Constructions of spherical 3-designs

Spherical t-designs are Chebyshev-type averaging sets on the d-sphere S-d subset of Rd+l which are exact for polynomials of degree at most t. This concept was introduced in 1977 by Delsarte, Goethals, and Seidel, who also found the minimum possible size of such designs, in particular, that the number of points in a 3-design on S-d must be at least n greater than or equal to 2d + 2. In this paper we give explicit constructions for spherical 3-designs on S-d consisting of n points for d = 1 and n greater than or equal to 4; d = 2 and n = 6, 8, greater than or equal to 10; d = 3 and n = 8, greater than or equal to 10; d = 4 and n = 10, 12, greater than or equal to 14; d greater than or equal to 5 and n greater than or equal to 5(d + 1)/2 odd or n greater than or equal to 2d + 2 even. We also provide some evidence that 3-designs of other sizes do not exist. We will introduce and apply a concept from additive number theory generalizing the classical Sidon-sequences. Namely, we study sets of integers S for which the congruence epsilon(1)x(1 )+ epsilon(2)x(2) + ... + epsilon(t)x(t )= 0 mod n, where epsilon(i) = 0, +/- 1 and x(i )is an element of S (i = 1,2,..., t), only holds in the trivial cases. We call such sets Sidon-type sets of strength t, and denote their maximum cardinality by s(n, t). We find a lower bound for s(n, 3), and show how Sidon-type sets of strength 3 can be used to construct spherical 3-designs. We also conjecture that our lower bound gives the true value of s(n, 3) (this has been verified for n less than or equal to 125).