Some constructions of spherical 5-designs☆

Abstract Spherical designs were introduced by Delsarte, Goethals, and Seidel in 1977. A spherical t-design in, Rn is a finite set X ⊂ Sn−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 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| ⩾ 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.

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