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

TL;DR

This paper proves that lattice tilings of integer lattices by Lee spheres of radius 2 do not exist in dimensions three and higher, confirming the Golomb-Welch conjecture under certain prime conditions and impacting the degree-diameter problem in graph theory.

Contribution

It establishes the nonexistence of such lattice tilings for all dimensions n ≥ 3, confirming the Golomb-Welch conjecture in specific cases and answering an open question in graph theory.

Findings

No lattice tilings by Lee spheres of radius 2 exist for n ≥ 3.

Confirms Golomb-Welch conjecture when 2n^2+2n+1 is prime.

Abelian Cayley graphs of diameter 2 and degree > 5 cannot meet the Moore bound.

Abstract

We prove the nonexistence of lattice tilings of $\mathbb{Z}^n$ by Lee spheres of radius $2$ for all dimensions $n\geq 3$. 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.