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Iranian Conference on Electrical Engineering

A family of LDPC codes for Binary Symmetric Channel with optimized rate

Hassan Tavakoli

Published May 1, 2017• 1 citation
Publisher

Abstract

In this paper, we present a family of irregular Low-Density Parity-Check (LDPC) code ensemble, which is designed for Binary Symmetric Channel (BSC) with maximum achievable rate. Our proposed method is based on solving the optimization problem of maximum achievable rate problem based on the related code-design conditions. In the proposed method, approximation is not used and optimization variables does not relaxed, which are used typically. Simulation results show some good codes close to the capacity.

DOI
10.1109/iraniancee.2017.7985338

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In this paper, we present a family of irregular Low-Density Parity-Check (LDPC) code ensemble, which is designed for Binary Symmetric Channel (BSC) with maximum achievable rate. Our proposed method is based on solving the optimization problem of maximum achievable rate problem based on the related code-design conditions. In the proposed method, approximation is not used and optimization variables does not relaxed, which are used typically. Simulation results show some good codes close to the capacity.
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A design of robust error-correcting codes that achieve reli able communication over various channels is of great theoretical and practical interest. Such codes are termed universal. This paper considers the universality of lowdensity parity-check (LDPC) code ensembles over families of memoryless binary-input output-symmetric (MBIOS) channels. Universality is considered both under belief-pr opagation (BP) and maximum-likelihood (ML) decoding. For the BP decoding case, we derive a density-evolution-based analytical method for designing LDPC code ensembles that are universal over various families of MBIOS channels. We also derive a necessary condition for universality of LDPC code ensembles under BP decoding; this condition is used to provide bounds on the universally achievable fraction of capacity. These results enable us to provide conditions for reliable / unreliable communications under BP decoding that are based on the Bhattacharyya parameter of the channel. For the ML decoding case, we prove that properly selected regular LDPC code ensembles are universally capacity-achieving for the set of equicapacity MBIOS channels and extend this result to punctured regular LDPC code ensembles. Index Terms Belief propagation (BP), Bhattacharyya parameter (B-parameter), density evolution (DE), linear programming (LP) bounds, low-density parity-check (LDPC) codes, maximum-likelihood (ML), memoryless binary-input outputsymmetric (MBIOS) channels, stability.
Existing design methods for irregular Low-Density Parity-Check (LDPC) codes over the additive white Gaussian noise channel are based on using asymptotic analysis tools such as density evolution in an optimization process. Such a process is computationally expensive particularly when a large number of constituent variable node degrees are involved in the design. In this paper, we propose a systematic approach for the design of irregular LDPC codes. The proposed method, which is based on a pre-computed upper bound on the fraction of edges connected to variable nodes of degree 3, is considerably less complex than the conventional optimization approach. Through a number of examples, we demonstrate that using our method, ensembles with performance very close to those devised based on optimization, can be designed. In addition to having very good performance, the number of constituent variable node degrees in the designed ensembles is only three or four. This, in some cases, is much smaller than the corresponding number for optimization-based designs with similar performance.
A method for estimating the performance of low-density parity-check (LDPC) codes decoded by hard-decision iterative decoding algorithms on binary symmetric channels (BSC) is proposed. Based on the enumeration of the smallest weight error patterns that cannot be all corrected by the decoder, this method estimates both the frame error rate (FER) and the bit error rate (BER) of a given LDPC code with very good precision for all crossover probabilities of practical interest. Through a number of examples, we show that the proposed method can be effectively applied to both regular and irregular LDPC codes and to a variety of hard-decision iterative decoding algorithms. Compared with the conventional Monte Carlo simulation, the proposed method has a much smaller computational complexity, particularly for lower error rates.
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.