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We prove the nonexistence of lattice tilings of $\mathbb{Z}^n$ by Lee spheresof radius $2$ for all dimensions $n\geq 3$. This implies that the Golomb-Welchconjecture 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 openquestion in the degree-diameter problem of graph theory: the order of anyabelian Cayley graph of diameter $2$ and degree larger than $5$ cannot meet theabelian Cayley Moore bound.

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