(δu(zq))
subject to1∫ 1
0λ∗s(x)dx
=∑i
Ti,u/M, εBPλ∗s(δu(zq)) < zq for 1 ≤ q ≤ Q
10: end for
11: end for
12: s∗ = argmin I(s), λu(x) = λ∗s∗(x), rvs∗ = rs∗, T = T s∗
13: end for
14: end for
degrees, which is considered in Algorithm 3.4. For example, when designing λu(x),
degree ru is changed from rmin to rmax and the optimal ru value is selected such that
the approximated value of the number of required iterations in (3.3) is minimized.
Such comparison is performed for all degrees of check nodes connected to variable
nodes at position u, that is, ru, . . . , ru+w−1 and then the optimal result is selected.
Note that the corresponding entries Tu,u, . . . , Tu+w−1,u of connectivity matrix T are
modified according to the selected check node degree. This algorithm permits non-
uniform check node degrees and non-zero entries of connectivity matrix T . Because it
35
adds a degree of freedom in the design of λu(x), Algorithm 3.4 has higher complexity
than Algorithm 3.3 but additional improvements can be achieved.
Unlike Algorithm 3.3, Algorithm 3.4 can be initialized with initial degrees (l, r) =
(3, 6) and the minimum variable node degree lmin = 3 because the average vari-
able node degree can be higher than 3 if degrees of connected check nodes are in-
creased. For example, let’s consider when designing λu(x) for (l, r) = (3, 6) and
w = 3. Recall that the number of edges∑
i Ti,u coming into variable nodes at po-
sition u should be equal to the number of sockets M/∫ 10 λu(x)dx and that there are
Ml/w = M edges between two connected positions. Thus, the average variable node
degree 1/∫ 10 λu(x)dx should be equal to 3 because the number of edges coming to
variable nodes at position u is∑
i Ti,u = 3M for the regular SC-LDPC ensemble.
However, as an example, if the degree of check nodes at position u increases from
6 to 7 and the number of edges between variable nodes and check nodes at position
u accordingly increases from M to 32M , the number of edges connected to variable
nodes at position u becomes 72M . Then the average variable node degree becomes
72 ≥ 3. Therefore, variable nodes of higher degrees can be introduced while maintain-
ing the minimum variable node degree lmin as 3. In Table 3.3, the BP thresholds of
the proposed irregular SC-LDPC ensembles from Algorithm 3.4 with initial degrees
(l, r) = (3, 6) and (l, r) = (4, 8) are included for a various ofL. The other input values
of Algorithm 4 are set by w = 3, lmin = 3, lmax = 10, rmin = 6, rmax = 8, Q = 1000,
and Imax = 10. As expected, the BP threshold is further improved by Algorithm 3.4
with initial degrees (l, r) = (4, 8) from the BP threshold of Algorithm 3.3. In addition,
Algorithm 3.4 with initial degrees (l, r) = (3, 6) outperforms the other cases for small
values of L such as L = 10 and L = 20.
Remark 2. The number of total edges in designed codes by Algorithm 3.4 depends
on the maximum check node degree rmax. To maintain the decoding complexity with
regular SC-LDPC codes, the maximum degree rmax is set by 8. Thus, the decoding
complexity of the codes to be evaluated in the following section is lower bounded by
36
that of regular SC-LDPC codes with (l, r) = (4, 8). Without the constraint of rmax, it
is observed that a better BP threshold is obtained but the result is not included for a
fair comparison with regular SC-LDPC codes.
3.3 Performance Evaluation
In this section, a method is introduced to improve the performance further and show
feasibility of the proposed design algorithms by comparing the finite-length perfor-
mance.
3.3.1 Performance Improvement Using Multi-Edge Type Check Nodes
SC-LDPC ensembles defined in Section 3.1.1 are referred to as randomly constructed
ensembles because variable node sockets at position u are connected to check nodes at
position v at random with probability Tv,u/∑
i Ti,u. However, as mentioned in previ-
ous works [35], [41], [56], randomly constructed SC-LDPC codes are inferior to SC-
LDPC codes with specific structures such as protograph-based SC-LDPC codes [35]
in terms of the finite-length performance. In Protograph-based SC-LDPC codes, the
multi-edge type (MET) structure [7] is imposed to both variable and check nodes. In
this chapter, however, SC-LDPC ensembles having the MET structure only for check
nodes in designing degree distributions are considered because variable node sockets
should be connected randomly with check nodes to utilize the one-dimensional DE.
It is assumed that edges connected to a check node consist of w edge types. An
edge of type t, which is connected to a check node at position v, is connected to a
variable node at position v − t + 1 for 1 ≤ t ≤ w. A degree type of check nodes is
represented by vector d = (d1, . . . , dw), where dt is the number of edges of type t.
Also, at position v, it is assumed that there are w degree types, which are represented
by each row of aw×w degree type matrix Sv. To be specific,Mc1w check nodes have a
degree type represented by d = Svk , (Svk,1, . . . , Svk,w), where Svk denotes the kth row
37
vector of Sv for k ∈ [1, w]. For example, the MET regular SC-LDPC ensemble with
r = 6 and w = 3 has Svk,j = 2 for all k and j, that is, all check nodes have the same
degree type vector (2, 2, 2) similar to protograph-based SC-LDPC codes. For regular
SC-LDPC ensembles when r/w is not an integer, degree types of check nodes are not
yet defined in terms of protograph-based ensembles. To define these degree types as
closely as possible to the structure of protograph-based ensembles, the first row Sv1 of
degree type matrix Sv for position v is defined as
Sv1 =( r mod w︷ ︸︸ ︷dr/we, . . . , dr/we,
w−(r mod w)︷ ︸︸ ︷br/wc, . . . , br/wc
).
In addition, the kth row Svk for 2 ≤ k ≤ w is obtained by right circular shifting the
first row Sv1 by k times.
The number of edges between check nodes at position v and variable nodes at
position v − t + 1 becomes Mc1w
∑k S
vk,t, which equals to Tv,v−t+1. The construc-
tion method of MET SC-LDPC codes is described as follows. First, the placement of
variable and check nodes is identical to that of randomly constructed SC-LDPC codes
in Section 3.1.1. Let’s label the variable and check node sockets at each position and
assign degree type vector Svk to Mc1w check nodes at position v for k ∈ [1, w]. Then,
the number of check node sockets connected to variable nodes at position v − t+ 1 is
Mc1w
∑k S
vk,t = Tv,v−t+1 and let these check node sockets be group t for t ∈ [1, w].
Let πu be a random permutation on [1,∑
i Ti,u] for position u. Divide πu into w dis-
joint subsets denoted by π1u, . . . , πwu such that the size of πtu becomes Tu+t−1,u. Then,
the jth check node socket in group t at position u+ t− 1 is connected to the πtv(j)th
variable node socket at position u.
Since the 1/w fraction of check nodes at position v has check node degree type Svk,
the node perspective degree distribution [7] of check nodes at position v is represented
as
Rv(x) =
w∑k=1
1
wxS
vk ,
38
where x = (x1, . . . , xw) and xSvk denotes Πw
t=1xSvk,t
t . From the node perspective de-
gree distribution, the edge perspective degree distribution of check nodes at position v
is obtained as
ρv(x) =
(∂Rv(x)
∂x1/∂Rv(1)
∂x1, . . . ,
∂Rv(x)
∂xw/∂Rv(1)
∂xw
)=
(w∑k=1
Svk,1rv
xSvk/x1, . . . ,
w∑k=1
Svk,wrv
xSvk/xw
).
Similar to the DE equations in (3.2), let x(`)u denote the average erasure probability
of messages at iteration ` emitted from variable nodes at position u. In addition, let
the tth element of a vector y(`)v
= (y(`)v,1, . . . , y
(`)v,w) be the average erasure probability
of messages at iteration ` emitted from check nodes at position v to variable nodes
at position v − t + 1. Set the initial conditions as x(0)u = ε for all u. Then, the DE
equations for MET SC-LDPC ensembles are described as
y(`)v
= 1− ρv(1− x(`)v−(w−1), . . . , 1− x
(`)v )
x(`+1)u = ελu
w∑t=1
Tu+t−1,uy(`)u+t−1,t∑
i Ti,u
(3.4)
where x(`)u = 0 for u < 0 and all `.
Algorithms 3.3 and 3.4 can be applied to MET SC-LDPC ensembles using the
DE equations in (3.4). Note that especially for Algorithm 4, the degree type matrix Sv
should be changed according to the change of check node degrees and the connectivity
matrix. Also, because the DE equations (3.4) are diffrerent from (3.2), the resulting
degree distributions also differ from those obtained from randomly constructed SC-
LDPC ensembles. Table 3.4 summarizes the BP thresholds of the regular SC-LDPC
ensembles with degrees (3, 6) and (4, 8) along with the proposed irregular SC-LDPC
ensembles from Algorithms 3.3 and 3.4 with the MET structure. The same result is
plotted in Figure 3.8. By comparing the results in Tables 3.3 and 3.4, is is verified that
39
Table 3.4: Comparison of the BP thresholds between the regular SC-LDPC ensembles
and proposed irregular SC-LDPC ensembles with the MET structure.
εBP
L = 10 L = 20 L = 30 L = 40 L = 100
SC-LDPC, Regular, (3, 6) 0.4962 0.4881 0.4881 0.4881 0.4881
SC-LDPC, Regular, (4, 8) 0.4984 0.4977 0.4977 0.4977 0.4977
SC-LDPC, Alg. 3, (4, 8) 0.5280 0.5079 0.5029 0.5018 0.4985
SC-LDPC, Alg. 4, (3, 6) 0.5693 0.5290 0.5099 0.5012 0.4979
SC-LDPC, Alg. 4, (4, 8) 0.5465 0.5205 0.5095 0.5058 0.4994
Shannon limit 0.6000 0.5500 0.5333 0.5250 0.5100
the BP threshold is improved by imposing the MET structure. Thus, the performances
of codes having the MET structure will be evaluated from now on.
