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

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.

11 pages

References in corpus (1)

• Renormalization and computation I: motivation and background

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