There is no lattice tiling of Z^n by Lee spheres of radius 2 for n≥ 3

08/26/2018

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by   Ka Hin Leung, et al.

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We prove the nonexistence of lattice tilings of Z^n by Lee spheres of radius 2 for all dimensions n≥ 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. Another consequence is that the order of any abelian Cayley graph of diameter 2 and degree larger than 5 cannot meet the abelian Cayley Moore bound.