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The main objective of this paper is to explore the precise relationshipbetween the Bethe free energy (or entropy) and the Shannon conditional entropyof graphical error correcting codes. The main result shows that the Bethe freeenergy associated with a low-density parity-check code used over a binarysymmetric channel in a large noise regime is, with high probability,asymptotically exact as the block length grows. To arrive at this result wedevelop new techniques for rather general graphical models based on the loopsum as a starting point and the polymer expansion from statistical mechanics.The true free energy is computed as a series expansion containing the Bethefree energy as its zero-th order term plus a series of corrections. It iseasily seen that convergence criteria for such expansions are satisfied forgeneral high-temperature models. We apply these general results to ensembles oflow-density generator-matrix and parity-check codes. While the application togenerator-matrix codes follows standard "high temperature" methods, the case ofparity-check codes requires non-trivial new ideas because the hard constraintscorrespond to a zero-temperature regime. Nevertheless one can combine thepolymer expansion with expander and counting arguments to show that thedifference between the true and Bethe free energies vanishes with highprobability in the large block

The Bethe Free Energy Allows to Compute the Conditional Entropy of Graphical Code Instances. A Proof from the Polymer Expansion