Linear Programming Bounds for Covering Radius of Spherical Designs

We apply polynomial techniques (i.e., techniques which invole polynomials) to obtain lower and upper bounds on the covering radius of spherical designs as function of their dimension, strength, and cardinality. In terms of inner products we improve the lower bounds due to Fazekas and Levenshtein and propose new upper bounds. Our approach to the lower bounds involves certain signed measures whose corresponding series of orthogonal polynomials are positive definite up to a certain (appropriate) degree. The upper bounds are based on a geometric observation and more or less standard in the field linear programming techniques.