Sharp Bounds for Optimal Decoding of Low Density Parity Check Codes

TL;DR

This paper extends lower bounds on the conditional entropy for irregular LDPC codes over symmetric channels, using an analysis of the second derivative and correlation properties to refine the interpolation method.

Contribution

It generalizes Montanari's lower bound to irregular LDPC ensembles and introduces a novel analysis of the second derivative related to codebit correlations.

Findings

Lower bounds match the replica formula for irregular LDPC codes.

Analysis of second derivative relates to codebit mutual information.

Channel symmetry helps control overlap fluctuations.

Abstract

Consider communication over a binary-input memoryless output-symmetric channel with low density parity check (LDPC) codes and maximum a posteriori (MAP) decoding. The replica method of spin glass theory allows to conjecture an analytic formula for the average input-output conditional entropy per bit in the infinite block length limit. Montanari proved a lower bound for this entropy, in the case of LDPC ensembles with convex check degree polynomial, which matches the replica formula. Here we extend this lower bound to any irregular LDPC ensemble. The new feature of our work is an analysis of the second derivative of the conditional input-output entropy with respect to noise. A close relation arises between this second derivative and correlation or mutual information of codebits. This allows us to extend the realm of the interpolation method, in particular we show how channel symmetry allows to control the fluctuations of the overlap parameters.