2012/11/04 by Schwartz, Moshe
#Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.1211.0658
We study necessary conditions for the existence of lattice tilings of \Rn 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≤ 250, apart from the known constructions, there are no lattice tilings of \Rn by (3,1,n)-quasi-crosses except for ten remaining cases, and no lattice tilings of \Rn by (3,2,n)-quasi-crosses except for eleven remaining cases.