The set of dimensions for which there are no linear perfect 2-error-correcting Lee codes has positive density

Abstract

The Golomb-Welch conjecture states that there are no perfect $e$-error-correcting Lee codes in ${\mathbb{Z}}^n$ ($PL(n,e)$-codes) whenever $n\ge 3$ and $e\ge 2$. A special case of this conjecture is when $e=2$. In a recent paper of A. Campello, S. Costa and the author of this paper, it is proved that the set $\mathcal{N}$ of dimensions $n\ge 3$ for which there are no linear $PL(n,2)$-codes is infinite and $\#\{n\in \mathcal{N}:n\le x\}\ge \frac{x}{3\ln (x)/2}(1+o(1))$. In this paper we present a simple and elementary argument which allows to improve the above result to $\#\{n\in \mathcal{N}:n\le x\}\ge \frac{4x}{25}(1+o(1))$. In particular, this implies that the set $\mathcal{N}$ has positive (lower) density in ${\mathbb{Z}}^{+}$.