Regular Two-Distance Sets

Abstract

This paper makes a deep study of regular two-distance sets. A set of unit vectors X in Euclidean space \({\mathbb {R}}^n\) is said to be regular two-distance set if the inner product of any pair of its vectors is either \(\alpha \) or \(\beta \), and the number of \(\alpha \)’s (and hence \(\beta \)’s) on each row of the Gram matrix of X is the same. We present various properties of these sets as well as focus on the case where they form tight frames for the underlying space. We then give some constructions of regular two-distance sets, in particular, two-distance frames, both tight and non-tight cases. We also supply an example of a non-tight maximal two-distance frame. Connections among two-distance sets, equiangular lines, and quasi-symmetric designs are also discussed. For instance, we give a sufficient condition for constructing sets of equiangular lines from regular two-distance sets, especially from quasi-symmetric designs satisfying certain conditions.