Abstract
In 1968, Golomb and Welch conjectured that Zn cannot be tiled by Lee spheres with a fixed radius r≥2 for dimension n≥3. This conjecture is equivalent to saying that there is no perfect Lee codes in Zn with radius r≥2 and dimension n≥3. Besides its own interest in discrete geometry and coding theory, 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 still far from being solved. In this paper, we introduce some new algebraic approaches to investigate the nonexistence of lattice tilings of Zn by Lee spheres, which is a special case of the Golomb–Welch conjecture. Using these new methods, we show the nonexistence of lattice tilings of Zn by Lee spheres of the same radius r=2 or 3 for infinitely many values of the dimension n. In particular, there does not exist lattice tilings of Zn by Lee spheres of radius 2 for all 3≤n≤100 except 8 cases.
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