As shown in Table 3.4, the BP threshold is improved for all values of L by Algo-
rithm 3.3 and further improved by Algorithm 4 with (l, r) = (4, 8) as expected. How-
ever, the degree of improvement becomes smaller as L increases because the regular
SC-LDPC ensemble already approaches the Shannon limit, which is clearly shown in
Figure 3.8. On the contrary, the improvement is evident especially for short and mod-
erate chain lengths. Likewise, for Algorithm 4 with initial degrees (l, r) = (3, 6), the
BP threshold is substantially improved for a small chain length such as L = 10 but
it approaches the BP threshold of the regular SC-LDPC ensemble. In the following
subsections, the finite-length performance will be evaluated using the window decoder
[53], [54] for moderate and large values of L and BP decoder for a short L, respec-
tively.
40
0.35 0.4 0.45 0.5
0.48
0.5
0.52
0.54
0.56
0.58
0.6
0.62
Figure 3.8: Design rate and BP threshold of the regular SC-LDPC and proposed irreg-
ular SC-LDPC ensembles for various value L.
3.3.2 Windowed Decoding of Proposed SC-LDPC Codes for Moderate
and Large L
In Figure 3.8, the effectiveness of the proposed design algorithms has been shown
through the BP threshold, which is obtained under the assumptions of infinite block-
length and the flooding scheduling. In this subsection, the finite-length performance
of proposed codes is confirmed using the windowed decoding scheduling. In general,
SC-LDPC codes with a large L are decoded by the windowed decoding algorithm
to mitigate the high decoding complexity and latency induced by long blocklengths.
The decoding complexity and latency of the window decoder with window size W are
reduced by W/L times compared to the BP decoder [53], [54].
Since the proposed design algorithms do not change the diagonal stairlike structure
of connectivity matrix T, proposed irregular SC-LDPC codes still maintain a suitable
structure to be decoded by the window decoder. To figure out the windowed decoding
performance, let’s define the threshold εWINu of position u as the maximum of ε for
which x(`)u goes to below target erasure probability δt, given that x(`)i = δt for 1 ≤ i ≤
41
5 10 15 20 25 30
0.48
0.485
0.49
0.495
0.5
0.505
0.51
0.515
0.52
Figure 3.9: Threshold εWINu of the regular SC-LDPC and the proposed irregular SC-
LDPC ensembles along with their folded versions for L = 40.
u − 1 and x(`)i = 1 for i ≥ u + W [54]. Then, the windowed threshold εWIN of an
ensemble is defined as the minimum value of εWINu for all u.
Figure 3.9 shows the threshold εWINu of the regular SC-LDPC ensemble and the
proposed irregular SC-LDPC ensembles for L = 40, δt = 10−6, and W = 10. It
shows that the threshold εWINu of the regular SC-LDPC ensemble is independent of
position u except the right boundary positions. On the contrary, for the irregular SC-
LDPC ensembles, εWINu is dependent on position u because degree distributions are
different for each position. Then, the windowed threshold εWIN of the proposed irreg-
ular SC-LDPC ensembles is determined at a strict global minimum point. For example,
the windowed threshold εWIN of the irregular SC-LDPC ensemble from Algorithm 3.3
is 0.4920, which is equal to the minimum value εWIN27 . In Table 3.5, the BP and win-
dowed thresholds are summarized, which shows that the windowed threshold εWIN of
the irregular SC-LDPC ensemble from Algorithm 3.3 is lower than its BP threshold.
For the irregular SC-LDPC ensemble from Algorithm 4, the performance degrada-
42
Table 3.5: BP and windowed thresholds of the regular and proposed irregular SC-
LDPC ensembles along with their folded versions for L = 40
εBP εWIN
SC-LDPC, Regular, (4, 8) 0.4977 0.4976
SC-LDPC, Alg. 3, (4, 8) 0.5018 0.4920
SC-LDPC, Alg. 4, (4, 8) 0.5058 0.4269
Folded SC-LDPC, Regular, (4, 8) 0.4977 0.4976
Folded SC-LDPC, Alg. 3, (4, 8) 0.5018 0.5011
Folded SC-LDPC, Alg. 4, (4, 8) 0.5058 0.5011
tion under windowed decoding is more noticeable since the threshold εWINu sharply
decreases from position 11 as shown in Figure 3.9.
The gap between the BP and windowed thresholds comes from different behaviors
of the wave-like decoding under each of decoding algorithms. While two decoding
waves occur from both of the left and right boundaries under BP decoding [36], only
one decoding wave occurs and propagates from the left boundary under windowed
decoding [54]. When obtaining εWINu , it is assumed that messages coming from left
positions i < u are sufficiently reliable by setting x(`)i = δt for 1 ≤ i ≤ u − 1.
Nevertheless, for the proposed irregular SC-LDPC ensembles, the result that εWINu
at the minimum point is significantly lower than the BP threshold implies that one
decoding wave is insufficient to show their maximum decoding performance. In other
words, considering the differences in the behavior of the wave-like propagation in
BP and windowed decodings, it is concluded that the proposed irregular SC-LDPC
ensembles require two decoding waves starting from both boundaries to avoid the
performance degradation.
Although two decoding waves can be triggered by two window decoders starting
from the right and left boundaries, respectively, the decoding method using two win-
dow decoders loses the latency gain of windowed decoding schemes. In order to over-
43
(a) Original SC-LDPC (b) Folded SC-LDPC
Figure 3.10: Tanner graphs of an original SC-LDPC code for L = 6 and w = 3 and
the corresponding folded SC-LDPC code.
come the problem, folded SC-LDPC codes [51] are used. As an example, Figure 3.10
shows the Tanner graphs of an original SC-LDPC code for L = 6 and w = 3 and
its corresponding folded SC-LDPC code. Given an original SC-LDPC code for even
L and odd w, the corresponding folded SC-LDPC code is obtained by folding in half
using the symmetric structure of SC-LDPC codes. Since the proposed irregular SC-
LDPC codes maintain the symmetric structure as well, it is possible to construct the
folded irregular SC-LDPC codes for all even values of L. In a folded SC-LDPC code
with even L and odd w, there exist M variable nodes at each position, 1 ≤ u ≤ L/2
and Mc check nodes at each position, 1 ≤ v ≤ (L+w− 1)/2. Given code parameters
λu, ρv,Tv,u of an original SC-LDPC ensemble, code parameters λFu , ρFv ,T
Fv,u of the
corresponding folded SC-LDPC ensemble become
λFu = λu for u ≤ L/2, ρFv = ρv for v ≤ (L+ w − 1)/2
TFv,u = Tv,u + Tv,L−u+1 for v ≤ (L+ w − 1)/2 and u ≤ L/2.
Consider an original SC-LDPC code and its corresponding folded SC-LDPC code
to compare two codes. First, the blocklength of the folded SC-LDPC code becomes
half of that of the original SC-LDPC code. For the design rate, original and folded en-
sembles have the same design rate because the ratio of variable nodes and check nodes
remains unchanged by folding. For example, the code-rate of both of codes in Fig-
ure 3.10 is 1/3. For a comparison of the decoding performance, first, consider the BP
44
decoding algorithm. Under BP decoding, the decoding wave from the right boundary
is not generated because low degree check nodes at the right boundary positions do not
exist in folded SC-LDPC codes as shown in Figure 3.10. Losing one decoding wave
leads to degradation in the BP decoding performance, but it can be compensated by
doubling the value of M , which results in the same blocklength with the original SC-
LDPC code. Next, consider the windowed decoding algorithm. Under windowed de-
coding, losing one decoding wave is not problematic since the window decoder utilizes
only one decoding wave. Rather, a folded SC-LDPC code has slightly improved per-
formance compared to its corresponding original one because the number of positions
to be passed through by the window decoder is decreased, which will be confirmed by
simulation. In addition, two ensembles have the same decoding latency and complex-
ity under windowed decoding since the constraint length of codes remains unaffected
by folding. In other words, folded SC-LDPC codes would be a preferable option under
both of BP and windowed decoding algorithms. Note that SC-LDPC codes having a
similar concept, which utilizes one decoding wave, are introduced in [53], [57], [58].
Furthermore, the windowed threshold of proposed irregular SC-LDPC ensembles
is improved by folding because the decoding behavior of folded SC-LDPC codes using
a conventional window decoder is equivalent to that of original SC-LDPC codes using
two window decoders. From the equivalence, the requirement that two decoding waves
are necessary to decode proposed irregular SC-LDPC codes is satisfied and then the
windowed threshold of folded irregular SC-LDPC ensembles is sufficiently close to
its BP threshold as shown in Table 3.5. To show the effect in detail, consider the case
when an original SC-LDPC code is decoded by two window decoders forW = 10 and
L = 40. Up to time 10 in the windowed decoding scheme, the edges covered by each
decoder do not overlap. From time 11, the edges covered by two decoders overlap and
variable nodes at position 11 are influenced by reliable messages propagated from the
right boundary. As a result, the threshold εWINu of folded irregular SC-LDPC ensem-
bles is improved from u = 11 as shown in Figure 3.9 and consequently the windowed
45
0.43 0.44 0.45 0.46 0.47 0.48 0.49
10-4
10-3
10-2
10-1
100
Blo
ck/b
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rasu
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rob
abil
itie
sSC-LDPC, Regular, (4,8)
Folded SC-LDPC, Regular, (4,8)
Folded SC-LDPC, Alg. 3.3, (4,8)
Folded SC-LDPC, Alg. 3.4, (4,8)
Figure 3.11: Block and bit erasure probabilities of the folded regular SC-LDPC code
and the folded irregular SC-LDPC codes for L = 40 under windowed decoding.
thresholds εWIN are not significantly degraded from their BP thresholds.
