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. 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). The codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides. TL DR: This work designs low-density parity-check codes that perform at rates extremely close to the Shannon capacity and proves 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.Ībstract: We design low-density parity-check (LDPC) codes that perform at rates extremely close to the Shannon capacity.
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