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International Conference on Signal Processing

Construction of short binary QC-LDPC codes: Cyclotomic coset approach

Yanjing Zhang, Lijun Zhang

Published Oct 1, 2012• 2 citations
Publisher

Abstract

A class of short binary (3, ρ)-regular quasi-cyclic low-density parity-check (QC-LDPC) code called cyclotomic coset (CC) code is constructed on the basis of cyclotomic coset. The length of the CC codes is less than 200 bits. The CC codes perform reasonably well over the additive white Gaussian noise (AWGN) channel with iterative decoding in terms of bit-error rate and decoding convergence.

DOI
10.1109/icosp.2012.6491846

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A class of short binary (3, ρ)-regular quasi-cyclic low-density parity-check (QC-LDPC) code called cyclotomic coset (CC) code is constructed on the basis of cyclotomic coset. The length of the CC codes is less than 200 bits. The CC codes perform reasonably well over the additive white Gaussian noise (AWGN) channel with iterative decoding in terms of bit-error rate and decoding convergence.
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.
This study considers the construction of four‐cycle free quasi‐cyclic low‐density parity‐check (QC‐LDPC) codes. These codes are based on the cyclotomic cosets of q modulo n where q is a prime power, n is a prime and ( n , q ) = 1. If s is the order of q modulo n , then n − 1 = sl for some integer l . Then there are l distinct cyclotomic cosets , 1 ≤ i ≤ l . Concatenation of the l circulant matrices formed using these sets gives an s × sl matrix M co ( n , q ). In addition, the Kronecker product of the transpose of ∪ i with itself modulo n gives an sl × sl matrix M kr ( n , q ). Replacing the entries of these matrices with their associated dispersed binary n × n matrices provides binary QC‐LDPC codes with girth at least six. Furthermore, for any prime power q ′ such that q ′ = n + 1 or q ′ ≥ 2 n , these two matrices can be used to construct q ′‐ary QC‐LDPC codes with girth at least six. These constructions produce large classes of four‐cycle free binary and non‐binary QC‐LDPC codes. Among the structured LDPC codes, the introduced technique, from code‐construction‐complexity perspective, which is an engineering factor, is at least among the best ones if not the best. Performance results are presented which show that the codes obtained perform well over additive white Gaussian noise channel with the iterative sum‐product decoding algorithm.
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.
Low-density parity check (LDPC) codes are an important class of codes with many applications. Two algebraic methods for constructing regular LDPC codes are derived -- one based on nonprimitive narrow-sense BCH codes and the other directly based on cyclotomic cosets. The constructed codes have high rates and are free of cycles of length four; consequently, they can be decoded using standard iterative decoding algorithms. The exact dimension and bounds for the minimum distance and stopping distance are derived. These constructed codes can be used to derive quantum error-correcting codes.