Tight bounds for LDPC codes under MAP decoding

We present a new approach to the analysis of codes on graphs under symbol-maximum a posteriori probability (MAP) decoding. The approach is based on Guerra\´s interpolation technique for spin glasses. The basic idea is to consider a smooth interpolation between the original decoding problem, and a much simpler system in which each bit is retransmitted without coding through a (different) "effective" channel. Quite interestingly, the resulting expressions are strictly related to the density-evolution analysis of belief propagation decoding. The fundamental quantities involved in the interpolation procedure are, for instance, the same as in density evolution, i.e. densities of messages. Motivated by heuristic statistical mechanics results, we conjecture our bounds to be tight.