Equiangular spherical semidesigns

We show that an equiangular tight frame of m unit vectors in the n –dimensional Euclidean space is a 4th order semidesign if and only if m = n ( n +1)/2. For an important example we present an equiangular tight frame of n + 1 vertices of a regular simplex inscribed into the unit sphere (the Mercedes–Benz frame). We construct a system of the projections of the midpoints of edges of a simplex onto the sphere and show that this system is a tight frame for any and is equiangular only if n = 2 and n = 7. In the last two cases, we find an equiangular 4th order semidesign. Bibliography: 6 titles.