Figure 3.11 shows the block and bit erasure probabilities of the folded regular SC-
LDPC code and the folded irregular SC-LDPC codes for L = 40, where the solid lines
and the dashed lines correspond to the block and bit erasure probabilities, respectively.
The number of variable nodes at each position M is set by 990. The decoder uses
the windowed decoding algorithm over the BEC with W = 10. The simulation result
confirms the performance improvement of the proposed irregular SC-LDPC codes in
the finite-length performance. Note that the performance of the folded irregular SC-
LDPC code from Algorithm 3.4 is superior to that from Algorithm 3.3 even though
two codes have the same windowed decoding threshold. This is because the thresholds
εWINu from Algorithm 4 are on average better than the thresholds εWIN
u from Algorithm
3.3 as shown in Figure 3.9. Lastly, the performance of the original regular SC-LDPC
code is also included to confirm the validity of folded codes under windowed decod-
46
0.44 0.445 0.45 0.455 0.46 0.465 0.47 0.475 0.4810
-4
10-3
10-2
10-1
100
Blo
ck/b
it e
rasu
re p
rob
abil
itie
sFolded SC-LDPC, Regular, (4,8)
Folded SC-LDPC, Alg. 3.3, (4,8)
Folded SC-LDPC, Alg. 3.4, (4,8)
Figure 3.12: Block and bit erasure probabilities of the folded regular SC-LDPC code
and the folded irregular SC-LDPC codes for L = 100 under windowed decoding.
ing. The folded regular SC-LDPC code has slightly improved performance due to the
decreased number of positions compared to the original one. In addition, Figure 3.12
shows similar results for L = 100, where the degree of performance improvement
is rather diminished compared to the case for L = 40 as expected in the threshold
analysis.
3.3.3 Comparing the BP Decoding Performance of SC-LDPC Codes for
Small L
For small L such as L = 10, there is no gain in the windowed decoding algorithm
and thus BP decoding is considered. As shown in Table 3.4 and Figure 3.8, the gap
between the BP threshold and the Shannon limit is large for the regular SC-LDPC
ensembles with L = 10. However, the gap is substantially reduced for the proposed
irregular SC-LDPC ensembles. For example, the gap between the Shannon limit and
47
0.46 0.48 0.5 0.52 0.54 0.56
10-4
10-3
10-2
10-1
100
Blo
ck e
rasu
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rob
abil
ity
SC-LDPC, Regular, (4,8)
SC-LDPC, Alg.3, (4,8)
SC-LDPC, Alg.4, (3,6)
SC-LDPC in [14]
Block irregular LDPC
Folded SC-LDPC, Alg.4, (3,6)
Figure 3.13: BP decoding performances of the regular SC-LDPC code, the proposed
irregular SC-LDPC codes, and the block irregular LDPC code with the same design
rate and blocklength 9,900.
the BP threshold of the irregular SC-LDPC ensemble from Algorithm 4 with initial
degrees (l, r) = (3, 6) for L = 10 is 0.0307. On the contrary, the gap to the Shannon
limit of the regular SC-LDPC ensemble with (l, r) = (4, 8) for L = 10 is 0.1016. The
gap less than 0.0307 is achievable when the chain length L is greater than 36, where
the code rate, BP threshold, and the gap to the Shannon limit are 0.4722, 0.4977, and
0.0301, respectively. However, the BP threshold of the proposed irregular SC-LDPC
ensemble approaches the Shannon limit sufficiently even for L = 10 and thus large
values of L are not required.
Since the BP decoder uses the flooding scheduling, it is necessary to compare with
block codes having the same blocklength. As an example, let’s consider block irregular
LDPC codes with code rate 0.4 to compare SC-LDPC codes with L = 10. The degree
distribution of the block irregular LDPC ensemble is optimized by the LP problem
[7] with the maximum variable node degree 10. The resulting degree distributions are
48
0.42 0.44 0.46 0.48 0.5 0.52 0.5410
-8
10-6
10-4
10-2
100
Blo
ck/b
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rasu
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robab
ilit
ies
Block irregular LDPC
Folded SC-LDPC, Alg. 3.4, (3,6)
Figure 3.14: BP decoding performances of the regular SC-LDPC code, the proposed
irregular SC-LDPC codes, and the block irregular LDPC code with the same design
rate and blocklength 1,500.
λ(x) = 0.3468x+0.1672x2+0.001x3+0.001x7+0.002x8+0.4855x9, ρ(x) = x5 and
the BP threshold is 0.5720. The BP threshold of the block irregular LDPC ensemble
is slightly higher than that of the irregular SC-LDPC ensemble from Algorithm 4 as
shown in Figure 3.8. However, the block irregular LDPC ensemble has a large fraction
of degree-two variable nodes, while there are no degree-two variable nodes in the
proposed irregular SC-LDPC codes.
Figure 3.13 shows the block erasure probability of the regular SC-LDPC code and
the irregular SC-LDPC codes for L = 10, M = 990, and the blocklength 9,900 under
BP decoding. The simulation result shows a substantial performance improvement of
the proposed codes. It is also shown that the irregular SC-LDPC code from Algorithm
4 outperforms the irregular SC-LDPC code with block length 10,000 proposed in [49].
In addition, the performances of the block irregular LDPC code constructed by the pro-
gressive edge growth (PEG) algorithm [59] with blocklength 9,900 and the folded ir-
49
regular SC-LDPC code from Algorithm 4 are included. For the folded SC-LDPC code,
the value M is set to 990× 2 so that the blocklength becomes 9,900. By doubling M ,
the performance of the folded irregular SC-LDPC code is improved and accordingly
the folded irregular SC-LDPC code from Algorithm 4 outperforms the block irregular
LDPC code. Moreover, the performance of the proposed irregular SC-LDPC code is
expected to be better in the error-floor region due to the design constraint on excluding
degree-two variable nodes. To confirm the performance in the error-floor region, the
performances of the codes with short blocklength 1,500 are compared in Figure 3.14.
The simulation result shows that the performance of the proposed irregular SC-LDPC
code is better than that of the block irregular LDPC code in the block error rate lower
than 10−4 and the bit error rate lower than 10−7 because of the error-floor phenomenon
of block irregular LDPC codes.
3.3.4 Applying the Proposed Algorithms to SC-RA Codes
In this subsection, it is shown that the proposed algorithms can be applied to other
coupled code structures. Especially, the SC-RA code ensemble [42] is considered due
to its superior decoding performance reported in [56]. It is known that SC-RA codes
outperform regular SC-LDPC codes with the same design rate, blocklength, and de-
coding complexity [42], [56]. The SC-RA ensemble is constructed by coupling the
L disjoint uncoupled (q, a)-regular RA ensemble. Specifically, the SC-RA ensemble
with q = a is considered in this chapter. Figure 3.15 shows a Tanner graph of the SC-
RA ensemble for L = 2, q = 3, and M = 4. There are M/2 variable nodes of degree
q at each position from 1 to L in the upper side and M/2 variable nodes of degree 2
at each position from 1 to L + q − 1 in the lower side. The variable nodes of degree
q at position u are connected to the check nodes at positions u, . . . , u + q − 1, which
implies that the possible number w of positions to be connected is equal to the variable
node degree q. On the other hand, the variable nodes of degree 2 are connected to the
check nodes at the same position. Let’s consider the SC-RA ensemble with the MET
50
Position 1
Figure 3.15: Tanner graph of the SC-RA ensemble for q = 3, L = 2, and M = 2.
Table 3.6: BP thresholds of the regular SC-RA and the SC-RA ensembles designed by
Algorithm 3.3
εBP
L = 10 L = 20 L = 30
SC-RA, Regular 0.5107 0.4949 0.4946
SC-RA, Alg. 3.3 0.5399 0.5128 0.5051
structure whose check nodes at position v are connected to variable nodes of degree
q based on a q × q degree type matrix Sv. Since w is equal to q, Sv becomes all one
matrix for all v. The edge connection between the variable nodes of degree q and the
check nodes is established using a method similar to that of the MET SC-LDPC en-
semble. The remaining construction method follows the process described in Section
3.3.1.
The irregular SC-RA ensemble with non-uniform degree distributions has irregu-
lar degree distribution λu(x) with a minimum degree of 3 for the variable nodes in the
upper side at position u, while the variable nodes of degree 2 remain unchanged in the
lower side. Using the design algorithm of Algorithm 3.3, the degree distributions of
the SC-RA ensemble with non-uniform degree distributions can be obtained. Table 3.8
shows that the BP threshold of the SC-RA ensemble with non-uniform degree distri-
butions designed by Algorithm 3.3 is superior to that of the regular SC-RA ensemble.
