Statistical Mechanics and error-correction Codes

TL;DR

This paper explores the deep connection between error-correction codes and spin glass models in statistical mechanics, revealing how decoding processes relate to finding ground states in physical systems.

Contribution

It establishes a novel theoretical framework linking error-correction decoding to spin glass models, providing insights into code optimality and decoding algorithms.

Findings

Minimum error probability decoding is equivalent to finding the ground state of a spin system.

Convolutional codes correspond to one-dimensional spin systems.

An exactly solvable spin-glass model relates to ideal error-free communication below channel capacity.

Abstract

I will show that there is a deep relation between error-correction codes and certain mathematical models of spin glasses. In particular minimum error probability decoding is equivalent to finding the ground state of the corresponding spin system. The most probable value of a symbol is related to the magnetization at a different temperature. Convolutional codes correspond to one-dimensional spin systems and Viterbi's decoding algorithm to the transfer matrix algorithm of Statistical Mechanics. A particular spin-glass model, which is exactly soluble, corresponds to an ideal code, i.e. a code which allows error-free communication if the rate is below channel capacity.