Error-correcting codes and phase transitions

TL;DR

This paper explores the connection between error-correcting codes and phase transitions in quantum statistical mechanics by constructing operator algebras linked to code parameters and analyzing their KMS states.

Contribution

It introduces a novel framework relating asymptotic bounds of codes to phase diagrams via operator algebras and fractal measures, bridging coding theory and quantum statistical mechanics.

Findings

Identification of code parameters with fractal dimensions.

Construction of Toeplitz algebras with time evolution.

Linking KMS states to Hausdorff measures on fractals.

Abstract

The theory of error-correcting codes is concerned with constructing codes that optimize simultaneously transmission rate and relative minimum distance. These conflicting requirements determine an asymptotic bound, which is a continuous curve in the space of parameters. The main goal of this paper is to relate the asymptotic bound to phase diagrams of quantum statistical mechanical systems. We first identify the code parameters with Hausdorff and von Neumann dimensions, by considering fractals consisting of infinite sequences of code words. We then construct operator algebras associated to individual codes. These are Toeplitz algebras with a time evolution for which the KMS state at critical temperature gives the Hausdorff measure on the corresponding fractal. We extend this construction to algebras associated to limit points of codes, with non-uniform multi-fractal measures, and to tensor products over varying parameters.