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Three Algebraic Methodsfor ConstructingNonbinary LDPCCodes basedon Finite Fields

Keke Liu, Zesong Fei, Jingming Kuang

Published Jan 1, 2008• 0 citations
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Abstract

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, p) 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. Keywords-nonbinaryLDPCcodes; finite fields; automorphism; Cycles-LimitedLDPCcodes

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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, p) 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. Keywords-nonbinaryLDPCcodes; finite fields; automorphism; Cycles-LimitedLDPCcodes
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.
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.
In this paper, we present two algebraic methods for constructing Quasi-Cyclic LDPC codes. The first method is based on the quadratic residues modulo p (where p is a prime), the second method is based on the multiplicative inverses in finite fields. The codes constructed based on these methods are efficiently encodable and have a girth at least 6. Simulation results show that with SPA decoding, the constructed codes based on our methods perform very well over the AWGN channel compared to some other types of famous codes, such as random Mackay code or extended EG-LDPC code recommended by NASA.
This paper presents an algebraic method for constructing structural nonbinary LDPC codes based on the prime difference over finite field. The approach gives a class of quasi-cyclic (QC)-LDPC codes with girth at least 6. Especially, the approach can construct a class of short length code and the decoding complexity is low. Simulation results show that the constructed codes decoded with iterative decoding perform well over the AWGN channel using BPSK transmission in terms of bit-error probability and rate of decoding convergence, collectively.