Constructions of Spherical 3-Designs

TL;DR

This paper presents explicit constructions of spherical 3-designs on spheres of various dimensions, providing minimal and specific point configurations, and discusses the non-existence of certain other sizes.

Contribution

The paper offers explicit constructions for spherical 3-designs across multiple dimensions and point counts, advancing understanding of their existence and structure.

Findings

Explicit constructions for d=1,2,3,4, and higher dimensions.

Minimal point counts for 3-designs in various dimensions.

Evidence suggesting non-existence of certain other sizes.

Abstract

Spherical t-designs are Chebyshev-type averaging sets on the d-sphere S^d 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>=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>=4; d=2 and n=6; 8; >= 10; d=3 and n=8; >=10; d = 4 and n = 10; 12; >= 14; d>=5 and n>=5(d+1)/2 odd or n>=2d+2 even. We also provide some evidence that 3-designs of other sizes do not exist.