51
Table 3.7: Coefficients of λu(x) for the SC-RA ensemble designed by Algorithm 3.3
for L = 20.u x2 x3 x4 x5 x6 x7 x8 x9
1 0 0 1 0 0 0 0 0
2 0.02391 0 0.880448 0.095642 0 0 0 0
3 0.025618 0 0.914607 0 0.059775 0 0 0
4 0.414463 0 0 0.04938 0 0 0 0.536157
5 0.327934 0 0 0 0.374095 0.297971 0 0
6 0.428571 0 0 0 0 0 0 0.571429
7 0.428571 0 0 0 0 0 0 0.571429
8 0.428571 0 0 0 0 0 0 0.571429
9 0.428571 0 0 0 0 0 0 0.571429
10 0.428571 0 0 0 0 0 0 0.571429
Note that the result obtained by Algorithm 3.4 is not included because the performance
improvement is insignificant compared to the results by Algorithm 3.3.
The design method of degree distributions of irregular SC-RA ensembles is con-
ceptually identical to that of irregular SC-LDPC ensembles. However, when a code
instance is generated from the designed ensemble, small stopping sets [7] consisting
of variable nodes of degree lower than q can be made with a high probability due to the
existence of variable nodes of degree 2. Therefore, it is practically important to avoid
such small stopping sets for finite-length codes. Figure 3.16 shows two types of stop-
ping sets to be avoided, where the stopping sets are represented by the hatched variable
nodes. Type 1 stopping sets consist of one variable node of degree i, 3 ≤ i < q, with
a set of variable nodes of degree 2 and all edges of a variable node degree i have at
least one edge pair connected to check nodes at the same position. Similarly, type 2
stopping sets consist of multiple variable nodes of degree i, 3 ≤ i < q, with a set of
variable nodes of degree 2 and all edges of the multiple variable nodes have at least one
52
Table 3.8: Comparing the BP thresholds of the regular SC-RA ensembles with the
ensembles obtained by the proposed algorithm
εBP
L = 10 L = 20 L = 30
SC-RA, Regular 0.5107 0.4949 0.4946
SC-RA, Alg. 3 0.5399 0.5128 0.5051
ci cj
Position u
d=3
Position u+1
d=2
(a) Type 1
Position u Position u+1 Position u+2
(b) Type 2
Figure 3.16: Two types of stopping sets produced by low degree variable nodes.
edge pair connected to check nodes at the same position. Other stopping sets such as
stopping sets consisting of variable nodes whose degrees are greater than or equal to q
are disregarded because the probability of occurrence of such stopping sets is identical
to that in the regular SC-RA ensemble.
Consider a loop consisting of Mc check nodes and Mc variable nodes of degree 2
at a position in a Tanner graph. If the edges between check nodes and variable nodes
of degree 2 are properly connected, only one loop of length 2Mc is produced at each
position. On the loop, define distance d in the loop of two check nodes as the number of
check nodes between two check nodes including themselves. For example, the distance
in the loop of check nodes ci and cj in Figure 3.16(a) is 3. Consider a case in which
two edges of a variable node are connected to two check nodes that are separated
by distance d in the loop at the same position. This inevitably produces a cycle of
length 2d. Further, if all edges of a variable node are contained in such cycles, a type
53
1 stopping set is produced.
However, type 1 stopping sets are easily avoided by imposing the MET structure on
variable nodes as well as check nodes. If all edges of a variable node are connected to
check nodes at different positions, type 1 stopping sets are avoided accordingly. Thus,
the MET structure is imposed on variable nodes such that each edge of a variable node
is connected to the check node at different positions. To be specific, the first row Su,i1
of the q × q degree type matrix Su,i for variable nodes of degree i at position u is
defined as
Su,i1 =( i mod q︷ ︸︸ ︷di/qe, . . . , di/qe,
q−(i mod q)︷ ︸︸ ︷bi/qc, . . . , bi/qc
)and the remaining rows are obtained by circularly shifting the first row. For example,
Su,i1 for degree i = 3 and q = 5 is given as Su,i1 = (1, 1, 1, 0, 0) which implies that all
edges of variable nodes of degree 3 are connected to check nodes located at different
positions and thus type 1 stopping sets can be avoided.
Even if all edges of a variable node are connected to check nodes at different
positions, type 2 stopping sets can be made. To avoid type 2 stopping sets, the PEG
algorithm [59] is used with an additional constraint. Let N lv be the neighborhood of
variable node v within depth l, which denotes the set of check nodes reached by a
computation tree spreading from variable node v within depth l. In the conventional
PEG algorithm, a cycle of length 2l+ 2 is avoided by connecting the edges of variable
node v to the check nodes not inN lv. However, the conventional PEG algorithm cannot
prevent the production of type 2 stopping sets if the girth of the PEG algorithm is lower
than the length of the cycles in the stopping set. For example, the length of the cycle
denoted by the bold line in Figure 3.16(b) is 10 and thus it is not avoided by the PEG
algorithm with a girth such as 6 or 8. Thus, it is difficult to prohibit connecting the
check nodes in N lv using the conventional computational tree. Instead, let’s consider
the expanded computation tree by d, which is defined as follows. At depth l of the
conventional computation tree from variable node v, there exist check nodes directly
connected to variable node v within depth l− 1, while the depth l of the expanded tree
54
0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.51
10-4
10-3
10-2
10-1
100
Blo
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bilit
ySC-LDPC, Regular, (4,8)SC-LDPC, Alg. 3.3, (4,8)SC-LDPC, Alg. 3.4, (4,8)SC-RA,RegularSC-RA, Alg. 3.3
Figure 3.17: Block erasure probability of the SC-RA codes for L = 20.
contains not only the directly connected check nodes but also the set of check nodes
within distance d in the loop from the directly connected check nodes. Also, only
the variable nodes of degree lower than q are included without other higher degree
variable nodes in the expanded computation tree. Let N l,dv be the set of check nodes
within depth l in the expanded computation tree by d from a variable node v. Then, by
connecting a new edge of variable node v to the check nodes not in N l,dv , a cycle of a
length up to (2l+ 2) + (l+ 1)d is avoided in the sub-graph consisting of check nodes
and low degree variable nodes. Because the expanded tree contains only low degree
variable nodes, this additional constraint can be applied for large l and d to avoid small
size of type 2 stopping sets. For example, the cycle in Figure 3.16(b) is avoided by
prohibiting a connection between variable node v and a check node in N 1,3v and thus
the type 2 stopping set can be avoided. Note that because this constraint aims to avoid
stopping sets consisting of low degree variable nodes, the conventional PEG algorithm
should also be applied at the same time.
55
0.44 0.45 0.46 0.47 0.48 0.49 0.5
10-4
10-3
10-2
10-1
100
Blo
ck e
rasu
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bilit
ySC-LDPC, Regular, (4,8)SC-LDPC, Alg. 3.3, (4,8)SC-LDPC, Alg. 3.4, (4,8)SC-RA,RegularSC-RA, Alg. 3.3
Figure 3.18: Block erasure probability of the SC-RA codes for L = 30.
3.3.5 Comparing Finite-Length Performances of SC-LDPC and SC-RA
Codes
The simulation results are included for the SC-LDPC codes and the SC-RA codes
for L = 20 and 30 in Figures 3.17 and 3.18. Each code instance for the SC-LDPC
ensembles is obtained with M = 990. To match their blocklengths with those of the
SC-LDPC codes, the SC-RA code instances are obtained from their ensembles with
M = 900. For L = 20, the blocklength of all the codes in Figure 3.17 is 19, 800. In
terms of the code rate, the SC-RA codes have an advantage because the code rate of
the SC-RA codes with q = 5 is slightly higher than that of the SC-LDPC codes with
w = 3 for the same L [42], [56]. Each code instance of the SC-RA codes with the
designed non-uniform degree distributions is obtained using the PEG algorithm with
an additional constraint that avoids a connection between variable node v and check
nodes in N 1,8v . As in the SC-LDPC codes, the SC-RA codes obtained by Algorithm
3.3 show better performance than the regular SC-RA codes for L = 10, 20, and 30.
Comparing all the results in Figures 3.17 and 3.18, it is confirmed that the SC-RA
56
codes obtained by Algorithm 3.3 are the best in terms of the finite-length performance.
57
Chapter 4
Rate-Loss Mitigation of SC-LDPC Codes without Degra-
dation of Finite-Length Performances
Compared to uncoupled block LDPC codes, the outstanding performance of SC-LDPC
codes comes from wave-like decoding [36], which is triggered by boundary low-degree
check nodes [44]. However, the design rate of SC-LDPC codes is reduced from that
of their corresponding uncoupled block LDPC codes. Thus, there have been many
works to deal with the rate-loss mitigation of SC-LDPC codes. First, the rate-loss can
be reduced to the desired level by simply adjusting the code parameters such as the
chain length L of SC-LDPC codes. Thus, in general, a large value of L is chosen to
design capacity approaching SC-LDPC codes [44], [48] and decoding is performed by
the windowed decoder [53], [54] to address long blocklength induced by a large value
of L. Increasing the chain length, however, causes a problem that the block error rate
(BLER) scales linearly with increasing L [41].
Other approaches to mitigate the rate-loss are attaching additional variable nodes
with regular [63], [64] or irregular degree distributions [51]. By adding extra variable
nodes to the boundary check nodes to increase the code dimension, the rate-loss can
be kept small even for a moderate value of L. Especially, in [51], the rate-loss is sub-
stantially mitigated by optimizing the degree distribution of additional variable nodes
58
using linear programming. However, it is observed that the SC-LDPC codes optimized
in [51] show degraded finite-length performance compared to the conventional SC-
LDPC codes even though their asymptotic performance is equivalent with respect to
the BP threshold.
The main purpose of this chapter is to propose a new optimization algorithm for
the degree distribution of attached additional variable nodes to minimize the rate-loss
without sacrificing the finite-length performance. First, I find out a design condition
affecting the finite-length performance using the expected graph evolution [41], [56].
