Construction of spherical 4- and 5-designs

Abstract

A finite subsetX of thed-dimensional unit sphereS ^d-1 is called a sphericalt-design, if and only if

$$\frac{1}{{\left| {S^{d - 1} } \right|}}\int_{S^{d - 1} } {f(x)d\omega (x)} = \frac{1}{{\left| x \right|}}\sum\limits_{x \in X} {f(x)} $$

holds for all polynomialsf(x) =f(x ₁,x ₂,...,x _d ) of degree at mostt. In 1984 Seymour and Zaslavsky proved the existence of sphericalt-designs for anyt andd, 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 fort = 2,d ∈ ℕ and |X| ≥n ₂ for somen ₂ ∈ ℕ (n ₂ is sharp). Here we will give an explicit construction fort = 4 and 5,d ∈ ℕ and |X| ≥n ₄ for somen ₄ ∈ ℕ.