Kolmogorov complexity and the asymptotic bound for error-correcting codes

TL;DR

This paper demonstrates that the asymptotic bound for error-correcting codes, previously uncomputable, can be computed using Kolmogorov complexity and an oracle, linking coding theory with thermodynamic phase concepts.

Contribution

It introduces a method to compute the asymptotic bound using Kolmogorov complexity and interprets it as a phase boundary in a thermodynamic analogy.

Findings

The asymptotic bound is computable with an oracle for code complexity.

The bound can be viewed as a phase transition curve.

A partition function involving complexity offers a thermodynamic interpretation.

Abstract

The set of all error--correcting block codes over a fixed alphabet with $q$ letters determines a recursively enumerable set of rational points in the unit square with coordinates $(R,\delta)$:= (relative transmission rate, relative minimal distance). Limit points of this set form a closed subset, defined by $R\le \alpha_q(\delta)$, where $\alpha_q(\delta)$ is a continuous decreasing function called asymptotic bound. Its existence was proved by the first--named author in 1981 ([Man1]), but no approaches to the computation of this function are known, and in [Man5] it was even suggested that this function might be uncomputable in the sense of constructive analysis. In this note we show that the asymptotic bound becomes computable with the assistance of an oracle producing codes in the order of their growing Kolmogorov complexity. Moreover, a natural partition function involving complexity allows us to interpret the asymptotic bound as a curve dividing two different thermodynamic phases of codes.