The paper is about bounds for LDPC and LDGM codes under MAP. A new method for analyzing low density parity check (LDPC) codes and low density generator

matrix (LDGM) codes under bit maximum a posteriori probability (MAP) decoding is

introduced. The method is based on a rigorous approach to spin glasses developed by Francesco

Guerra. It allows to construct lower bounds on the entropy of the transmitted message conditional

to the received one. Based on heuristic statistical mechanics calculations, we conjecture

such bounds to be tight. The result holds for standard irregular ensembles when used over

binary input output symmetric channels.

The method is first developed for Tanner graph ensembles with Poisson left degree distribution.

It is then generalized to ‘multi-Poisson’ graphs, and, by a completion procedure, to

arbitrary degree distribution.