On almost perfect linear Lee codes of packing radius 2

TL;DR

This paper investigates the existence of nearly perfect linear Lee codes with packing radius 2, demonstrating that for certain dimensions, the optimal packing density cannot be achieved, thus supporting the Golomb-Welch conjecture.

Contribution

It classifies linear Lee codes with near-optimal packing density for radius 2, proving non-existence in specific dimensions, advancing understanding of Lee code packing limits.

Findings

Optimal packing density cannot be achieved for certain dimensions.

Supports the Golomb-Welch conjecture for linear codes.

Provides classification results for linear Lee codes with specific densities.

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 $r\geq2$ and $n\geq 3$. Recently, Leung and the second author proved that if $C$ is linear, then the Golomb-Welch conjecture is valid for $r=2$ and $n\geq 3$. 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 $n\equiv 0,3,4 \pmod{6}$, this packing density can never be achieved.