Structured quasi‐cyclic low‐density parity‐check codes based on cyclotomic cosets

Abstract

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.