A computability challenge: asymptotic bounds and isolated error-correcting codes

Yuri I. Manin

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arXiv · 1107.4246

A computability challenge: asymptotic bounds and isolated error-correcting codes

Yuri I. Manin

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Abstract

Consider the set of all error--correcting block codes over a fixed alphabet with $q$ letters. It determines a recursively enumerable set of points in the unit square with coordinates $(R,\delta)$:= {\it (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 {\it asymptotic bound.} Its existence was proved by the author in 1981, but all attempts to find an explicit formula for it so far failed. In this note I consider the question whether this function is computable in the sense of constructive mathematics, and discuss some arguments suggesting that the answer might be negative.

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Yuri I. Manin. 2011-07-21. A computability challenge: asymptotic bounds and isolated error-correcting codes. https://arxiv.org/abs/1107.4246

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