Tight bounds for LDPC and LDGM codes under MAP decoding

Details Tight bounds for LDPC and LDGM codes under MAP decoding Tight bounds for LDPC and LDGM codes under MAP decoding

2004-07-25

arxiv.org/abs/cs/0407060v2

IEEE Trans. on Inf. Theory, vol.51, pp. 3221-3246 (2005)

Computer Science

Information Theory

28 pages, 9 eps figures; Second version contains a generalization of the previous result

Scientific paper

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.

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