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

In this study, a method is presented to construct column weight 3 (CW3) low-density parity-check (LDPC) codes using three-partite graphs. Let G_b be a bipartite graph and N_g be the set of all minimum length cycles in G_b. Using G_b and N_g, a three-partite graph denoted G(G_b, N_g), or simply G_t, is formed. Let T be the set of length 3 cycles in G_t and T^a be the set of three element subsets of vertices in G_t such that each of these subsets form a subgraph with no edges in G_t and has precisely one element in each section of G_t. Furthermore, let H be the binary matrix in which the set of rows represent the set of vertices of G_t, the columns represent the elements of V:= T ∪ T^a, and h_ij = 1 if and only if the ith vertex of G_t 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 T^a, 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.