On products of graphs for LDPC codes

The notion of product block codes, whose generator matrix is the tensor product of the constituent generator matrices, is well established. Typically, they have good performance and a decoding algorithm with complexity on the order of the complexity of the decoding algorithms of the constituent codes. There has been much recent attention on the construction of bipartite graphs for low density parity check codes whose parity check matrices are the incidence matrices of right versus left vertices of the graph. The relation of the properties of the incidence matrix to code performance is difficult to establish precisely, although some guidelines are available. Two types of incidence matrix constructions are given here that show promise. In the first instance, a combinatorial construction is given, and, secondly, two types of graph products are considered for their application to LDPC codes.