Nonexistence of Certain Spherical Designs of Odd Strengths and Cardinalities

A spherical τ -design on S is a finite set such that, for all polynomials f of degree at most τ , the average of f over the set is equal to the average of f over the sphere S. In this paper we obtain some necessary conditions for the existence of designs of odd strengths and cardinalities. This gives nonexistence results in many cases. Asymptotically, we derive a bound which is better than the corresponding estimation ensured by the Delsarte-GoethalsSeidel bound. We consider in detail the strengths τ = 3 and τ = 5 and obtain further nonexistence results in these cases. When the nonexistence argument does not work, we obtain bounds on the minimum distance of such designs.