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International Conference on Information Science and Control Engineering

A novel family of nonbinary LDPC codes over finite fields

Lisheng Zhang, Fanxin Zeng, Lirui Tan

Published Jan 1, 2012• 0 citations
Publisher

Abstract

In this paper, a family of nonbinary LDPC codes over a finite field GF(q) is presented and its properties are derived. In comparison with the second method presented by Ref. [4], the simulation results show that in 16-ary (75,150), 64-ary (441,882) and 128-ary (1270,2540) LDPC code, the proposed LDPC codes have a slight higher coding gain, 0.25dB, 0.2dB, and 0.15 dB at the BER= 10-4, respectively, which demonstrates that the proposed nonbinary LDPC codes are valid.

DOI
10.1049/cp.2012.2438

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In this paper, a family of nonbinary LDPC codes over a finite field GF(q) is presented and its properties are derived. In comparison with the second method presented by Ref. [4], the simulation results show that in 16-ary (75,150), 64-ary (441,882) and 128-ary (1270,2540) LDPC code, the proposed LDPC codes have a slight higher coding gain, 0.25dB, 0.2dB, and 0.15 dB at the BER= 10-4, respectively, which demonstrates that the proposed nonbinary LDPC codes are valid.
Although a great deal of research effort has been expended on the design, construction, encoding, decoding, performance analysis, and applications of binary LDPC codes in communication and storage systems, very little has been done on nonbinary LDPC codes in these respects. The first study of nonbinary LDPC codes was conducted by Davey and MacKay in 1998. In their paper, they generalized the SPA for decoding binary LDPC codes to decode q -ary LDPC codes, called QSPA . Later, in 2000, MacKay and Davey introduced a fast-Fourier-transform (FFT)-based QSPA to reduce the decoding computational complexity of QSPA. This decoding algorithm is referred to as FFT-QSPA. MacKay and Davey's work on FFT-QSPA was further improved by Barnault and Declercq in 2003 and Declercq and Fossorier in 2007. Significant works on the design, construction and analysis of nonbinary LDPC codes didn't appear until the middle of 2000. Results in these works are very encouraging. They show that nonbinary LDPC codes have a great potential to replace widely used RS codes in some applications in communication and storage systems. This chapter is devoted to nonbinary LDPC codes. Just like binary LDPC codes, nonbinary LDPC codes can be classified into two major categories: (1) random-like nonbinary codes constructed by computer under certain design criteria or rules; and (2) structured nonbinary codes constructed on the basis of algebraic or combinatorial tools, such as finite fields and finite geometries.
In this paper, we present three algebraic methods for constructing structured nonbinary LDPC codes over finite fields, among which the first method is used to construct quasi-cyclic codes with girth at least 6 based on the automorphisms of finite fields, the second method gives a class of (4,ρ) quasi-cyclic codes with girth at least 8, the third method gives a class of codes with cycles limited. Simulation results show that the constructed codes perform very well over AWGN channel, and they have better performances or far lower computational complexities than the corresponding random Mackay codes or codes algebraically constructed by Lin.
Nonbinary low-density parity-check (LDPC) codes over finite fields GF(q) have a better performance than those of the binary low-density parity-check codes, at short and medium block lengths, but the decoder of GF(q)-LDPC has more complexity. In this paper, a new stopping criterion is proposed. Based on the changes of the variance of variable node¿s extrinsic information, the criterion can predict a decoding failure and decide whether the iteration of the decoder should be stopped. The simulation results show that the stopping criterion can greatly reduces the average number of required iterations of GF(q)-LDPC decoder with negligible performance loss.
In this paper, we present three algebraic methods for constructing structured nonbinary LDPC codes over finite fields, among which the first method is based on the multiplicative inverses of nonzero elements in finite fields and gives a class of quasi-cyclic codes with girth 6, the second method gives a class of (3, rho) quasi-cyclic codes with girth 8, the third method gives a class of structured codes with cycles limited. The codes given in examples perform well over AWGN channel and have better performances or far lower computational complexities than the corresponding random Mackay codes or codes algebraically constructed by Shu Lin.