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Expurgated Bound Estimation of LDPC Codes

Yang Xiao, Moon Ho Lee

Published Jan 1, 2006• 0 citations
Publisher

Abstract

The expurgated bound is proposed to evaluate good LDPC codes in this paper. To get the expurgated bound for LDPC codes needs to know the weights' distribution of LDPC codes to be evaluated, while the calculation of the weights' distribution is NP problem. To solve the problem, an expurgated bound of LDPC codes based on generator matrices is developed. Since the weights' distribution is obtained by the tested generator matrix of given LDPC codes, the expurgated bound of the given codes can reflect the BER performance of the codes. Simulations verify the results to be valid

DOI
10.1109/icosp.2006.345811

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The expurgated bound is proposed to evaluate good LDPC codes in this paper. To get the expurgated bound for LDPC codes needs to know the weights' distribution of LDPC codes to be evaluated, while the calculation of the weights' distribution is NP problem. To solve the problem, an expurgated bound of LDPC codes based on generator matrices is developed. Since the weights' distribution is obtained by the tested generator matrix of given LDPC codes, the expurgated bound of the given codes can reflect the BER performance of the codes. Simulations verify the results to be valid
Since the weight and distance computations of LDPC codes is a NP problem, most presented evaluations of LDPC codes are based on the girth numbers of the codes, and the computation for the distance of the codes is also depended on the girth numbers. There has been no simple way to obtain the distance of LDPC codes because of the code length. Based on generation matrices, this paper develops an algorithm to estimate the upper bound of minimum weights and minimum distance of LDPC codes. The paper establishes the upper bounds of minimum weights and distances of LDPC codes by using the vectors of generation matrices, which is different from the presented probabilitistic search methods for the weights' and distances of LCPC codes. The proposed algorithm can greatly reduce the searching time of weights and distances. Applying the algorithm we find that some LDPC codes get good BER performance although they are only free girth 4 but of great distances. Simulations verify the algorithms to be valid.
To find the minimum distance computation of LDPC codes is a NP problem, there has been no simple way to obtain the minimum distance of LDPC codes because of the code length being very long. To provide a solution to deal with this hard problem, this paper develops an algorithm to estimate the weights and distance of LDPC codes based on generator matrices. The paper establishes the upper bounds of weights and distances of LDPC codes by using the vectors of generator matrices, which is different from the existed probabilistic search methods for the weights' and distances of LCPC codes. The proposed algorithm can greatly reduce the searching time of weights and distances. Applying the algorithm we can obtain some LDPC codes of great distances and being free girth 4. Simulations verify the algorithms to be valid.
Research on LDPC codes is in the ascendant nowadays.Our paper has done some work on the proposition ofperformance of different codes in the same family constrained by weight distribution polynomials.Different codes of the same weight distribution are listed,and the results indicate the upper performance bound is determined by density evolution theory and the lower performance bound by the Fill-Shift construction method.Furthermore,the codes in the set of elementary matrix transforma- tion share the same error-correction performance but different coding complexity.We can get a conclusion that codes constrained by weight distribution polynomial can be divided into many groups,performance among different groups is not the same,it depends on the loops length and distribution,and performance in the same group is identical.What we can do now is that to find the best group and do suitable transformations if given weight distribution polynomials.Through this paper,we can see that a new idea for code constructing is also presented.If the relations between big girth of LDPC codes and weight distribution polynomial are discovered, new codes can be achieved only by the weight distribution polynomial,the currently used method given in references for code con- structing will be less useful.We can think that we have found a completely different method to construct LDPC codes.
LDPC codes provide a lot of advantages,such as low complexity and excellent error-correcting performance.Thus they have been adopted as a channel coding scheme in OFDMA PHY layer of WiMAX802.16e standard.In this paper,we use the minimum distance and code weight distribution to study the performance of LDPC codes.The ANC algorithm is studied deeply and is used to calculate the minimum distance of LDPC codes.Estimation results verify the correctness of ANC algorithm,and also show the influence of the minimum distance and code weight distribution on the error-correcting perfor-mance of LDPC codes.The results provide a reference for selecting LDPC codes in practical design.
Low-density parity-check (LDPC) codes achieve outstanding performance by the sum-product decoder. Its performance approaches the maximum-likelihood (ML) decoding performance at high SNR region. Upper bounds on the error probability of the ML decoded LDPC codes can be computed using the weight distribution. Recently, X.-Y. Hu et al. presented an approximate algorithm for estimating the weight distribution of LDPC codes. However, the weight distribution computed by their method depends on the iterative decoding algorithm, and is not always estimated exactly. In this paper, we propose a probabilistic method for computing the low-part weight distribution of LDPC codes. Using our method, the approximate weight distribution of LDPC codes can be computed with high accuracy