Decay of Correlations in Low Density Parity Check Codes: Low Noise Regime

TL;DR

This paper proves that in low noise regimes, correlations between code bits in low-density parity check codes decay exponentially with graph distance, leading to identical GEXIT functions for belief propagation and optimal decoders.

Contribution

It establishes exponential decay of correlations in LDPC codes at low noise levels, connecting correlation decay to decoder performance equivalence.

Findings

Correlation decays exponentially with graph distance at low noise.

GEXIT functions of belief propagation and optimal decoders are identical in this regime.

Results apply to fixed low-density parity check codes over Gaussian noise channels.

Abstract

Consider transmission over a binary additive white gaussian noise channel using a fixed low-density parity check code. We consider the posterior measure over the code bits and the corresponding correlation between two codebits, averaged over the noise realizations. We show that for low enough noise variance this average correlation decays exponentially fast with the graph distance between the code bits. One consequence of this result is that for low enough noise variance the GEXIT functions (further averaged over a standard code ensemble) of the belief propagation and optimal decoders are the same.