Tight spherical designs, I

Abstract

Let $R^{d}$ be Euclidean space of dimension $d$ and $\Omega_{d}$ the set of unit vectors in $R^{d}$ . A non-empty finite set $X\subseteqq\Omega_{d}$ is called a sPherical t-design in $\Omega_{d}$ if $\sum_{a\in X}W(\alpha)=0$ for all homogeneous harmonic polynomials $W$ on $R^{d}$ of degree 1, 2, $\cdots,$ $t$ . This is equivalent to the condition that the k-th moments of $X$ are invariant under orthogonal transformations of $R^{d}$ for $k=0,1,2,$ $\cdots,$ $t$ . These designs were studied by Delsarte, Goethals and Seidel They proved that the cardinality of a design is bounded below;