Spherical two-distance sets

A set S of unit vectors in n-dimensional Euclidean space is called spherical two-distance set, if there are two numbers a and b so that the inner products of distinct vectors of S are either a or b. It is known that the largest cardinality g(n) of spherical two-distance sets does not exceed n(n + 3)/2. This upper bound is known to be tight for n = 2, 6, 22. The set of mid-points of the edges of a regular simplex gives the lower bound L(n) = n(n + 1)/2 for g(n). in this paper using the so-called polynomial method it is proved that for nonnegative a + b the largest cardinality of S is not greater than L(n). For the case a + b < 0 we propose upper bounds on vertical bar S vertical bar which are based on Delsarte's method. Using this we show that g(n) = L(n) for 6 < n < 22, 23 < n < 40, and g(23) = 276 or 277. (C) 2008 Elsevier Inc. All rights reserved.