Nonexistence of certain spherical designs of odd strengths and cardinalities

A spherical tau-design on Sn-1 is a finite set such that, for all polynomials f of degree at most tau, the average of f over the set is equal to the average of f over the sphere Sn-1. 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-Goethals-Seidel bound. We consider in detail the strengths tau = 3 and tau = 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.