In general, the expected graph evolution shows decoding behaviors of codes at a given
channel parameter, whereas the BP threshold represents the limiting point of decodable
channel parameters. According to the analysis of the decoding behavior of SC-LDPC
codes [41], the decoding failure rarely occurs in the initial phase of the decoding pro-
cess. However, if too many additional variable nodes are added at the boundary, a local
minimum of the number of degree-1 check nodes has appeared in the initial phase of
the decoding process, which causes performance degradation. Thus, the local mini-
mum constraint is defined such that a local minimum of the number of degree-1 check
nodes should not exist in the initial phase.
By using the concept of differential evolution algorithms [65], the optimized de-
gree distribution of additional variable nodes satisfying the local minimum constraint
can be obtained. Also, the rate-loss is mitigated further by employing a protograph
structure for additional variable nodes. From the experimental results, the proposed
SC-LDPC codes with the optimized degree distribution show nearly the same decoding
performance as the conventional SC-LDPC codes while the rate-loss is significantly
mitigated. It makes it possible to construct capacity approaching SC-LDPC codes even
for a moderate value of L. In addition, the gain in the design rate can be translated into
a performance improvement when a comparison is performed under the same design
rate.
The remainder of the chapter is organized as follows. Section 4.1 introduces the
59
SC-LDPC code structure and its density evolution equations. In Section 4.2, a new
optimization method for the degree distribution of additional variable nodes are pro-
posed and the finite-length performance is evaluated. In Section 4.3, the performances
of the proposed SC-LDPC codes and conventional SC-LDPC codes are compared in
the various settings.
4.1 Code Structure of SC-LDPC Codes
4.1.1 (l, r, L) SC-LDPC Ensemble
The conventional SC-LDPC ensemble in [44] with variable and check node degrees
(3, 6) and chain length L is represented by the following base matrix
2L︷ ︸︸ ︷
1 1
1 1. . .
1 1. . . 1 1
. . . 1 1 1 1
1 1 1 1
1 1
where its size is (L+ 2)× 2L. The degrees (l, r) = (3, 6) are considered as a running
example in this chapter. Each column and row of the base matrix correspond to a
variable node and a check node in the protograph, respectively, and 2L variable nodes
are grouped into L positions, where each position consists of two variable nodes. A
parity check matrix of a code instance is obtained by lifting the base matrix [68] with
lifting factor z. Let M denote the number of variable nodes at each position, i.e.,
M = 2z.
As illustrated in [44], the code structure of SC-LDPC codes that the degrees of the
first and last two check nodes are lower than six gives the following properties. First,
60
it induces the rate-loss from the design rate of uncoupled LDPC codes. Second, low
degrees of boundary check nodes trigger the wave-like propagation of reliable infor-
mation, where two decoding waves are propagated from both ends toward the inside on
the Tanner graph. However, two decoding waves are unnecessary in the windowed de-
coding scheme which utilizes only one decoding wave. Thus, the modified SC-LDPC
ensemble, denoted by (l, r, L) ensemble, is introduced in [51] by folding the last check
node. For example, the base matrix of the (3, 6, L) ensemble is represented as
2L︷ ︸︸ ︷
1 1
1 1. . .
1 1. . . 1 1
. . . 1 1 1 1
1 1 2 2
(4.1)
whose size is (L+ 1)× 2L. Since the decoding wave cannot be triggered by the right
boundary check nodes, whose degrees are changed to six, only one decoding wave is
propagated from the left boundary in the graph. However, the rate-loss is halved com-
pared to the conventional SC-LDPC ensemble for a given L. Considering the number
of variable and check nodes in (4.1), the design rate of the (3, 6, L) SC-LDPC ensem-
ble is given as
R(3,6,L) = 1− (L+ 1)z
2Lz=
1
2− 1
2L(4.2)
where the first term is the design rate of the uncoupled (3, 6) regular LDPC ensemble
and the second term 1/(2L) corresponds to the rate-loss, denoted by ∆R(3,6,L). Note
that the design rate and asymptotic properties under BP decoding of the (3, 6, L) SC-
LDPC ensemble represented by (4.1) are equivalent to those of the conventional SC-
LDPC ensemble with 2L due to the symmetric structure of the conventional SC-LDPC
ensemble.
61
4.1.2 (l, r, L, λ(x)) SC-LDPC Ensemble
To mitigate the rate-loss of the (l, r, L) SC-LDPC ensemble, the (l, r, L, λ(x)) SC-
LDPC ensemble was proposed by attaching additional variable nodes with degree
distribution λ(x) to the (l, r, L) ensemble [51]. The part of the (l, r, L) ensemble in
the (l, r, L, λ(x)) ensemble is called the main region, that is, the (l, r, L, λ(x)) en-
semble consists of the main region and attached additional variable nodes. For the
(3, 6, L, λ(x)) ensemble, additional variable nodes are attached to the first 2z check
nodes in the lifted Tanner graph with lifting factor z such that the degree of the 2z
check nodes becomes six. Let MA denote the number of additional variable nodes
in the lifted Tanner graph and lavg denote the average degree of additional variable
nodes, i.e., lavg = 1/∫ 10 λ(x)dx. Then, the number of sockets of additional variable
nodes is given as MAlavg. Since all sockets of additional variable nodes are randomly
connected to the first 2z check nodes, the following equation is satisfied as
MAlavg = 4z + 2z.
Then, the design rate of the (3, 6, L, λ(x)) ensemble is given as
R(3,6,L,λ(x)) = 1− (L+ 1)z
2Lz +MA=
1
2− 1
2
1− 3/lavgL+ 3/lavg
(4.3)
where the rate-loss ∆R(3,6,L,λ(x)) is (1− 3/lavg)/(2L+ 6/lavg), which is lower than
∆R(3,6,L) for lavg > 0. The rate-loss ∆R(3,6,L,λ(x)) is reduced as MA increases, or
equivalently, lavg decreases. Let γA be the rate-loss mitigation ratio of ensemble A
defined by
γA ,∆R(3,6,L) −∆RA
∆R(3,6,L),
which is used as a performance measure for ensemble A.
62
4.1.3 DE Equations of the (3, 6, L, λ(x)) Ensemble
Consider a base matrix B represented as
B =
2L+1︷ ︸︸ ︷
4 1 1
2 1 1. . .
1 1. . . 1 1
. . . 1 1 1 1
1 1 2 2
, (4.4)
which is a concatenation of the base matrix in (4.1) and [4 2 0 . . . 0]T. In terms of
the protograph representation, the base matrix B corresponds to the (3, 6, L, x5) en-
semble with z additional variable nodes of degree six. However, the base matrix B
is used not only to represent a specific ensemble, but also to derive the DE equations
of the (3, 6, L, λ(x)) ensemble with general λ(x) in a simple form. Note that, when
constructing a code instance from the (3, 6, L, λ(x)) ensemble, the base matrix B is
not lifted.
Let Ni denote the set of column indices of non-zero elements in the ith row in
B and Mj denote the set of row indices of non-zero elements in the jth column in
B. Also, let y(`)i,j denote the average erasure probability of messages sent from the
ith check node to the jth variable node in B at iteration `. Likewise, let x(`)i,j denote
the average erasure probability of messages sent from the jth variable node to the ith
check node in B at iteration ` and x(`)j denote the erasure probability of the jth variable
node in B at iteration `. Note that the first variable node in B corresponds to additional
variable nodes. Set the initial conditions as x(0)i,j = ε if Bi,j 6= 0. Then, the evolutions
63
of y(`)i,j , x(`)i,j , and x(`)j can be expressed as
y(`)i,j = 1−
∏j′∈Ni
(1− x(`−1)i,j′
)Bi,j′−δj,j′
x(`)i,j =
ελ
(∑i′
Bi′,jy(`)
i′,j∑i′′Bi′′,j
), for (i, j) = (1, 1) and (i, j) = (2, 1)
ε∏
i′∈Mj
(y(`)i′,j
)Bi′,j−δi,i′, otherwise
x(`)j =
εL
(∑i′
Bi′,jy(`)
i′,j∑i′′Bi′′,j
), for j = 1
ε∏
i′∈Mj
(y(`)i′,j
)Bi′,j, otherwise
(4.5)
where δ denotes the Kronecker delta function and L(x) =∫ x0 λ(z)dz/
∫ 10 λ(z)dz. The
BP threshold ε(3,6,L,λ(x)) is defined as the maximum ε for which x(`)j goes to zero for
all j as ` increases. Here, the BP threshold ε(3,6,L) of the (3, 6, L) ensemble is obtained
in the same way by setting the erasure probability of messages from additional variable
nodes equal to zero, i.e., x(`)i,j = 0 for j = 1. Note that ε(3,6,L) for L ≥ 11 is almost the
same as εSC , 0.4881 [36], [44]1.
4.2 Optimizing Methods of Degree Distribution λ(x)
4.2.1 Minimizing the Rate-Loss While Maintaining the BP Threshold
The design objective of the (3, 6, L, λ(x)) SC-LDPC ensemble is to optimize degree
distribution λ(x) that minimizes ∆R(3,6,L,λ(x)), or equivalently, minimizes lavg while
minimizing performance degradation from the (3, 6, L) SC-LDPC ensemble. The first
approach proposed in [51] is minimizing lavg with the constraint of ε(3,6,L,λ(x)) = εSC.
The optimization problem is solved by linear programming and the resulting degree1Since the BP threshold of the conventional SC-LDPC ensemble saturates the MAP threshold of
underlying LDPC codes, 0.4881, for L ≥ 22, the same effect occurs for the (3, 6, L) ensemble for
L ≥ 11.
