Design of Provably Good Low-Density Parity Check Codes

We design sequences of low-density parity check codes that provably perform at rates extremely close to the Shannon capacity. The codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides. Our theoretical analysis of the codes is based on 1]. Additionally, based on the assumption that the underlying communication channel is symmetric, we prove that the probability densities at the message nodes of the graph satisfy a certain symmetry. This enables us to derive a succinct description of the density evolution for the case of a belief propagation decoder. Furthermore, we prove a stability condition which implies an upper bound on the fraction of errors that a belief propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution. Our codes are found by optimizing the degree structure of the underlying graphs. We develop several strategies to perform this optimization. We also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds.