Nonexistence of Certain Spherical Designs of Odd Strengths and Cardinalities

A spherical τ -design on S n-1 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 n-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 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.