Error-correcting Codes on a Bethe-like Lattice

We analyze Gallager codes by employing a simple mean-field approximation thatdistorts the model geometry and preserves important interactions between sites. The method naturally recovers the probability propagation decodingalgorithm as an extremization of a proper free-energy. We find a thermodynamic phase transition that coincides with information theoreticalupper-bounds and explain the practical code performance in terms of the free-energy landscape.