Maximal 2-distance sets containing the regular simplex

A finite subset X of the Euclidean space is called an m-distance set if the number of distances between two distinct points in X is equal to m. An m-distance set X is said to be maximal if any vector cannot be added to X while maintaining the m-distance condition. We investigate a necessary and sufficient condition for vectors to be added to a regular simplex such that the set has only 2 distances. We construct several d-dimensional maximal 2-distance sets that contain a d-dimensional regular simplex. In particular, there exist infinitely many maximal non-spherical 2-distance sets that contain both the regular simplex and the representation of a strongly resolvable design. The maximal 2-distance set has size 2s(2)(s + 1), and the dimension is d = (s - 1)(s + 1)(2) -1, where s is a prime power. (C) 2020 Elsevier B.V. All rights reserved.