No lattice tiling of ${\mathbb{Z}}^n$ by Lee Sphere of radius 2

Abstract

We prove the nonexistence of lattice tilings of ${\mathbb{Z}}^n$ by Lee spheres of radius $2$ for all dimensions $n\ge 3$. This implies that the Golomb-Welch conjecture is true when the common radius of the Lee spheres equals $2$ and $2n^2+2n+1$ is a prime. As a direct consequence, we also answer an open question in the degree-diameter problem of graph theory: the order of any abelian Cayley graph of diameter $2$ and degree larger than $5$ cannot meet the abelian Cayley Moore bound.

Compared with the first version, we have extended the introduction part and corrected several typos

References in corpus (3)

• Nonexistence of perfect $2$-error-correcting Lee codes in certain dimensions
• 50 Years of the Golomb--Welch Conjecture
• On the nonexistence of linear perfect Lee codes