Concentration to Zero Bit-Error Probability for Regular LDPC Codes on the Binary Symmetric Channel: Proof by Loop Calculus

TL;DR

This paper proves that regular LDPC codes achieve zero bit-error probability on the binary symmetric channel below a certain noise threshold, using a novel approach based on loop calculus and free entropy convexity.

Contribution

The authors introduce a new method combining loop calculus and convexity of free entropy to analyze LDPC codes, applicable to various degree distributions and channels.

Findings

Bit-error probability converges to zero below the noise threshold.

The method links bit-error probability to the derivative of Bethe free entropy.

Versatile approach potentially extendable to other LDPC code ensembles.

Abstract

In this paper we consider regular low-density parity-check codes over a binary-symmetric channel in the decoding regime. We prove that up to a certain noise threshold the bit-error probability of the bit-sampling decoder converges in mean to zero over the code ensemble and the channel realizations. To arrive at this result we show that the bit-error probability of the sampling decoder is equal to the derivative of a Bethe free entropy. The method that we developed is new and is based on convexity of the free entropy and loop calculus. Convexity is needed to exchange limit and derivative and the loop series enables us to express the difference between the bit-error probability and the Bethe free entropy. We control the loop series using combinatorial techniques and a first moment method. We stress that our method is versatile and we believe that it can be generalized for LDPC codes with general degree distributions and for asymmetric channels.