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IEEE Transactions on Communications

A unified approach to the construction of binary and nonbinary quasi-cyclic LDPC codes based on finite fields

Shumei Song, Bo Zhou, Shu Hwa Lin, Khaled Abdel-Ghaffar

Published Jan 1, 2009• 104 citations
Publisher

Abstract

A unified approach for constructing binary and nonbinary quasi-cyclic LDPC codes under a single framework is presented. Six classes of binary and nonbinary quasi-cyclic LDPC codes are constructed based on primitive elements, additive subgroups, and cyclic subgroups of finite fields. Numerical results show that the codes constructed perform well over the AWGN channel with iterative decoding.

DOI
10.1109/tcomm.2009.0901.060129

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In the late 1950's and early 1960's, finite fields were successfully used to construct linear block codes, especially cyclic codes, with large minimum distances for correcting random errors with algebraic decoding, such as Bose-Chaudhuri-Hocqenghem (BCH) and Reed-Solomon (RS) codes. Recently it has been shown that finite fields can also be used successfully to construct binary quasi-cyclic (QC)-LDPC codes that perform very well not only over the AWGN channel but also over the binary erasure channel with iterative decoding, besides being efficiently encodable. This paper is concerned with constructions of nonbinary QC-LDPC codes based on finite fields
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.
A new class of quasi-cyclic LDPC codes whose parity-check matrices are arrays of circulant permutation matrices constructed based on cyclic subgroups of finite fields is presented. This class of codes contains several known classes of algebraic quasi-cyclic LDPC codes as subclasses. Experimental results show that the codes constructed perform very well over the AWGN channel when decoded with iterative decoding based on belief propagation. This class of new QC-LDPC codes contains a subclass of codes which have large minimum distances. Combinatorial expressions for the ranks of the parity-check matrices of a subclass of codes constructed based on fields of characteristic two are given.