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IEEE Transactions on Communications

Computing the Minimum Distance of Nonbinary LDPC Codes

Lei Liu, Jie Huang, Wuyang Zhou, Shengli Zhou

Published Jul 1, 2012• 8 citations
Publisher

Abstract

Finding the minimum distance of low-density-parity-check (LDPC) codes is an NP-hard problem. Different from all existing works that focus on binary LDPC codes, we in this paper aim to compute the minimum distance of nonbinary LDPC codes, motivated by the fact that operating in a large Galois field provides one important degree of freedom to achieve both good waterfall and error-floor performance. Our method is based on the existing nearest nonzero codeword search (NNCS) method, but several modifications are incorporated for nonbinary LDPC codes, including the modified error impulse pattern, the dithering method, and the nonbinary decoder. Numerical results on the estimated minimum distances show that a code's minimum distance can be increased by careful selection of nonzero elements of the parity check matrix, or by increasing the mean column weight, or by increasing the size of the Galois field. These results support observations that have been made based on simulated performance in the literature. Finally, we provide an upper bound on the minimum distance for nonbinary quasi-cyclic LDPC codes.

DOI
10.1109/tcomm.2012.050812.110073a

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Finding the minimum distance of low-density-parity-check (LDPC) codes is an NP-hard problem. Different from all existing works that focus on binary LDPC codes, we in this paper aim to compute the minimum distance of nonbinary LDPC codes, motivated by the fact that operating in a large Galois field provides one important degree of freedom to achieve both good waterfall and error-floor performance. Our method is based on the existing nearest nonzero codeword search (NNCS) method, but several modifications are incorporated for nonbinary LDPC codes, including the modified error impulse pattern, the dithering method, and the nonbinary decoder. Numerical results on the estimated minimum distances show that a code's minimum distance can be increased by careful selection of nonzero elements of the parity check matrix, or by increasing the mean column weight, or by increasing the size of the Galois field. These results support observations that have been made based on simulated performance in the literature. Finally, we provide an upper bound on the minimum distance for nonbinary quasi-cyclic LDPC codes.
Low-density parity-check (LDPC) codes in their broader-sense definition are linear codes whose parity-check matrices have fewer 1s than 0s. Finding their minimum distance is therefore in general an NP-hard problem. We propose a randomized algorithm called nearest nonzero codeword search (NNCS) approach to tackle this problem for iteratively decodable LDPC codes. The principle of the NNCS approach is to search codewords locally around the all-zero codeword perturbed by minimal noise, anticipating that the resultant nearest nonzero codewords will most likely contain the minimum-Hamming- weight codeword whose Hamming weight is equal to the minimum distance of the linear code. This approach has its roots in Berrou et al.'s error-impulse method and a form of Fossorier's list decoding for LDPC codes.
In order to construct nonbinary low-density parity-check(LDPC) codes with good waterfall and error floor properties,a method to improve the minimum Hamming distance of a class of nonbinary quasi-cyclic(QC) LDPC codes is presented.Two upper bounds on the minimum Hamming distance are derived for the nonbinary QC LDPC codes whose parity-check matrices consist of weight-2 columns.A special base matrix design allowing linear encoding in a parallel manner is also proposed.In addition,the design principals are given,according to which,the parallel and efficiently encodable codes reaching the upper bound on the minimum Hamming distance can be developed.Simulation shows that the proposed codes have both good waterfall and error floor properties.
Low Density Parity Check (LDPC) codes over nonbinary Galois Fields GF(q) are a generalization of the industrial standard binary LDPC codes for forward error correction in communication and information systems.The nonbinary codes can achieve significantly better performance for short and moderate block lengths.A lot of works concerning "good" LDPC codes parity check matrix construction has been published so far.However, it is well known that efficient partially parallel hardware decoder architectures are allowed only for codes with blockwise partitioned structure of the parity check matrix, called structured codes.In this paper we present a versatile algorithm for construction of codes that are both nonbinary and structured.The proposed algorithm aims at optimizing the code graph (Tanner graph) by reducing the existence of small cycles with low external connectivity, while at the same time selecting appropriate nonzero coefficients from the Galois Field under interest.The algorithm can be used for code construction of any field order, block length and code rate.
Minimum distance is an important parameter of a linear error correcting code. For improved performance of binary Low Density Parity Check (LDPC) codes, we need to have the minimum distance grow fast with n, the codelength. However, the best we can hope for is a linear growth in dmin with n. For binary LDPC codes, the necessary and sufficient conditions on the LDPC ensemble parameters, to ensure linear growth of minimum distance is well established. In the case of non-binary LDPC codes, the structure of logarithmic weight codewords is different from that of binary codes. We have carried out a preliminary study on the logarithmic bound on the the minimum distance of non-binary LDPC code ensembles. In particular, we have investigated certain configurations which would lead to low weight codewords. A set of simulations are performed to identify some of these configurations. Finally, we have provided a bound on the logarithmic minimum distance of nonbinary codes, using a strategy similar to the girth bound for binary codes. This bound has the same asymptotic behaviour as that of binary codes.
The paper concerns Low Density Parity Check (LDPC) codes over nonbinary Galois Fields GF(q), that are a generalization of the industrial standard binary LDPC codes for forward error correction in communication and information systems. The semi-regular nonbinary LDPC codes can outperform their binary counterparts in the case of short to moderate code word lengths. The methods for the good code construction based on a computer search techniques are already well investigated. However, there is a lack of systematic results for optimal weight distribution of columns of parity check matrices for codes over different field order and different rate. In this article we present a Monte Carlo like algorithm for column weight optimization devoted to nonbinary semi-regular codes along with optimization results that facilitate the codes construction computer-based methods.
In this work, we present an upper bound on the minimum distance of array low-density parity-check (LDPC) codes. An array LDPC code is a quasi-cyclic LDPC code specified by two integers q and m, where q is an odd prime and m ≤ q. In the literature, the minimum distance of these codes (denoted by d(q,m)) has been thoroughly studied for m ≤ 5. Both exact results, for small values of q and m, and general (i.e., independent of q) bounds have been established. For m ≤ 6, the best known minimum distance upper bound, derived by Mittelholzer (IEEE Int. Symp. Inf. Theory, Jun./Jul. 2002), is d(q, 6) ≤ 32. In this work, we derive an improved upper bound of d(q, 6) ≤ 20 by using the concept of a template support matrix of a codeword. The bound is tight with high probability in the sense that we have not been able to find codewords of strictly lower weight for several values of q using a minimum distance probabilistic algorithm. Finally, we provide new specific minimum distance results for m ≤ 6 and low-to-moderate values of q ≤ 79.