Recursive method for generating column weight 3 low‐density parity‐check codes based on three‐partite graphs

Abstract

In this study, a method is presented to construct column weight 3 (CW3) low-density parity-check (LDPC) codes using three-partite graphs. Let Gb be a bipartite graph and Ng be the set of all minimum length cycles in Gb. Using Gb and Ng, a three-partite graph denoted G(Gb, Ng), or simply Gt, is formed. Let T be the set of length 3 cycles in Gt and Ta be the set of three element subsets of vertices in Gt such that each of these subsets form a subgraph with no edges in Gt and has precisely one element in each section of Gt. Furthermore, let H be the binary matrix in which the set of rows represent the set of vertices of Gt, the columns represent the elements of V:= T ∪ Ta, and hij = 1 if and only if the ith vertex of Gt belongs to the jth three element set in V. Then H is a CW3 binary matrix. Using the Tanner graph representing H, a recursive construction for CW3 LDPC codes is provided. Applying a simple restriction on T and Ta, codes free of length 4 cycles are generated. Euclidean and finite geometry codes are used as the base codes for generating new CW3 LDPC codes. Results are presented which show that these new codes perform well in an additive white Gaussian noise (AWGN) channel with the iterative sum-product decoding algorithm.