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 [1]. 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. Index Terms-Belief propagation, irregular low-density paritycheck codes, low-density parity-check codes, turbo codes.

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Abstract-We introduce a simple erasure recovery algorithm for codes derived from cascades of sparse bipartite graphs and analyze the algorithm by analyzing a corresponding discrete-time random process. As a result, we obtain a simple criterion involving the fractions of nodes of different degrees on both sides of the graph which is necessary and sufficient for the decoding process to finish successfully with high probability. By carefully designing these graphs we can construct for any given rate and any given real number a family of linear codes of rate which can be encoded in time proportional to ln(1 ) times their block length . Furthermore, a codeword can be recovered with high probability from a portion of its entries of length (1 + ) or more. The recovery algorithm also runs in time proportional to ln(1 ). Our algorithms have been implemented and work well in practice; various implementation issues are discussed.

We present randomized constntctions of linear-time encodable and decodable codes that can transmit over 10SSY channels at rates extremely close to capacity. The encoding and decoding algorithms for these codes have fast and simple soft ware implementations.Partial implementations of our algorithms are faster by orders of magnitude than the best software implementations of any previous algorithm for this problem. We expect these codes will be extremely useful for applications such as real-time audio and video transmission over the Internet, where 10SSYchannels are common and fast decoding is a requirement.Despite the simplicity of the algorithms, their design and analysis are mathematically intricate. The design requires the careful choice of a random irregular bipartite graph, where the structure of the irregular graph is extremely important. We model the progress of the decoding algorithm by a set of differential equations. The solution to these equations can then be express~as polynomials in one variable with coef-icients determined by the graph structure. Based on these polynomials, we design a graph structure that guarantees successful decoding with high probability.