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IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC)

Search for Good Irregular Low-Density Parity-Check Codes Via Graph Spectrum

Dawei Yin, Xiaojing Zhang, Xichao Shu, Guiying Yan, Guanghui Wang

Published Sep 13, 2021• 0 citations
Publisher

Abstract

Research on the expander code shows that for a regular low-density parity-check (LDPC) code, the Tanner graph’s spectrum determines its properties, such as the minimum distance and the size of stopping sets. In this study, we demonstrate theoretically and experimentally that the performance of irregular LDPC codes is related to the graph spectrum. Our observations may provide an efficient metric to search for good irregular LDPC codes.

DOI
10.1109/pimrc50174.2021.9569696

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Research on the expander code shows that for a regular low-density parity-check (LDPC) code, the Tanner graph’s spectrum determines its properties, such as the minimum distance and the size of stopping sets. In this study, we demonstrate theoretically and experimentally that the performance of irregular LDPC codes is related to the graph spectrum. Our observations may provide an efficient metric to search for good irregular LDPC codes.
LDPC(Low Density Parity Check) codes are described by bipartite graphs with bit nodes and parity-check nodes. Tanner derived minimum distance bounds of the regular LDPC code in terms of the eigenvalues of the associated adjacency matrix. In this paper we generalize the Tanner's results. We derive minimum distance bounds applicable to both regular and blockwise-irregular LDPC codes. The first bound considers the relation between bit nodes in a minimum-weight codeword, and the second one considers the connectivity between parity nodes adjacent to a minimum-weight codeword. The derived bounds make it possible to describe the distance property of the code in terms of the eigenvalues of the associated matrix.
A low-density parity-check (LDPC) code can be described by a Tanner graph composed of symbol nodes and check nodes. Short cycles in the Tanner graph should be avoided because such cycles will degrade the performance of the decoder and hence increase the error rate. In most of the circumstances, cycles of length 4 can be totally eliminated in the Tanner graph when the LDPC code has been designed properly. Locally, for each variable node in the Tanner graph, the girth may take the value of 6 or 8, or even higher. In this study, the authors attempt to increase the local girths of a Tanner graph with an aim to improving the error performance of the code. The authors propose and analyse the ‘degree spectrum’ of the symbol nodes connecting to each of the check nodes. Based on the analysis, the authors can determine if the local girths can be further increased. The authors also propose simple ways to construct codes with higher local girths. Finally, the authors show the characteristics and the error performance of the codes constructed with the proposed methods.
The asymptotic distance distribution of regular low-density parity-check (LDPC) codes derived by Litsin and Sherlev (2002) is generalised for the case of irregular LDPC codes with constant check node degree. Numerical analysis of simple irregular LDPC codes with weight-two variable nodes shows that their minimum Hamming distance can be increased linearly with the code length when the maximum variable node degree of the code is kept below a critical value. This result is consistent with the stability condition of the LDPC codes, at least for the simple irregular case.
A low-density parity-check (LDPC) code is a linear block code described by a sparse parity-check matrix, which can be efficiently represented by a bipartite Tanner graph. The standard iterative decoding algorithm, known as belief propagation, passes messages along the edges of this Tanner graph. Density evolution is an efficient method to analyze the performance of the belief propagation decoding algorithm for a particular LDPC code ensemble, enabling the determination of a decoding threshold. The basic problem addressed in this work is how to optimize the Tanner graph so that the decoding threshold is as large as possible. We introduce a new code optimization technique which involves the search space range which can be thought of as minimizing randomness in differential evolution or limiting the search range in exhaustive search. This technique is applied to the design of good irregular LDPC codes and multi-edge type LDPC codes.