On almost perfect linear Lee codes of packing radius 2

Abstract

More than 50 years ago, Golomb and Welch conjectured that there is no perfect Lee codes $C$ of packing radius $r$ in ${\mathbb{Z}}^n$ for $r\ge 2$ and $n\ge 3$. Recently, Leung and the second author proved that if $C$ is linear, then the Golomb-Welch conjecture is valid for $r=2$ and $n\ge 3$. In this paper, we consider the classification of linear Lee codes with the second-best possibility, that is the density of the lattice packing of ${\mathbb{Z}}^n$ by Lee spheres $S(n,r)$ equals $\frac{|S(n,r)|}{|S(n,r)|+1}$. We show that, for $r=2$ and $n\equiv 0,3,4(\mathrm{mod}6)$, this packing density can never be achieved.

The extended abstract of an earlier version of this paper was presented in the 12th International Workshop on Coding and Cryptography (WCC) 2022