CONSTRUCTION OF SPHERICAL 4-DESIGNS AND 5-DESIGNS

A finite subset X of the d-dimensional unit sphere S(d-1) is called a spherical t-design, if and only if 1/\S(d-1)\ integral S(d-1) f(x)d-omega(x) = 1/\x\ SIGMA-x is-an-element-of X f(x) holds for all polynomials f(x) = f(x1, x2,..., x(d)) of degree at most t. In 1984 Seymour and Zaslavsky proved the existence of spherical t-designs for any t and d, but for sufficiently large \X\. Since spherical designs can be used for numerical integration, it is of interest to give explicit constructions. Mimura gave a construction for t = 2, d is-an-element-of N and \X\ greater-than-or-equal-to n2 for some n2 is-an-element-of N (n2 is sharp). Here we will give an explicit construction for t = 4 and 5, d is-an-element-of N and \X\ greater-than-or-equal-to n4 for some n4 is-an-element-of N.