Perfect $2$-codes over arbitrary alphabets

Abstract

The classification of perfect $e$-codes over an arbitrary alphabet of size $q$ is complete for $e>2$. In the case of non prime power $q$, it is conjectured that no perfect $2$-codes exist. We confirm this conjecture in a number of situations, including the case where $q=2^{\hat{I}}\pm p^2$ with $p$ prime, $\hat{I}\pm$ and ${\hat{I}}^2$ positive integers, and either $\hat{I}\pm \le 20$, or $\hat{I}\pm$ sufficiently large.

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