Bounds on three- and higher-distance sets

Authors:
Oleg R. Musin;Hiroshi Nozaki
Affiliations:
Department of Mathematics, University of Texas at Brownsville, 80 Fort Brown, Brownsville, TX 78520, USA;Graduate School of Information Sciences, Tohoku University, Aramaki-Aza-Aoba 6-3-09, Aoba-ku, Sendai 980-8579, Japan
Venue:
European Journal of Combinatorics
Year:
2011

Citing 5
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Abstract

A finite set X in a metric space M is called an s-distance set if the set of distances between any two distinct points of X has size s. The main problem for s-distance sets is to determine the maximum cardinality of s-distance sets for fixed s and M. In this paper, we improve the known upper bound for s-distance sets in the n-sphere for s=3,4. In particular, we determine the maximum cardinalities of three-distance sets for n=7 and 21. We also give the maximum cardinalities of s-distance sets in the Hamming space and the Johnson space for several s and dimensions.