Abstract :
Spherical designs were introduced by Delsarte, Goethals, and Seidel in 1977. A spherical t-design in, Rn is a finite set X subset of Sn−1 with the property that for every polynomial p with degree less-than-or-equals, slant t, the average value of p on X equals the average value of p on Sn−1. This paper contains some existence and nonexistence results, mainly for spherical 5-designs in R3. Delsarte, Goethals, and Seidel proved that if X is a spherical 5-design in R3, then X greater-or-equal, slanted 12 and if Xz.sfnc; = 12, then X consists of the vertices of a regular icosahedron. We show that such designs exist with cardinality 16, 18, 20, 22, 24, and every integer greater-or-equal, slanted 26. If X is a spherical 5-design in Rn, then X greater-or-equal, slanted n(n + 1); if X = n(n + 1), then X has been called tight. Tight spherical 5-designs in Rn are known to exist only for n = 2, 3, 7, 23 and possibly n = u2 − 2 for odd u greater-or-equal, slanted 7. Any tight spherical 5-design in Rn must consist of n(n + 1)/2 antipodal pairs of points. We show that for n greater-or-equal, slanted 3, there are no spherical 5-designs in Rn consisting of n(n + 1)/2 + 1 antipodal pairs of points.
