基于数组的高性能中速非二进制LDPC码构造

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本文档深入探讨了"Construction of High-Performance Array-Based Non-Binary LDPC Codes With Moderate Rates"这一主题，发表于2016年1月的IEEE COMMUNICATIONS LETTERS期刊。作者Shancheng Zhao和Xiao Ma针对低密度奇偶校验（Low-Density Parity-Check, LDPC）编码的研究取得了重要进展，尤其关注非二进制（Non-Binary, NBLDPC）码在中等速率下的高效构建。阵列理论在提高LDPC编码性能中发挥着关键作用。研究者首先提出了一种基于阈值和特定错误敏感子结构的掩码矩阵（Masking Matrix, M）构造方法。这种设计旨在优化编码的效率，通过数学上的分析来确定矩阵的结构，以抵御潜在的错误传播。接着，他们利用Golomb rulers生成一系列列选择集（Column Selection Sets, CSS），这些集合用于进一步构建不同的校验矩阵。这种方法确保了每个校验矩阵的独特性，从而可能降低循环结构带来的负面影响，提升编码的纠错能力。在构建过程中，作者特别关注了循环分布，这是一种衡量码性能的重要指标，因为它直接影响到码的错误检测和纠正能力。通过对每个校验矩阵的循环分布进行计算和比较，他们能够选出具有最优循环特性的一个作为最终的编码方案。为了验证其设计的有效性，文中展示了几个非二进制高性能LDPC码的错误性能结果，并与现有的其他编码进行了对比分析。结果显示，基于阵列的非二进制LDPC码在中等速率下具有显著的优势，不仅具有更高的编码效率，而且在实际应用中的纠错性能表现出色。这篇研究论文为设计高性能的非二进制LDPC码提供了一种创新且实用的方法，尤其是在处理中等速率通信系统时，阵列理论和精心选择的构造策略对于提升编码的稳健性和可靠性至关重要。对于从事编码理论、通信系统设计以及信息编码研究的专家来说，这是一项具有实际应用价值的重要研究成果。

IEEE COMMUNICATIONS LETTERS, VOL. 20, NO. 1, JANUARY 2016 13

Construction of High-Performance Array-Based Non-Binary

LDPC Codes With Moderate Rates

Shancheng Zhao and Xiao Ma

Abstract—Based on arrays, excellent low-density parity-check

(LDPC) codes with high rates have been constructed. In this let-

ter, we propose to construct high-performance nonbinary LDPC

(NBLDPC) codes with moderate rates based on arrays. The pro-

posed construction combines masking and column selection. First,

we construct a masking matrix M based on asymptotic thresh-

olds and certain error-prone substructures. Second, we generate

a number of column selection sets (CSS) based on Golomb rulers.

Based on M, we construct a parity-check matrix for each CSS.

Third, we compute the cycle distribution of each parity-check

matrix and select the one with the best cycle distribution as output.

To assess the effectiveness of the proposed construction, error per-

formances of several NBLDPC codes are presented. Comparisons

with other codes are also carried out to show the advantages of our

construction.

Index Terms—Array-based non-binary LDPC codes, inevitable

cycles, masking, moderate rates, short-to-moderate lengths.

I. INTRODUCTION

ON-BINARY low-density parity-check (NBLDPC)

codes were ﬁrst introduced in [1] based on modulo

arithmetics and were redeﬁned over ﬁnite ﬁelds in [2]. When

combined with higher-order modulations or used in burst-error

channels, NBLDPC codes with short-to-moderate block

lengths usually outperform binary LDPC codes of the same

bit lengths. Hence, in short-to-moderate block length regime,

NBLDPC codes are attractive candidates. In [3], Poulliat et al.

proposed to select non-zero elements to improve the error-ﬂoor

of NBLDPC codes with column weight 2, which are also

called NBLDPC cycle codes. Construction of NBLDPC cycle

codes with minimum lengths was proposed in [4]. In [5],

modular Golomb rulers were used to construct cyclic NBLDPC

codes.

From the implementation point of view, protograph-based

NBLDPC codes are preferred. Weight enumerators and asymp-

totic thresholds of protograph-based NBLDPC codes were

computed in [6], where good moderate-rate NBLDPC codes

were also constructed. To design long LDPC codes, asymp-

totic thresholds of [6] provide good guidelines. However, to

design ﬁnite-length codes, these thresholds should be used in

companion with other factors, such as cycle distribution and

Manuscript received June 26, 2015; revised October 15, 2015; accepted

October 19, 2015. Date of publication October 26, 2015; date of current version

January 7, 2016. This work was supported in part by the 863 Program under

Grant 2015AA01A709, and in part by the NSF of China Grant 61501206 and

Grant 91438101. The associate editor coordinating the review of this paper and

approving it for publication was E. Paolini.

S. Zhao is with the College of Information Science and Technology, Jinan

University, Guangzhou 510632, China (e-mail: zhaoday@mail2.sysu.edu.cn).

X. Ma is with the Department of ECE, Sun Yat-sen University, Guangzhou

510275, China (e-mail: maxiao@mail.sysu.edu.cn).

Digital Object Identiﬁer 10.1109/LCOMM.2015.2494581

minimum weight. In [4], [7], NBLDPC codes with large girths

were constructed. Algebraic construction of NBLDPC codes

based on ﬁnite ﬁelds was proposed in [8]. This construction was

extended by Li et al. in [9] based on masking and quasi-cyclic

(QC) labeling to construct NB QC LDPC codes. Simulation

results showed that their codes perform better than some of the

protograph-based NBLDPC codes [6]. More on the construc-

tion and decoding of NBLDPC codes can be found in [10]–[13]

and the references therein.

High-rate array-based LDPC codes with thousands of bits

were shown to perform well [14]. In this letter, we attempt to

construct high-performance array-based NBLDPC codes with

moderate rates. Both constrained and unconstrained NBLDPC

codes are considered [6]. Our construction combines mask-

ing with column selection. Firstly, we construct a protograph

G with γ check nodes and ρ variable nodes. The adjacent

matrix M of G is taken as the masking matrix. We only

consider protograph without parallel edges. To construct uncon-

strained NBLDPC codes, the protograph having a competitive

asymptotic threshold [6] is selected. To construct constrained

NBLDPC codes, both asymptotic thresholds [6] and error-

prone substructures of the protograph are considered. The

error-prone substructures we considered are submatrices that

incur inevitable cycles of length 4i (i = 3, 4, 5) [15]. Secondly,

we generate a number of column selection sets (CSS) under the

constraint that the CSS or a subset of the CSS is a Golomb ruler.

Thirdly, for each CSS, we construct a parity-check matrix based

on column selection and masking. Finally, the cycle distribution

of each parity-check matrix is computed and the matrix with the

best cycle distribution is selected.

Compared with related work in [4], [5], [10], [16], the pro-

posed method is broader in viewpoint of construction. For

example, the proposed method can be used to reproduce the

NBLDPC cycle codes constructed in [4], [16]. Note that in the

proposed method asymptotic thresholds and error-prone sub-

structures are used. This is different from the construction in

[7], [9]. To assess the effectiveness of our construction, the

performance of several NBLDPC codes with moderate rates is

analyzed. Comparisons with other codes are carried out, which

show the superior performances of our codes. Particularly, our

64-ary constrained NBLDPC code performs 0.15 dB better than

a longer code recently constructed in [9].

II. A

RRAY-BASED NON-BINARY LDPC CODES

A. Array-Based Non-Binary LDPC Codes

An NBLDPC code is deﬁned as the null space of a sparse

parity-check matrix H over the ﬁnite ﬁeld F

. We assume that

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