  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.
