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IEEE International Symposium on Information Theory (ISIT)

Two-dimensional generalized Reed-Solomon codes: A unified framework for quasi-cyclic LDPC codes constructed based on finite fields

Chao Chen, Baoming Bai, Xinmei Wang

Published Jun 1, 2010• 3 citations
Publisher

Abstract

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).

DOI
10.1109/isit.2010.5513621

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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).
In this letter, we first propose a general framework for constructing quasi-cyclic low-density parity-check (QCLDPC) 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. Experimental studies show that the constructed QC-LDPC codes perform well with the sum-product algorithm (SPA).
This paper shows that several attractive classes of quasi-cyclic (QC) low-density parity-check (LDPC) codes can be obtained from affine planes over finite fields. One class of these consists of duals of one-generator QC codes. Presented here for codes contained in this class are the exact minimum distance and a lower bound on the multiplicity of the minimum-weight codewords. Further, it is shown that the minimum Hamming distance of a code in this class is equal to its minimum additive white Gaussian noise (AWGN) pseudoweight. Also discussed is a class consisting of codes from circulant permutation matrices, and an explicit formula for the rank of the parity-check matrix is presented for these codes. Additionally, it is shown that each of these codes can be identified with a code constructed from a constacyclic maximum distance separable code of dimension222. The construction is similar to the derivation of Reed–Solomon (RS)-based LDPC codes presented by Chenand DjurdjevicExperimental results show that a number of high rate QC-LDPC codes with excellent error performance are contained in these classes.
Designs and constructions of quasi-cyclic (QC) LDPC codes for the AWGN channel are presented. The codes are constructed based on the conventional parity-check matrices of Reed-Solomon (RS) codes and are referred to as RS-QC-LDPC codes. Several classes of RS-QC-LDPC codes are given. Cycle structural properties of the Tanner graphs of codes in these classes are analyzed and specific methods for constructing codes with girth at least eight and reducing their short cycles are presented. The designed codes perform well in both waterfall and low error-rate regions.
In this paper, we exploit construction of Reed-Solomon (RS) based Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes using protograph combining two existing QC-LDPC codes construction and experiment the performance of the codes. Two constructions have benefits: One is the construction RS based QC-LDPC codes whose girth is at least 8 and another is protograph based QC-LDPC codes to increase the upper bounds of minimum Hamming distance. We construct the protographs with three strategies and two of them have coding gain in sense of bit error rate compare to existing Reed Solomon based QC-LDPC codes.
In this paper, we propose construction of Reed-Solomon(RS) based Quasi-Cyclic Low-Density Parity-Check(QC-LDPC) codes using protograph combining two existing QC-LDPC codes construction. One is the construction RS based QC-LDPC codes whose girth is at least 8 and another is protograph based QC-LDPC codes to increase the upper bounds of minimum Hamming distance. We construct the protographs to increase the upper bound of minimum Hamming distance for Proto-RS-QC-LDPC codes and simulate some experiment to show coding gain in sense of BER compare to existing QC-LDPC codes.
This paper presents a simple and very flexible method for constructing quasi-cyclic (QC) low density paritycheck (LDPC) codes based on finite fields. The code construction is based on two arbitrary subsets of elements from a given field. Some well known constructions of QC-LDPC codes based on finite fields and combinatorial designs are special cases of the proposed construction. The proposed construction in conjunction with a technique, known as masking, results in codes whose Tanner graphs have girth 8 or larger. Experimental results show that codes constructed using the proposed construction perform well and have low error-floors. Also presented in the paper is a reduced-complexity iterative decoding scheme for QC-LDPC codes based on the section-wise cyclic structure of their parity-check matrices. The proposed decoding scheme is an improvement of an earlier proposed reduced-complexity iterative decoding scheme.