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Journal of Communications Software and Systems

Performance Study of a Class of Irregular Near Capacity Achieving LDPC Codes

F. Vatta, Alessandro Soranzo, Massimiliano Comisso, Giulia Buttazzoni, Fulvio Babich

Published Jan 1, 2021• 0 citations
Publisher

Abstract

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 $\lambda(x)$ of minimum degree $i_1 \gt 2$: being, in this case, $\lambda^{'} (0)=\lambda_2=0$, 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 $\lambda_2 \neq 0$. However, in this latter case, the block error probability might converge to a constant, as shown in the aforementioned paper.

DOI
10.24138/jcomss-2020-0009

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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 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 about the usefulness of some recently published low complexity upper bounds on belief-propagation decoding thresholds for irregular low-density parity-check (LDPC) codes to analyze their performance on memoryless binary input - additive white Gaussian noise (BI-AWGN) channels, with sum-product decoding. Irregular LDPC codes are known to perform better than regular ones, and to exhibit, like them, the so called “threshold phenomenon”, being the threshold defined as the maximum noise level such that an arbitrarily small bit-error probability can be achieved as the block length tends to infinity. We use a simplified analysis of the belief-propagation decoding algorithm, i.e., consider a Gaussian approximation for message densities under density evolution, and a simple algorithmic method, defined recently, to estimate the decoding thresholds for regular and irregular LDPC codes on BI-AWGN channels.
This paper investigates the performance of irregular low-density parity-check (LDPC) codes on memoryless BI-AWGN (Binary Input - Additive White Gaussian Noise) channels, with sum-product decoding. Objective of this work is to study the relationship between an LDPC code performance and some parameters specifying the code itself, such as the coefficients of its degree distributions. In fact, these coefficients where shown in Di et al.'s 2006 paper to determine the growth rate of the minimum distance of an LDPC code, which can be only sublinear in the block length in some well defined conditions of the degree distribution pair.
The optimal performance-complexity tradeoff for error-correcting codes at rates strictly below the Shannon limit is a central question in coding theory. This paper proposes a numerical approach for the minimization of decoding complexity for long-block-length irregular low-density parity-check (LDPC) codes. The proposed design methodology is applicable to any binary-input memoryless symmetric channel and any iterative message-passing decoding algorithm with a parallel-update schedule. A key feature of the proposed optimization method is a new complexity measure that incorporates both the number of operations required to carry out a single decoding iteration and the number of iterations required for convergence. This paper shows that the proposed complexity measure can be accurately estimated from a density-evolution and extrinsic-information transfer chart analysis of the code. A sufficient condition is presented for convexity of the complexity measure in the variable edge-degree distribution; when it is not satisfied, numerical experiments nevertheless suggest that the local minimum is unique. The results presented herein show that when the decoding complexity is constrained, the complexity-optimized codes significantly outperform threshold-optimized codes at long block lengths, within the ensemble of irregular codes.
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 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.