Bounds on three- and higher-distance sets

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.