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International Conference on Wireless Communications, Networking and Mobile Computing

Construction of QC LDPC codes Based on Cyclic Subgroup over GF(q)

Xiaotao Deng, Jun Gao, Gaoqi Dou

Published Sep 1, 2009• 0 citations
Publisher

Abstract

In this paper, we proposed a novel method for constructing quasi-cyclic low-density parity-check (QC LDPC) codes based on cyclic subgroup over GF(q) . LDPC codes constructed by this method are quasi-cyclic (QC) and they perform very well over the additive white Gaussian noise (AWGN) with sum product algorithm (SPA). The main advantage is that QC LDPC codes with a variety of block lengths and rates can be easily constructed with no cycles of length four or less. Simulation results show that the proposed QC LDPC codes perform slight better than the random regular LDPC codes for short to moderate block lengths. Since the codes are QC, they can be encoded using simple shift registers with linear complexity.

DOI
10.1109/wicom.2009.5301069

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In this paper, we proposed a novel method for constructing quasi-cyclic low-density parity-check (QC LDPC) codes based on cyclic subgroup over GF(q) . LDPC codes constructed by this method are quasi-cyclic (QC) and they perform very well over the additive white Gaussian noise (AWGN) with sum product algorithm (SPA). The main advantage is that QC LDPC codes with a variety of block lengths and rates can be easily constructed with no cycles of length four or less. Simulation results show that the proposed QC LDPC codes perform slight better than the random regular LDPC codes for short to moderate block lengths. Since the codes are QC, they can be encoded using simple shift registers with linear complexity.
A construction method of Quasi-Cyclic Low-Density Parity-Check(QC-LDPC) codes based on cyclic subgroups of finite field is proposed.The method owns the advantages such as easier construction,more flexible adjustment on code-length and code-rate and lower encoding/ decoding complexity.Moreover,a regular QC-LDPC(5334,4955)code suitable for optical communication systems is constructed by this construction method.The simulation results show that the code can gain good performance under the condition of additive white Gaussian noise(AWGN)channel with iterative decoding sum-product algorithm(SPA).The error-correction performance of the QC-LDPC(5334,4955)code is better than that of the classic RS(255,239) code which is widely used in optical communication system.Therefore,the constructed QC-LDPC(5334,4955)code can be better suitable for the long-haul and high-speed optical communication systems.
Encoding of computer-generated random low-density parity-check(LDPC) codes can be rather complex.QC-LDPC codes have encoding advantage over random LDPC codes.Encoding of an QC-LDPC code can be implemented using shift-registers with linear complexity.The usual algorithm for the decoding of LDPC codes is the iterative sum-product algorithm(SPA),operating on a Tanner graph of the code.It is well known that cycle-free Tanner graph with SPA algorithm yeilds optimal decoding.The girth of the cycle -free Tanner graph is infinite.In this paper,we present an algebraic method for constructing binary QC-LDPC codes based on the rowcolumn constraint.The codes constructed by this method have girths at least 6.Experimental results show that the constructed LDPC codes decoded with SPA over the AWGN channel can perform as well as standard LDPC codes in terms of bit-error probability and block error probability.
We propose a construction method for constructing quasi-cyclic low-density parity-check (QC LDPC) codes based on subgroup's coset. Our construction method is available not only for the prime circulants size, but also for the nonprime circulants size in some conditions. And it is showed that these conditions are easy to satisfy. Regular QC LDPC codes with various lengths and rates can be easily constructed with girth at least 6. Simulation results show that they are have almost the same performance as random regular LDPC codes over AWGN channel.
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.
Quasi-cyclic (QC) low-density parity-check (LDPC) codes which are known as QC-LDPC codes, have many applications due to their simple encoding implementation by means of cyclic shift registers. In this paper, we construct QC-LDPC codes from group rings. A group ring is a free module (at the same time a ring) constructed in a natural way from any given ring and any given group. We present a structure based on the elements of a group ring for constructing QC-LDPC codes. Some of the previously addressed methods for constructing QC-LDPC codes based on finite fields are special cases of the proposed construction method. The constructed QC-LDPC codes perform very well over the additive white Gaussian noise (AWGN) channel with iterative decoding in terms of bit-error probability and block-error probability. Simulation results demonstrate that the proposed codes have competitive performance in comparison with the similar existing LDPC codes. Finally, we propose a new encoding method for the proposed group ring based QC-LDPC codes that can be implemented faster than the current encoding methods. The encoding complexity of the proposed method is analyzed mathematically, and indicates a significate reduction in the required number of operations, even when compared to the available efficient encoding methods that have linear time and space complexities.