Sharp Bounds for Optimal Decoding of Low-Density Parity-Check Codes
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."
Keywords: Density evolution ; input-output entropy ; interpolation method ; low-density parity-check (LDPC) codes ; overlap parameter ; replica solution ; Error-Correcting Codes ; Linear Block-Codes ; Spin-Glass Model ; Ldpc Codes ; Performance ; Channels
Record created on 2010-11-30, modified on 2016-08-09