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 ) . s 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 | S | which are based on Delsarteʹs method. Using this we show that g ( n ) = L ( n ) for 6 < n < 22 , 23 < n < 40 , and g ( 23 ) = 276 or 277.