Finite-connectivity systems as error-correcting codes

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 $\overrightarrow{K}\infty$ 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.