Design of systematic LDPC codes using density evolution based on tripartite graph

In this paper we consider the design of systematic low-density parity-check (LDPC) codes using density evolution. We show that systematic LDPC codes can be well represented by tripartite graphs, where the edges in the code graph are divided into two types, source edges and redundancy edges. Using the tripartite code graph representation, we consider the degree distributions for source edges and redundancy edges separately. In particular, we discuss the necessary conditions for the two types of edges to be viewed as a single type in terms of specifying the LDPC codes properly. Then we discuss the relationship between the conventional bipartite graph representation and the tripartite graph representation. For the design of systematic LDPC codes, we show that the tripartite representation can specify the ensembles that are not available for the bipartite representation. Furthermore, we show that codes optimized by deploying the tripartite representation can achieve better performance with respect to the bipartite representation.