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We design sequences of low-density parity check codes that provably perform at rates extremely close to the Shannon capacity. These codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides. We further show that under suitable conditions the message densities fulfil a certain symmetry condition which we call the consistency condition and we present a stability condition which is the most powerful tool to date to bound/determine the threshold of a given family of low-density parity check codes.
Richardson et al. (Thu,) studied this question.