LAUSR

Abstract For t ∈ { 1 , 2 , … } fixed, a natural class of spherical designs is given by the vectors v 1 , … , v n in F d = R d , C d (not all zero) which give equality in the bound ∑ j = 1 n ∑ k = 1 n | 〈 v j , v k 〉 | 2 t ≥ c t ( F d ) ( ∑ l = 1 n ‖ v l ‖ 2 t ) 2 , where c t ( F d ) is a known constant. These spherical ( t , t ) -designs integrate a space of homogeneous polynomials of degree 2t, and are variously known as real spherical half-designs of order 2t, complex (projective) t-designs, complex spherical semi-designs, and as tight frames when t = 1 . Little is known about the minimal number of vectors n for such a design. Here we report on the results of a numerical search for ( t , t ) -designs with a minimal number of vectors. In some cases, we obtain the designs explicitly as an orbit of a unitary action of a finite group on the sphere. We also list all the currently known ( t , t ) -designs. It is shown that many of these belong to a family of designs which we construct from the complex reflection groups. This family includes several new spherical ( t , t ) -designs with a small number of vectors.