Nonexistence results for spherical 3-designs of small cardinalities

A spherical τ-design on S^n-1 is a finite set such that, for all polynomials f of degree at most τ, the average of f over the set is equal to the average of f over the sphere S^n-1 (Delsarte et al., 1977). We obtain some restrictions on the structure of spherical τ-designs with the smallest possible cardinalities in terms of the strength τ and the dimension n. In fact, we investigate the distributions of the inner products for such designs. Applications for τ=3 are shown