Kolmogorov complexity and the asymptotic bound for error-correcting codes

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 $R \leq \alpha_q(\delta)$, where $\alpha_q(\delta)$ is a continuous decreasing function called the asymptotic bound. Its existence was proved by the first-named author in 1981 (Renormalization and computation I: Motivation and background), but no approaches to the computation of this function are known, and in A computability challenge: Asymptotic bounds and isolated errorcorrecting codes, 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.