Design of capacity-approaching irregular low-density parity-check codes

We design low-density parity-check (LDPC) codes that perform at rates extre mely close to the Shannon capacity The codes are built from highly irregula r bipartite graphs with carefully chosen degree patterns on both sides. Our theoretical analysis of the codes is based on [1], Assuming that the under lying communication channel is symmetric, we prove that the probability den sities at the message nodes of the graph possess a certain symmetry, Using this symmetry property we then show that, under the assumption of no cycles , the message densities always converge as the number of iterations tends t o infinity. Furthermore, we prove a stability condition which implies an up per bound on the fraction of errors that a belief-propagation decoder can c orrect when applied to a code induced from a bipartite graph with a given d egree distribution. Our codes are found by optimizing the degree structure of the underlying gr aphs. We develop several strategies to perform this optimization. We also p resent some simulation results for the codes found which show that the perf ormance of the codes is very close to the asymptotic theoretical bounds.