64
0.44 0.45 0.46 0.47 0.48 0.49
10-4
10-3
10-2
10-1
100
Blo
ck e
rasu
re p
roba
bilit
y
Figure 4.1: BLER of the (3, 6, 20) conventional SC-LDPC code and the
(3, 6, 20, λ1(x)) SC-LDPC code in [51].
distribution λ1(x) is obtained as [51]
λ1(x) = 0.0193x2 + 0.3439x3 + 0.5310x6 + 0.1058x7.
Then, the average variable node degree of λ1(x) is computed as lavg = 5.5117 and
γ(3,6,20,λ1(x)) = 0.5564, i.e., 55.6% of the rate-loss is mitigated. Since the BP thresh-
old ε(3,6,L,λ1(x)) is unchanged from the BP threshold ε(3,6,L) under the BP threshold
constraint, the (3, 6, L, λ1(x)) ensemble is equivalent to the (3, 6, L) ensemble in the
sense of the asymptotic performance. However, the equivalent asymptotic performance
does not guarantee the equivalent finite-length performance.
Let P(3,6,L) and P(3,6,L,λ(x)) be BLERs of the (3, 6, L) and (3, 6, L, λ(x)) ensem-
bles, respectively. Figure 4.1 shows the BLERs of the (3, 6, 20) and (3, 6, 20, λ1(x))
ensembles with z = 500. As shown in Figure 4.1, the finite-length performance of the
(3, 6, 20, λ1(x)) ensemble is degraded from that of the (3, 6, 20) ensemble. To investi-
gate the reason of performance degradation, define the local BLER PA(3,6,L,λ(x)) as the
65
probability that at least one of additional variable nodes are not recovered. If all addi-
tional variable nodes are recovered during the decoding process of the (3, 6, L, λ(x))
ensemble, the probability that at lease one of the remaining variable nodes in the main
region are not recovered is equivalent to P(3,6,L). Thus, P(3,6,L,λ(x)) is given as
P(3,6,L,λ(x)) = PA(3,6,L,λ(x)) +
(1− PA
(3,6,L,λ(x))
)P(3,6,L)
≈ P(3,6,L), for PA(3,6,L,λ(x)) � P(3,6,L)
where the last approximation is satisfied if the local BLER of additional variable nodes
is sufficiently lower than that of the (3, 6, L) ensemble. In other words, the local per-
formance of additional variable nodes should be better than the performance of the
main region in order to maintain the finite-length performance. However, Figure 4.1
shows that PA(3,6,L,λ1(x))
is rather higher than P(3,6,L) for ε ≤ 0.457, which leads to
performance degradation.
4.2.2 Minimizing the Rate-Loss with Target Local Threshold
In order to find degree distribution λ(x) satisfying the requirement PA(3,6,L,λ(x)) �
P(3,6,L), let’s define the local threshold εA(3,6,L,λ(x)) of the (3, 6, L, λ(x)) ensemble as
the maximum ε for which the erasure probability x(`)1 of additional variable nodes goes
to below sufficiently small such as 10−3 as ` increases. As the BP threshold ε(3,6,L,λ(x))
represents the asymptotic performance of the overall codeword, the local threshold
εA(3,6,L,λ(x))is a performance measure specialized for additional variable nodes. For
λ1(x), the local threshold εA(3,6,L,λ1(x)) is equal to ε(3,6,L) = 0.4881, which implies
that λ1(x) is not a proper degree distribution to satisfy the requirement PA(3,6,L,λ(x)) �
P(3,6,L).
A differential evolution algorithm in Algorithm 4.1 is proposed to find a degree
distribution achieving target local threshold εA > ε(3,6,L). Similar to the differential
evolution algorithms in [66], [67], Algorithm 4.1 is initialized with a set of randomly
generated degree distributions and generate the improved set of degree distributions
66
through the mutation, crossover, and selection steps. In the selection step, a trial degree
distribution is selected if the trial degree distribution satisfies two constraints related
with the local threshold and the average variable node degree, respectively. Due to the
constraint on the average variable node degree, the degree distribution with a lower
average degree is generated as the generation progresses. With input parameters Np =
100, G = 6,000, lmin = 3, lmax = 10, pc = 0.88, and εA = ε(l,r,L) + 10−3, the
resulting degree distribution λ2(x) is obtained as
λ2(x) = 0.0026x3 + 0.6757x4 + 0.2943x7 + 0.0273x8
and γ(3,6,20,λ2(x)) = 0.5390, which is lower than γ(3,6,20,λ1(x)) = 0.5564. In Fig-
ure 4.2, PA(3,6,20,λ2(x))
is shown, where the local BLER is improved compared to the
results for λ1(x) at the cost of slightly reduced rate-loss mitigation ratio. In specific,
the requirement PA(3,6,20,λ2(x))
� P(3,6,20) is satisfied for ε close to the target local
threshold εA. However, the requirement is not satisfied for low ε such as ε = 0.45,
which shows the necessity of further improved optimizing methods.
4.2.3 Minimizing the Rate-Loss with Target Local Threshold and with-
out Local Minimum of r1
The BP and local thresholds are the upper limits of the channel parameters for which
the decoding is successful. On the contrary, the expected graph evolution [20] shows
the decoding behavior for a given channel parameter, which can be utilized in analysis
of the finite-length performance. In [41], the expected number of degree-1 check nodes
under peeling decoding is obtained by solving a system of differential equations. Also,
in [56], an upper bound of the expected number of degree-1 check nodes is obtained
from the DE equations, which requires less computational complexity compared to
solving a system of differential equations. Due to the low computational complexity, a
constraint obtained from the expected graph evolution can be used for the optimization
algorithm for the degree distribution of additional variable nodes.
67
0.44 0.45 0.46 0.47 0.48 0.49
10-4
10-3
10-2
10-1
100B
lock
era
sure
pro
babi
lity
Figure 4.2: BLERs of the (3, 6, 20) code, the (3, 6, 20, λ1(x)) code, and the pro-
posed (3, 6, 20,B1, 0.05) code along with local BLERs of the (3, 6, 20, λ1(x)),
(3, 6, 20, λ2(x)), (3, 6, 20, λ3(x)), (3, 6, 20,B1, 0.05) codes.
Let r(`)1 (ε) be the upper bound of the normalized number of degree-1 check nodes
connected to variable nodes in the main region of the (3, 6, L, λ(x)) ensemble at iter-
ation ` with channel parameter ε under BP decoding. Then r(`)1 (ε) is given as [56]
r(`)1 (ε) =
2L+1∑j=2
x(`−1)j −
2L+1∑j=2
x(`)j
where x(`)j is obtained by the DE equations in (3.2). Figures 4.3 and 4.4 show the
evolution of r(`)1 (ε) at ε = εSC − 2× 10−2 = 0.4681 and ε = εSC − 10−2 = 0.4781,
respectively. For the (3, 6, 20) ensemble, there exist three phases in the evolution of
r(`)1 (ε); the initial phase, the critical phase, and the third phase. In [41], the scaling
law of SC-LDPC codes is derived with the assumption that the decoding failure rarely
occurs in the initial phase. The assumption is justified by the property that the value
68
20 40 60 80 100 120 1400
0.05
0.1
0.15
0.2
5 10 15 20 25
0.05
0.06
0.07
Figure 4.3: Evolution of r(`)1 (ε) of the (3, 6, 20) and (3, 6, 20, λ(x)) SC-LDPC ensem-
bles with λ1(x), λ2(x), and λ3(x) at ε = 0.4681.
of r(`)1 in the initial phase is steadily decreasing to the value of the critical phase.
However, for the (3, 6, 20, λ1(x)) and (3, 6, 20, λ2(x)) ensembles, a local minimum
appears in the initial phase, which becomes additional factor of decoding failures and
accordingly the finite-length performance is predicted to be degraded for ε < εSC.
The predication of the finite-length performance actually requires not only the ex-
pected value of r1 but also the second order moments of r1 [41]. However, computing
the second order moments takes a large amount of computations, which is undesirable
for the optimization process. Moreover, the expected value of r1 is sufficient to find
a design condition causing decoding failure. In oder to verify that the emerged local
minimum becomes a factor of decoding failure, the evolution of r(`)1 (0.4781) is plotted
as a function of x(`)1 for the (3, 6, 20, λ1(x)) ensemble in Figure 4.5. In addition, the
empirical probability Pr(failure at x(`)1 ) is plotted, which is defined as the probability
that the average erasure probability of additional variable nodes is equal to x1 when the
decoder declares decoding failure in the finite-length code simulation. In Figure. 4.5,
69
50 100 150 200 2500
0.05
0.1
0.15
0.2
10 20 30 400.02
0.03
0.04
0.05
Figure 4.4: Evolution of r(`)1 (ε) of the (3, 6, 20) and (3, 6, 20, λ(x)) SC-LDPC ensem-
bles with λ1(x), λ2(x), and λ3(x) at ε = 0.4781.
Pr(failure at x(`)1 ) is plotted with the different y-axis, where Pr(failure at x
(`)1 ) is ob-
tained from 105 transmissions of the (3, 6, 20, λ1(x)) ensemble with z = 2,000. By
comparing two plots in Figure 4.5, it is confirmed that the decoding failure is likely
to occur at the local minimum of r(`)1 (ε), which implies that the local minimum is a
significant factor of decoding failure.
The local minimum constraint is defined such that r(`)1 (ε) should not have a local
minimum in the initial phase for ε < ε(3,6,L). By including the local minimum con-
straint in the selection step of Algorithm 4.12, the optimal degree distribution λ3(x)
can be obtained with target local threshold εA = εSC + 2× 10−3 as
λ3(x) = 0.01067x3 + 0.63926x4 + 0.35007x9
and γ(3,6,20,λ3(x)) = 0.5088. Figures 4.3 and 4.4 show that a local minimum of r(`)1 (ε)
does not exist for the (3, 6, 20, λ3(x)) ensemble. Accordingly, the local performance2To reduce the optimization complexity, the local minimum constraint is applied at two points ε =
εSC − 2× 10−2 and ε = εSC − 10−2.
70
10-2 10-10.02
0.025
0.03
0.035
0.04
0
1
2
3
4
510-3
Figure 4.5: Plots of r(`)1 (ε) and Pr(failure at x(`)1 ) against x(`)1 at ε = 0.4781 for the
(3, 6, 20, λ1(x)) SC-LDPC ensemble.
PA(3,6,L,λ3(x))
for degree distribution λ3(x) satisfies the requirement PA(3,6,L,λ3(x))
�
P(3,6,L) as shown in Figure 4.2. However, at the cost of the additional constraints im-
posed on λ3(x), the rate-loss mitigation ratio is reduced from γ(3,6,20,λ1(x)) = 0.5564
to γ(3,6,20,λ3(x)) = 0.5088. In the next subsection, a method is introduced to achieve
higher value of γ by employing the protograph structure for attached additional vari-
able nodes.
4.2.4 Optimizing the (3, 6, L,B, αA) Ensemble
As mentioned in Section 4.1.3, the base matrix in (4.4) is used to represent the DE
equations of the (3, 6, L, λ(x)) ensemble in a simple form. However, if the base matrix
in (4.4) is lifted with lifting factor z, the resulting codes correspond to the (3, 6, L, x5)
ensemble with z additional variable nodes of degree six. In other words, the node and
edge distributions can be represented by a base matrix instead of λ(x). Let B denote
the attached base matrix and the (3, 6, L,B) SC-LDPC ensemble denote the ensemble
71
consisting of the (3, 6, L) ensemble and additional variable nodes determined by B.
For example, B = [4 2 0 . . . 0]T for the base matrix in (4.4). By changing the non-zero
entries of B, the (3, 6, L,B) ensemble can be optimized. For B = [4 2 0 . . . 0]T, how-
ever, the number of non-zero entries to be optimized is only two. In order to increase
the search space of B, let’s consider the pre-lifting base matrix of (4.1) with lifting
factor z1, where z = z1× z2. And then attach B of size 2z1× zA to the pre-lifted base
matrix as
1 P
BΠ1,1 Π1,2
Π2,1 Π2,2 Π2,3 Π2,4
Π3,1 Π3,2 Π3,3 Π3,4. . .
Π4,3 Π4,4. . .. . .
(4.6)
where P = [P1, . . . , PzA ] and Πi,j is a random permutation matrix of size z1 × z1.
The matrix size of (4.6) is (1 + (L+ 1)z1)× (1 + zA + 2Lz1). In addition, a degree-1
variable node is included using the precoding technique [69] while the last variable
node in B is punctured to maintain the design rate. The design rate of the (3, 6, L,B)
ensemble is given as
R(3,6,L,B) = 1− (L+ 1)z1z22Lz1z2 + zAz2
=1
2− 1
2
1− zA2z1
L+ zA2z1
.
Compared to the (3, 6, L, λ(x)) ensemble, the (3, 6, L,B) ensemble has advan-
tages such as a practical advantage of using the protograph structure and a possibility
to improve the local performance by employing degree-1 variable nodes. However, the
number of additional variable nodes is restricted to the multiples of zA after lifting the
base matrix (4.6), which leads to the restriction on the range of the rate-loss mitigation
ratio γ(3,6,L,B).
To increase the available values of the rate-loss mitigation ratio, let’s consider
72
puncturing αA fraction of additional variable nodes. After lifting the base matrix in
(4.6) with lifting factor z2, the number of degree-1 variable nodes and additional vari-
able nodes in B are z2 and MA = zAz2, respectively. Then, αA fraction of the z2
degree-1 variable nodes and the first (zA − 1)z2 additional variable nodes are punc-
tured, where punctured variable nodes are selected randomly. The number of ran-
domly punctured variable nodes is p1 = αAzAz2 and the resulting design rate of the
(3, 6, L,B, αA) ensemble is given as
R(3,6,L,B,αA) = 1− (L+ 1)z1z2 − p1(2Lz1z2 + zAz2)− p1
=1
2− 1
2
1− zA2z1− αA
zA2z1
L+ zA2z1− αA
zA2z1
. (4.7)
From the DE equations for protograph based ensembles [44], the optimization of
B and P can be performed in the similar way of the optimization of λ(x). Using the
differential evolution algorithm, the optimal B, P, and αA achieving the minimum
rate-loss are obtained. For z1 = 5, zA = 5, and αA = 0.05, the obtained base matrices
B and P satisfying the design constraints are given as
B1 =
0 1 2 0 1
1 1 0 1 1
0 0 0 1 2
2 0 0 0 1
0 0 1 0 2
0 0 0 0 1
0 1 1 1 0
0 1 0 0 2
0 2 1 1 0
2 0 0 2 0
P =[
0 0 1 0 2].
73
15 20 25 30 35 40 45 500.46
0.47
0.48
0.49
0.5
0.51
Figure 4.6: Design rates of the (3, 6) regular LDPC ensemble and (3, 6, L),
(3, 6, L, λ1(x)), and (3, 6, L,B1, 0.05) SC-LDPC ensembles for various values of L.
The corresponding rate-loss mitigation ratio γ(3,6,20,B1,0.05)becomes 0.5390, which
is higher than γ(3,6,20,λ3(x)) = 0.5088 and close to γ(3,6,20,λ1(x)) = 0.5564. Figure 4.6
shows the design rates of the (3, 6, L), (3, 6, L, λ1(x)), and (3, 6, L,B1, 0.05) ensem-
bles for each value ofL. It shows that the design rate of the proposed (3, 6, L,B1, 0.05)
ensemble is substantially improved from that of the (3, 6, L) ensemble and almost the
same as that of the (3, 6, L, λ1(x)) ensemble for each value of L. However, as shown
in Figure 4.2, the BLER of the (3, 6, 20,B1, 0.05) ensemble is almost the same as that
of the (3, 6, 20) ensemble unlike the (3, 6, 20, λ1(x)) ensemble. Table 4.1 displays γ
together with the ratio of the BLER of ensemble A to that of the (3, 6, 20) ensemble
at ε = 0.45. While the (3, 6, 20, λ1(x)) ensemble shows the significant performance
degradation, the performance loss is not noticeable for the (3, 6, 20,B1, 0.05) ensem-
ble. Even though the degrees (l, r) = (3, 6) are mainly considered in this chapter as an
example, the proposed approach is valid for other SC-LDPC codes as long as boundary
check nodes have low degree.
74
Table 4.1: Comparing the (3, 6, 20, λ1(x)) and (3, 6, 20,B1, 0.05) SC-LDPC ensem-
bles with respect to γ and the ratio of BLERs at ε = 0.45.
Ensemble A γA P(3,6,20)/PA
(3, 6, 20, λ1(x)) 0.5564 4.545
(3, 6, 20,B1, 0.05) 0.5390 1.064
0.45 0.46 0.47 0.48 0.49
10-4
10-3
10-2
10-1
100
Blo
ck e
rasu
re p
roba
bilit
y
Figure 4.7: Comparison of BLERs of C(L) and P(L) for various values of z to show
independence of z.
4.3 Performance Comparison
In this section, the performance of the proposed (3, 6, L,B1, 0.05) SC-LDPC ensem-
ble is evaluated in the various settings. Let C(L) be the conventional (3, 6, L) SC-
LDPC ensemble and P(L) be the proposed (3, 6, L,B1, 0.05) SC-LDPC ensemble,
respectively. Two ensembles C(L) and P(L) are mainly compared in this section..
75
0.445 0.45 0.455 0.46 0.465 0.47 0.475 0.4810-4
10-3
10-2
10-1
100
Blo
ck e
rasu
re p
roba
bilit
y
Figure 4.8: Comparison of BLERs of C(L) and P(L) for various values of L to show
independence of L and the decoding algorithm.
4.3.1 Independence of L, z, and Decoding Algorithms for Optimized Re-
sults
In the previous section, the attached base matrix B1 is optimized for L = 20 and
the finite-length performance is evaluated under BP decoding. In this subsection, it is
shown that the optimized B1 is valid for different values of L and z as well as the
windowed decoding algorithm. First, the decoding performances of C(20) and P(20)
with different values of z are shown in Figure 4.7, where the performances of each
pair of two codes with the same value of z are almost the same. In addition, BLERs of
C(L) and P(L) are shown in Figure 4.8 for the various values of L using windowed
decoding with window size W = 12. For the windowed decoding scheme of the P(L)
ensemble, the window decoder receives the first WM variable nodes in the (3, 6, L)
ensemble together with MA additional variable nodes at the first windowed decoding
time. After the first windowed decoding time, the windowed decoding scheme ofP(L)
follows the conventional windowed decoding scheme. As shown in Figure 4.8, the
76
performances of C(L) and P(L) are almost the same regardless of L and the decoding
scheme. In fact, the independence of L is expected from the fact that the attached
additional variable nodes affect only the initial decoding phase, which is independent
to L.
