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Lecture notes in computer science

A Class of Low-Density Parity-Check Codes Constructed Based on Reed-Solomon Codes with Two Information Symbols

Ivana Djurdjevic, Jun Xu, Khaled Abdel-Ghaffar, Shu Lin

Published Jan 1, 2003• 24 citations
Publisher
DOI
10.1007/3-540-44828-4_12

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Yongsoo Hwang, Youngjoo Chung, Moongu Jeon

Quantum Information Processing•Jan 3, 2013•1 citation
In this paper, we first propose a general framework for constructing quasi-cyclic low-density parity-check (QC-LDPC) codes based on a two-dimensional (2-D) maximum distance separable (MDS) code. Two classes of QC-LDPC codes are defined, whose parity-check matrices are transposes of each other. We then use a 2-D generalized Reed-Solomon (GRS) code to give a concrete construction. The decoding parity-check matrices have a large number of redundant parity-check equations while their Tanner graphs have a girth of at least 6. The minimum distances of the codes are very respectable as far as LDPC codes are concerned. We further show that many existing constructions of QC-LDPC codes based on finite fields in the literature can be unified under this construction. Experimental studies show that the constructed QC-LDPC codes perform well with the sum-product algorithm (SPA).
Motivated by the success of constructing quasi-cyclic low-density parity-check (QC-LDPC) codes based on Reed-Solomon codes, a construction method for QC-LDPC codes is presented in this paper, using the check matrices of quadratic residues codes. Three QC-LDPC codes are obtained by utilizing two quadratic residues codes of lengths less than 100, and the resultant QC-LDPC codes are regular, high-rate and free of length-4 cycles. Simulation results show that the proposed QC-LDPC codes achieves similar performance as the random LDPC codes, but they have less short cycles. In addition, the proposed (279, 158) QC-LDPC code outperforms its counterpart constructed from Reed-Solomon codes at high signal-to-noise ratios.
An efficient symbol-flipping based decoding algorithm designed for nonbinary Generalized Low-Density Parity-Check (GLDPC) codes is proposed. By extending the concept of the Weighted Bit Flip Voting (WBFV) algorithm designed for binary Hamming-code based GLDPC codes, the symbol-flipping decoding algorithm can be beneficially employed for decoding the family of GLDPC codes constructed from nonbinary constituent codes, such as nonbinary Bose Chaudhuri Hocquenghem (BCH) codes or Reed Solomon (RS) codes. The simulation results demonstrate that improvements of 1 dB and 2.7 dB are achieved by the proposed coding scheme in comparison to the more conventional binary GLDPC codes using the WBFV decoding algorithm, when using the Galois Field GF(32) for communicating over AWGN and uncorrelated Rayleigh fading channels, respectively.