Tilings in Lee metric

Gravier et al. proved [S. Gravier, M. Mollard, Ch. Payan, On the existence of three-dimensional tiling in the Lee metric, European J. Combin. 19 (1998) 567–572] that there is no tiling of the three-dimensional space R3R3 with Lee spheres of radius at least 2. In particular, this verifies the Golomb–Welch conjecture for n=3n=3. Špacapan, [S. Špacapan, Non-existence of face-to-face four-dimensional tiling in the Lee metric, European J. Combin. 28 (2007) 127–133], using a computer-based proof, showed that the statement is true for R4R4 as well. In this paper we introduce a new method that will allow us not only to provide a short proof for the four-dimensional case but also to extend the result to R5R5. In addition, we provide a new proof for the three-dimensional case, just to show the power of our method, although the original one is more elegant. The main ingredient of our proof is the non-existence of the perfect Lee 2-error correcting code over ZZ of block size n=3,4,5n=3,4,5.