Some constructions of spherical 5-designs

Spherical designs were introduced by Delsarte, Goethals, and Seidel in 1977. A spherical t-design in, Rⁿ is a finite set X ⊂ S^n-1 with the property that for every polynomial p with degree ≤ t, the average value of p on X equals the average value of p on S^n-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| ≥ 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 ≥ 26. If X is a spherical 5-design in Rⁿ, then |X| ≥ 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 ≥ 7. Any tight spherical 5-design in Rⁿ must consist of n(n + 1) 2 antipodal pairs of points. We show that for n ≥ 3, there are no spherical 5-designs in Rⁿ consisting of n(n + 1) 2 + 1 antipodal pairs of points.