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