Abstract.
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.
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Received January 30, 1997, and in revised form November 29, 1997.
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Boyvalenkov, P., Danev, D. & Nikova, S. Nonexistence of Certain Spherical Designs of Odd Strengths and Cardinalities . Discrete Comput Geom 21, 143–156 (1999). https://doi.org/10.1007/PL00009406
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DOI: https://doi.org/10.1007/PL00009406