Sharp Bounds for MAP Decoding of General Irregular LDPC Codes

Consider communication over a binary input memoryless output symmetric channel with LDPC codes and MAP decoding. Recently Montanari proved that the replica solution is a lower bound to the conditional entropy for a class of LDPC ensembles. Here we extend this lower bound to any irregular LDPC ensemble for the BEC, BIAWGNC, BSC. Our work combines an analysis of the second derivative of the conditional entropy with respect to the noise and the interpolation method