4.3.2 Comparison in the Same Design Rate
Until now, the performances of C(L) and P(L) are compared for the same value of
L since I focus on whether performance degradation occurs by attaching additional
variable nodes or not. In this subsection, the performances of two ensembles C(L) and
P(L) having the same design rate are compared. To achieve a given design rate, C(L)
requires a large value of L compared to P(L). From the design rates in (4.2) and (4.7),
the relationship between L1 and L2 satisfying ∆RC(L1) = ∆RP(L2) is given as
L1 =L2 + zA
2z1− αA
zA2z1
1− zA2z1− αA
zA2z1
=40
19L2 + 1 (4.8)
where αA = 0.05 and zA/z1 = 1 for P(L). For example, the design rates of C(43)
and C(86) ensembles are almost the same as those of P(20) and P(40), respectively.
In Figure 4.9, BLERs of each pair of the ensembles are compared for z = 500 and
using windowed decoding, which shows that P(L) is superior to C(L) for each pair.
The performance improvement of P(L) comes from the small value of L since the
BLER scales linearly with the chain length L [41]. It is expected that the performance
improvement observed in Figure 4.9 continues for all pairs of L1 and L2 satisfying
(4.8) since L1 is a linear function of L2.
Also, the random puncturing technique in [61] can be used to match the design
rate. Let α be the fraction of punctured bits for each position. For additional variable
nodes of P(L), the total fraction of punctured bits becomes αA+α. In [61], the design
rate R(α) and the BP threshold ε(α) of a punctured code with puncturing fraction α
are expressed by the design rate R(0) and the BP threshold ε(0) of the unpuctured
77
0.44 0.445 0.45 0.455 0.46 0.465 0.47 0.475 0.4810-4
10-3
10-2
10-1
100
Blo
ck e
rasu
re p
roba
bilit
y
Figure 4.9: Comparison of BLERs of C(L) and P(L) for different values of L with
target design rate 0.5.
mother code as
R(α) =R(0)
1− α
ε(α) = 1− 1− ε(0)
1− α. (4.9)
Using (4.9) and the design rates of C(L) and P(L) in (4.2) and (4.7), the required
value of α to achieve target design rate R∗ can be obtained for each value of L. Ac-
cording to the obtained value of α, the BP thresholds of punctured C(L) and P(L) can
also be obtained. Figure 4.10 shows numerically calculated BP thresholds of C(L) and
P(L) to achieve target design rate R∗ = 0.5 by puncturing. For P(L), the required
α is lower than that of C(L) due to the rate-loss mitigation, and accordingly the BP
threshold of P(L) is much close to that of the unpunctured ensemble. The high BP
threshold directly influences the finite-length performance as shown in Figure 4.11,
where the punctured P(60) code is the best in terms of the finite-length performance.
78
20 30 40 50 60 70 80 90 1000.44
0.45
0.46
0.47
0.48
0.49
Figure 4.10: BP thresholds of C(L) and P(L) to achieve target rate 0.5 by puncturing.
0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48
10-4
10-3
10-2
10-1
100
Blo
ck e
rasu
re p
roba
bilit
y
Figure 4.11: Comparison of BLERs of punctured C(L) and P(L) with target design
rate 0.5.
79
Algorithm 4.1 Design algorithm of λ(x) with target local threshold εA
Input: Np, G, lmin, lmax, pc, εA
1: Initialization: Generate a set of degree distributions, {λ1(x), . . . , λNp(x)}, ran-
domly with minimum degree lmin and maximum degree lmax, where λi(x) =lmax∑i=lmin
λixi−1.
2: for g = 1 : G do
3: Mutation: For each i ∈ {1, . . . , Np}, generate a mutant polynomial mi(x) =lmax∑j=lmin
mi,jxi−1 as
mi(x) = λr1(x) + 0.5(λr2(x)− λr3(x))
where r1, r2, and r3 are randomly-chosen distinct values in the range
{1, . . . , NP }. If a coefficient of mi(x) is negative, it is set to be zero.
4: Crossover: For each i ∈ {1, . . . , Np}, generate a trial degree distribution
ti(x) =lmax∑j=lmin
ti,jxi−1 as
ti,j =
λi,j with probability pc
mi,j with probability 1− pc
for j ∈ {lmin, . . . , lmax}. If ti(1) is not 1, let’s select non-zero coefficients of
ti(x) randomly and adjust the coefficients to satisfy ti(1) = 1.
5: Selection: Using the DE equation (3.2), for each ti(x), check the following
constraints:
i) x(`)1 goes to below 10−3 at the target local threshold εA.
ii) 1/∫ 10 ti(x)dx < 1/
∫ 10 λi(x)dx.
If the constraints are satisfied, then set λi(x) = ti(x).
6: end for
7: Select λi(x) such that 1/∫ 10 λi(x)dx is the minimum.
80
Chapter 5
Conclusions
In this dissertation, research on the design methods of irregular SC-LDPC codes was
presented.
In Chapter 2, some preliminaries of LDPC codes were briefly overviewed. Basic
concepts, decoding, and analysis of LDPC codes were presented. The protograph-
based LDPC codes were also introduced.
In Chapter 3, an LP model was established to optimize irregular SC-LDPC codes
with non-uniform degree distributions. Since it is difficult to apply LP optimization
methods directly to irregular SC-LDPC codes, the following methods are used to ef-
fectively improve the performance of irregular SC-LDPC codes: Local optimization,
deriving one-dimensional DE equation, and selection of a proper objective function.
Based on the methods, a new class of capacity-approaching irregular SC-LDPC codes
is obtained over wide range of chain length L. For given L, it is verified that the pro-
posed irregular SC-LDPC codes have improved BP thresholds compared to regular
SC-LDPC codes. In addition, a method have been discussed to improve the perfor-
mance further by imposing the MET structure and an issue raised when the windowed
decoding is applied to the proposed codes. Simulation results confirm that the proposed
irregular SC-LDPC codes outperform regular SC-LDPC codes for moderate and large
L under windowed decoding. Moreover, it is shown that the propose design algorithm
81
can give a way to construct capacity-approaching block irregular LDPC codes without
degree two variable nodes.
In Chapter 4, optimization algorithms to mitigate the rate-loss of SC-LDPC codes
without degradation of finite-length performance were proposed. The proposed algo-
rithms are based on the differential evolution algorithm and impose a constraint using
the expected graph evolution to maintain the finite-length performance. Compared to
the SC-LDPC codes optimized with the BP threshold constraint, the proposed SC-
LDPC codes show almost the same finite-length performance with the conventional
SC-LDPC codes while the rate-loss is significantly mitigated. Since the proposed al-
gorithms sufficiently reduce the rate-loss, this method opens a new way to construct
capacity approaching SC-LDPC codes with a moderate value of L.
82
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91
초록
이학위논문에서는, i)선형계획법을이용하여비균일차수분포를갖는 spatially-
coupled low-density parity check (SC-LDPC)부호의설계방법및 ii)차동진화알고
리즘을이용하여 SC-LDPC부호의부호율손실완화방법들이연구되었다.
먼저, 선형 계획법을 이용하여 비균일 차수 분포를 갖는 SC-LDPC 부호 설계
방법을 제안한다. 일반적으로, SC-LDPC 부호의 밀도 진화 수식이 다 차원적이기
때문에 비균일 차수 분포를 갖는 SC-LDPC 부호를 낮은 복잡도로 설계하는 것은
어렵다. 제안 하는 방법은 차수 분포의 지역별 설계, 입출력 메세지 관계의 계산,
적절한 목적함수 선정과 같은 세가지 방법론에 기반하여 위와 같은 문제를 해결하
고 있다. 이러한 방법들을 이용하여 이진 소실 채널에서 비균일 SC-LDPC 부호의
차수 분포들을 낮은 복잡도의 선형 계획법으로 설계할 수 있다. 제안하는 비균일
SC-LDPC부호는점근적그리고유한한길이에서모두균일 SC-LDPC부호에보다
더 좋은 성능을 보인다. 또한 제안하는 비균일 SC-LDPC 부호가 동일 길이를 갖는
최적화된 비균일 블록 LDPC 부호보다 성능이 더 좋음을 확인하였다. 이는 제안하
는 설계 알고리즘이 임계값에 다가가는 블록 LDPC 부호 설계하는 새로운 종류의
방법을제공한다는것을의미한다.
두 번째로, 유한한 길이에서 성능 열화가 없는 SC-LDPC 부호의 부호율 손실
완화방법이제안된다. SC-LDPC부호의부호율손실은해결되어야할중요한문제
로써부호율손실을완화하기위한한가지방법으로경계확인노드에비균일차수
분포를 갖는 추가적인 변수 노드를 연결 하는 것이 제안되었다. 이전 연구에서는
비균일차수분포를 BP임계값이손실되지않는제한조건을갖는선형계획법으로
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최적화 되었다. 하지만 그러한 방법으로 얻은 비균일 차수 분포는 유한한 길이에
서의 성능열화를 야기시킨다. 이러한 문제를 해결하기 위해서 평균 그래프 진화를
이용하여 새로운 제한 조건을 제안하고 이 제한 조건을 차동 진화 알고리즘에 기
반한제안하는설계알고리즘에반영하였다.이를통해서 SC-LDPC부호의부호율
손실은 54% 완화되면서 유한한 길이의 손해가 없는 차수 분포를 얻었다. 또한, 설
계된 부호와 기존 부호를 동일 부호율에서 비교한 다면 부호율 손실 완화는 성능
개선으로이어진다는것을보여주었다.
주요어:밀도진화,오류정정부호,선형계획법,저밀도패리티체크부호,비균일
차수분포,부호율손실,공간결합저밀도패리티체크부호
학번: 2013-20743
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