  We apply polynomial techniques (linear programming) 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. Upper bounds are based on a geometric observation
and more or less standard linear programming techniques.
