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 $q=2^{\alpha }{p}^{\beta }$ with $p$ prime, $\alpha$ and $\beta$ positive integers, and either $\alpha \le 20$, or $\alpha$ sufficiently large.