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AEIT International Annual Conference

Improving the minimum distance of QC-LDPC codes by removing cycles

Massimo Battaglioni, Marco Baldi, Giovanni Cancellieri

Published Sep 23, 2020• 3 citations
Publisher

Abstract

We focus on regular and irregular quasi-cyclic low-density parity-check codes and analyze their codewords. We show that any codeword can be decomposed into component codewords with smaller weight belonging to some component codes. Component codewords can be associated to cycles in the Tanner graph of the code. We transfer these theoretical results into the code design, finding that the removal of some specific classes of cycles yields benefits in terms of minimum distance and error rate performance.

DOI
10.23919/aeit50178.2020.9241207

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We focus on regular and irregular quasi-cyclic low-density parity-check codes and analyze their codewords. We show that any codeword can be decomposed into component codewords with smaller weight belonging to some component codes. Component codewords can be associated to cycles in the Tanner graph of the code. We transfer these theoretical results into the code design, finding that the removal of some specific classes of cycles yields benefits in terms of minimum distance and error rate performance.
Short cycles in the Tanner graph of a low-density parity-check (LDPC) code may cause a severe performance degradation. In this paper, we investigate the cycle properties of quasi-cyclic LDPC (QC-LDPC) codes. We first analyze a necessary and sufficient condition for a cycle of a given length to exist, by using the sequence representation of a parity-check matrix for a QC-LDPC code. We then derive bounds which are necessary conditions for a QC-LDPC code to have a given girth in terms of its parameters. Our necessary conditions are applicable to any regular or irregular QC-LDPC codes as well as they improve the existing bounds for many classes of regular QC-LDPC codes.
In this paper, we propose a new approach to construct quasi-cyclic low-density parity-check (QC-LDPC) codes using Hamming codewords of weight three, namely the HM-QC-LDPC codes. Thanks to the distance property of Hamming codes, length four cycles are avoided from the Tanner graph of the constructed codes, resulting in codes of girth six. This approach can be further extended to construct QC-LDPC codes of girth eight. Our numerical results show that the proposal has a better cycle profile than the progressive-edge-growth (PEG) constructed codes, and consequently prevails in performance. Moreover, the constructed QC-LDPC code prevails the LDPC code of the IEEE802.22 standard by 0.2 dB at the bit error rate (BER) of 10-8when the codeword length is within hundreds of bits, making it a promising candidate for scenarios where strict decoding latency is imposed.
Short cycles in the Tanner graph of a given LDPC code has negative effect on the performance of iterative decoding. To improve the performance, we presented an approach for constructing Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes whose associated Tanner graph has large girth and a small amount of short cycles. Each column block of the parity check matrix was designed by adding circulant permutation sub-matrices one after another, provided the current parity check matrix has minimal number of short cycles. The condition for the proposed QC-LDPC codes to have girth at least 6 was also analyzed. Regular and irregular QC-LDPC codes with various lengths and rates can be constructed. Simulations show that the proposed QC-LDPC codes have less short cycles and better performance than existing progressive edge growth based QC-LDPC codes.
Quasi-cyclic (QC) low-density parity-check (LDPC) codes have several appealing properties regarding decoding, storage requirements and encoding aspects. In this paper, we focus on the QC LDPC codes over GF(q) whose parity-check matrices have fixed column weight j=2. By investigating two subgraphs in the Tanner graphs of the corresponding base matrices, we derive two upper bounds on the minimum Hamming distance for this class of codes. In addition, a method is proposed to construct QC LDPC codes over GF(q), which have good Hamming distance distributions. Simulations show that our designed codes have good performance.
A novel construction algorithm for quasi-cyclic low-density parity-check codes is proposed,with which we can obtain the improved girth of the corresponding Tanner graph by generalizing Yang et al.'s constraint for any cycle detected in the base matrix to detect and break the cycles in the parity check matrix.Compared with the existing cycle elimination(CE) algorithm,a better error performance can be achieved and cycle-8 free.Simulation results show that the proposed algorithm for constructing the irregular LDPC code of rate 1/2 achieves the performance of codeword error rate of 2×10-6at Eb/N0=1.5dB with 80 iterations by the belief-propagation decoding algorithm.
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.