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

摘要

In this study, a method is presented to construct column weight 3 (CW3) low-density parity-check (LDPC) codes using three-partite graphs. Let be a bipartite graph and be the set of all minimum length cycles in . Using and , a three-partite graph denoted (, ), or simply , is formed. Let be the set of length 3 cycles in and be the set of three element subsets of vertices in such that each of these subsets form a subgraph with no edges in and has precisely one element in each section of . Furthermore, let be the binary matrix in which the set of rows represent the set of vertices of , the columns represent the elements of := ∪ , and = 1 if and only if the th vertex of belongs to the th three element set in . Then is a CW3 binary matrix. Using the Tanner graph representing , a recursive construction for CW3 LDPC codes is provided. Applying a simple restriction on and , 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.