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IET Communications

Towards optimal edge weight distribution and construction of field‐compatible low‐density parity‐check codes over GF( q )

Guojun Han, Yong Liang Guan, Lingjun Kong, Kheong Sann Chan, Kui Cai

Published Oct 30, 2014• 2 citations

Abstract

Non‐binary low‐density parity‐check (NB‐LDPC) codes can be directly constructed by using algebraic methods, or indirectly constructed by mapping well‐designed binary parity‐check matrices to non‐binary parity‐check matrices. Given the Tanner graph (TG) of a NB‐LDPC code, the selection of edge weights in the TG significantly affects the performance of the NB‐LDPC code. The authors introduce an edge weight distribution (EWD) parameter for the TG of NB‐LDPC codes. By utilising particle swarm optimisation (PSO), the EWD is optimised and it has been demonstrated that the optimal EWD approaches a two‐element distribution for large field size and high average variable‐node degree. With the optimised EWD, the authors construct a class of field‐compatible LDPC (FC‐LDPC) codes over GF( q ) whose parity‐check matrices only include elements 0, 1 and 2, and can be encoded and decoded over different field sizes. The simulations demonstrate that the performance of the proposed FC‐LDPC codes improves monotonically with increasing field size, and significantly outperforms that of the corresponding algebraic NB‐LDPC codes or NB‐LDPC codes generated with uniform distribution of non‐zero elements over GF( q ).

DOI: 10.1049/iet-com.2014.0216

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Introduction The construction of binary low-density parity-check (LDPC) codes simply involves replacing a small number of the values in an all-zeros matrix by 1s in such a way that the rows and columns have the required degree distribution. In many cases, randomly allocating the entries in H will produce a reasonable LDPC code. However, the construction of H can affect the performance of the sum–product decoder, significantly so for some codes, and also the implementation complexity of the code. While there is no one recipe for a “good” LDPC code, there are a number of principles that inform the code designer. The first obvious decisions are which degree distribution to choose and how to construct the matrix with the chosen degrees, i.e. pseudo-randomly or with some sort of structure. Whichever construction is chosen, the features to consider include the girth of the Tanner graph and the minimum distance of the code. In this chapter we will discuss those properties of an LDPC code that affect its iterative decoding performance and then present the common construction methods used to produce codes with the preferred properties. Following common practice in the field we will call the selection of the degree distributions for an LDPC code code design and the methods to assign the locations in the parity-check matrix for the 1 entries code construction .

Low-density parity-check (LDPC) codes defined over non-binary (NB) finite field GF(2q)(q >1) achieve better error-correcting performance than binary LDPC codes when the codeword length is moderate. The decoder complexity increases very fast with the order of the field, 2q, although the error-correcting performance also improves. To reduce the complexity for practical applications, NB-LDPC codes over lower-order finite fields are of great interest. Previous designs have been focusing on decoders over either the smallest NB field GF (4) or much larger fields, such as GF (32) or higher. Prior optimization techniques are either not applicable or do not lead to efficient designs for codes over GF (8) or other lower-order fields. In this paper, an efficient architecture is developed for the check node processing, which is the most complicated step in NB-LDPC decoding, for the Min-max algorithm over lower-order fields. By utilizing the properties of finite field elements, the max/min comparison results are shared and the number of comparators needed is reduced significantly. Compared to the best prior design, the proposed check node processing has 18.5% smaller area and shorter critical path for an example code over GF (8).

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It is well known that random-like low-density parity-check (LDPC) codes over the extension fields GF(2/sup m/) of GF(2), for m>1, tend to outperform their binary counterparts of comparable length and rate. At the same time, structured LDPC codes offer the advantage of reduced implementation and storage complexity, so that it is of interest to investigate mathematical design methods for codes on graphs over fields of large order. We propose a new class of combinatorially developed codes obtained by properly combining Reed-Solomon (RS) type parity-check matrices and sparse parity-check matrices based on permutation matrices. The proposed codes have large girth and minimum distance. In order to further reduce the decoding complexity of the proposed scheme, we introduce a new decoding algorithm based on matrix representations of the underlying field, which trades performance for complexity. The particular field representation described in this abstract is based on a power basis generated by a companion matrix of a primitive polynomial of the field GF(2/sup m/). It is observed that the choice of the primitive polynomial influences the cycle distribution of the code graph.