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

TL;DR

This paper proves that the set of dimensions where no linear 2-error-correcting Lee codes exist has positive density, improving previous bounds and contributing to the understanding of the Golomb-Welch conjecture.

Contribution

It provides a simpler, elementary proof that the set of such dimensions has positive density, strengthening earlier results.

Findings

The set of dimensions with no linear 2-error-correcting Lee codes has positive lower density.

The new bound improves previous estimates from a logarithmic to a linear proportion.

The result supports the conjecture that perfect Lee codes are rare or nonexistent in higher dimensions.

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 $n\geq 3$ 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 $\mathbb{Z}^+$.