On the Non-existence of Lattice Tilings by Quasi-crosses

We study necessary conditions for the existence of lattice tilings of $\R^n$ by quasi-crosses. We prove non-existence results, and focus in particular on the two smallest unclassified shapes, the $(3,1,n)$-quasi-cross and the $(3,2,n)$-quasi-cross. We show that for dimensions $n\leq 250$, apart from the known constructions, there are no lattice tilings of $\R^n$ by $(3,1,n)$-quasi-crosses except for ten remaining cases, and no lattice tilings of $\R^n$ by $(3,2,n)$-quasi-crosses except for eleven remaining cases.