Tight spherical designs, I

Let Rᵈ be Euclidean space of dimension d and Ωd the set of unit vectors in Rᵈ . A non-empty finite set XΩd is called a sPherical t-design in Ωd if ∑a∈ XW(α)=0 for all homogeneous harmonic polynomials W on Rᵈ of degree 1, 2, ⋯, t . This is equivalent to the condition that the k-th moments of X are invariant under orthogonal transformations of Rᵈ for $k=0,1,2,$ ⋯, t . These designs were studied by Delsarte, Goethals and Seidel They proved that the cardinality of a design is bounded below;