## Journal of Differential Geometry

### 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.

#### Article information

Source
J. Differential Geom., Volume 97, Number 1 (2014), 91-108.

Dates
First available in Project Euclid: 9 July 2014

https://projecteuclid.org/euclid.jdg/1404912104

Digital Object Identifier
doi:10.4310/jdg/1404912104

Mathematical Reviews number (MathSciNet)
MR3229051

Zentralblatt MATH identifier
1318.94116

#### Citation

Manin, Yuri; Marcolli, Matilde. Kolmogorov complexity and the asymptotic bound for error-correcting codes. J. Differential Geom. 97 (2014), no. 1, 91--108. doi:10.4310/jdg/1404912104. https://projecteuclid.org/euclid.jdg/1404912104