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

TLDR

Theoretical analysis of the codes builds on Richardson and Urbanke’s 2000 work on LDPC codes. The study designs LDPC codes that operate at rates near Shannon capacity. The authors construct highly irregular bipartite graphs and optimize their degree distributions using multiple strategies to achieve near‑capacity performance. They prove symmetry of message densities, convergence under cycle‑free graphs, a stability condition bounding correctable errors, and demonstrate via simulations that the codes approach theoretical limits.

Abstract

We design low-density parity-check (LDPC) codes that 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 the work of Richardson and Urbanke (see ibid., vol.47, no.2, p.599-618, 2000). Assuming that the underlying communication channel is symmetric, we prove that the probability densities 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 to infinity. 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.