The Covering Radius of Spherical Designs

How far can an arbitrary point of the unit sphere Ω d of R d be away from a finite set of points X of Ω d ? The largest possible such distance is called the covering radius of X . The set X is said to be a spherical t -design if the average over X of every polynomial in d variables of total degree at most t is equal to the average over Ω d of the same polynomial. In the particular case when X is a 2 s + 1-design or an antipodal 2 s -design we obtain upper bounds on the covering radius of X . We derive an asymptotic upper bound for spherical t -designs with large t and d fixed. We use simple probabilistic arguments based on the analogy with the covering problems in Hamming metric for binary codes which are orthogonal arrays of strength t .