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

Existence Proofs of Some EXIT Like Functions

Abstract — The Extended BP (EBP) Generalized EXIT (GEXIT) function introduced in [4] plays a fundamental role in the asymptotic analysis of sparse graph codes. For transmission over the binary erasure channel (BEC) the analytic properties of the EBP GEXIT function are relatively simple and well understood. The general case is much harder and even the existence of the curve is not known in general. We introduce some tools from non-linear analysis which can be useful to prove the existence of EXIT like curves in some cases. The main tool is the Krasnoselskii-Rabinowitz (KR) bifurcation theorem. I