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A variation of Gallager error-correcting codes is investigated using statistical mechanics. In codes of this type, a given message is encoded into a codeword that comprises Boolean sums of message bits selected by two randomly constructed sparse matrices. The similarity of these codes to Ising spin systems with random interaction makes it possible to assess their typical performance by analytical methods developed in the study of disordered systems. The typical case solutions obtained via the replica method are consistent with those obtained in simulations using belief propagation decoding. We discuss the practical implications of the results obtained and suggest a computationally efficient construction for one of the more practical configurations.

We investigate the performance of parity check codes using the mapping onto Ising spin systems proposed by Sourlas [Nature (London) 339, 693 (1989); Europhys. Lett. 25, 159 (1994)]. We study codes where each parity check comprises products of K bits selected from the original digital message with exactly C checks per message bit. We show, using the replica method, that these codes saturate Shannon's coding bound for K-->infinity when the code rate K/C is finite. We then examine the finite temperature case to assess the use of simulated annealing methods for decoding, study the performance of the finite K case, and extend the analysis to accommodate different types of noisy channels. The connection between statistical physics and belief propagation decoders is discussed and the dynamics of the decoding itself is analyzed. Further insight into new approaches for improving the code performance is given.