On the nonexistence of linear perfect Lee codes

TL;DR

This paper proves that linear perfect Lee codes do not exist for many dimensions and radii, advancing understanding in coding theory and discrete geometry.

Contribution

It introduces new algebraic methods to establish the nonexistence of linear perfect Lee codes for specific radii and infinitely many dimensions.

Findings

No linear perfect Lee codes of radius 2 in $\

Nonexistence results for radii 2 and 3 in many dimensions.

Except for 8 cases, no such codes exist for 3 ≤ n ≤ 100.

Abstract

In 1968, Golomb and Welch conjectured that there does not exist perfect Lee code in $\mathbb{Z}^{n}$ with radius $r\ge2$ and dimension $n\ge3$. Besides its own interest in coding theory and discrete geometry, this conjecture is also strongly related to the degree-diameter problems of abelian Cayley graphs. Although there are many papers on this topic, the Golomb-Welch conjecture is far from being solved. In this paper, we prove the nonexistence of linear perfect Lee codes by introducing some new algebraic methods. Using these new methods, we show the nonexistence of linear perfect Lee codes of radii $r=2,3$ in $\mathbb{Z}^n$ for infinitely many values of the dimension $n$. In particular, there does not exist linear perfect Lee codes of radius $2$ in $\mathbb{Z}^n$ for all $3\le n\le 100$ except 8 cases.