Proof of replica formulas in the high noise regime for communication using LDGM codes

We consider communication over a binary input memoryless output symmetric channel with low density generator matrix codes and optimal maximum a posteriori decoding. It is known that the problem of computing the average conditional entropy, over such code ensembles in the asymptotic limit of large block length, is closely related to computing the free energy of a mean field spin glass in the thermodynamic limit. Tentative explicit formulas for these quantities have been derived thanks to the replica method (of spin glass theory) and are generally conjectured to be exact. In this contribution we show that the replica solution is indeed exact in the high noise regime, where it coincides with density evolution equations. Our method uses ideas coming from high temperature expansions in spin glass theory.

Published in:
2008 Ieee Information Theory Workshop, 416-420
Presented at:
IEEE Information Theory Workshop, Porto, PORTUGAL, May 05-08, 2008
Ieee Service Center, 445 Hoes Lane, Po Box 1331, Piscataway, Nj 08855-1331 Usa

 Record created 2010-11-30, last modified 2018-03-17

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