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한국통신학회논문지

On the Design of Block Lengths for Irregular LDPC Codes Based on the Maximum Variable Degree

Kyuhyuk Chung

Published Nov 1, 2010• 0 citations
OpenAlex

Abstract

This paper presents the design of block lengths for irregular low-density parity-check (LDPC) codes based on the maximum variable degree d v,max . To design a block length, the performance degradation of belief-propagation (BP) decoding performance from upper bounds on the maximum likelihood (ML) decoding performance is used as an important factor. Since for large block lengths, the performance of irregular LDPC codes is very close to the Shannon limit, we focus on moderate block lengths (5×10²≤N≤4×10³). Given degree distributions, the purpose of our paper is to find proper block lengths based on the maximum variable degree d v,max . We also present some simulation results which show how a block length can be optimized.

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This paper presents the design of block lengths for irregular low-density parity-check (LDPC) codes based on the maximum variable degree d v,max . To design a block length, the performance degradation of belief-propagation (BP) decoding performance from upper bounds on the maximum likelihood (ML) decoding performance is used as an important factor. Since for large block lengths, the performance of irregular LDPC codes is very close to the Shannon limit, we focus on moderate block lengths (5×10²≤N≤4×10³). Given degree distributions, the purpose of our paper is to find proper block lengths based on the maximum variable degree d v,max . We also present some simulation results which show how a block length can be optimized.
Low-density parity-check (LDPC) codes can approach the Shannon limit performance closely, and are becoming one of the most promising channel codes in the Error Control Coding area.The performance of an LDPC code with given length is mainly affected by the sizes of some combinatorial characteristics of its corresponding bipartite graph and the distribution of variable degrees and check node degrees.With the help of the density evolution algorithm, randomly designed LDPC codes, with carefully chosen degree distribution pairs, have been shown to achieve better Shannon capacity performance than their regular counterparts when decoded using the iterative belief propagation (BP) decoding algorithm.However, regular LDPC codes outperform their irregular counterparts in term of their error floors.Therefore, construction of LDPC codes which have both attractive error rate performance and low error floor performance has become an attractive topic.Currently, optimization of variable node degree distribution and check node degree distribution for an LDPC code with given length, which could help the code to achieve good Shannon limit performance, has been extensively studied.In this dissertation, we demonstrate some approaches of improving the error floor performance for a given LDPC code with desired degree distribution pair.The contribution mainly includes three parts.In the first part, the relationship between cycles and unfavorable combinatorial
In this paper, we derive an upper bound for the average block error probability of a standard irregular low-density parity-check (LDPC) code ensemble under the maximum-likelihood (ML) decoding. Moreover, we show that the upper bound asymptotically decreases polynomially with the code length. Furthermore, when we consider several regular LDPC code ensembles as special cases of standard irregular ones over an additive white Gaussian noise channel, we numerically show that the signal-to-noise ratio (SNR) thresholds at which the proposed bound converges to zero as the code length tends to infinity are smaller than those for a bound provided by Miller et al. We also give an example of a standard irregular LDPC code ensemble which has a lower SNR threshold than a given regular LDPC code ensemble.
This paper investigates about the usefulness of a recently published low complexity upper bound on beliefpropagation decoding thresholds for a class of irregular lowdensity parity-check (LDPC) codes. In particular, the class considered is characterized by variable node degree distributions λ(x) of minimum degree i1{i_1}i1​ >2: being, in this case, λ′\lambda^{\prime}λ′(0)=λ2=0\lambda_{2}=0λ2​=0, this is useful to design LDPC codes presenting a linear minimum distance growth with the block length with probability 1, as shown in Di et al.’s 2006 paper. These codes unfortunately cannot reach capacity under iterative decoding, since the achievement of capacity requires λ2≠0\lambda_{2}\neq0λ2​=0. However, in this latter case, the block error probability might converge to a constant, as shown in the aforementioned paper.
This paper investigates the performance of a class of irregular low-density parity-check (LDPC) codes through a recently published low complexity upper bound on their beliefpropagation decoding thresholds. Moreover, their performance analysis is carried out through a recently published algorithmic method, presented in Babich et al. 2017 paper. In particular, the class considered is characterized by variable node degree distributions λ(x)\lambda(x)λ(x) of minimum degree i1>2i_1 \gt 2i1​>2: being, in this case, λ′(0)=λ2=0\lambda^{'} (0)=\lambda_2=0λ′(0)=λ2​=0, this is useful to design LDPC codes presenting a linear minimum distance growth with the block length with probability 1, as shown in Di et al.’s 2006 paper. These codes unfortunately cannot reach capacity under iterative decoding, since the achievement of capacity requires λ2≠0\lambda_2 \neq 0λ2​=0. However, in this latter case, the block error probability might converge to a constant, as shown in the aforementioned paper.
This paper estimates belief-propagation (BP) decoding performance of moderate-length irregular low-density parity-check (LDPC) codes with sphere bounds. We note that for moderate-length(10³≤ N ≤ 4×10³) irregular LDPC codes, BP decoding performance, which is much worse than maximum likelihood (ML) decoding performance, is well matched with one of loose upper bounds, i.e., sphere bounds. We introduce the sphere bounding technique for particular codes, not average bounds. The sphere bounding estimation technique is validated by simulation results. It is also shown that sphere bounds and BP decoding performance of irregular LDPC codes are very close at bit-error-rates (BERs) P b of practical importance(10?≤ P b ≤10 -4 ).
In this paper, low density parity check (LDPC) codes are considered. Design techniques are examined and new LDPC codes with improved error performance are proposed for irregular LDPC codes. Especially for BPSK modulation and AWGN channel, superior LDPC codes are successfully designed for R = 0.5 and N = 1000. For the designed superior irregular codes, it can be said that not only the performance of the codes is good when the code block length goes to infinity, but also they approach to the channel capacity and for code rate R = 0.5, they perform better than their counterparts given in the literature, when the code block length is finite (N = 1000). For irregular codes, when the degree of the bit node that possess the maximum number of branches is large, then that code approaches to the channel capacity. During the optimization of the encoder and decoder blocks, a combination of differential evolution technique, bit filling and message passing algorithms are considered to reach superior codes. The best superior code proposed in the paper is an irregular LDPC code with rate R = 0.5 which is only 0.02 dB away from the channel capacity.