Upper Energy Bounds for Spherical Designs of Relatively Small Cardinalities

TL;DR

This paper derives upper energy bounds for spherical designs with small cardinalities using linear programming and Hermite interpolation, showing these designs are nearly energy optimal.

Contribution

It introduces a new method to bound the potential energy of spherical designs near the Delsarte-Goethals-Seidel limit, demonstrating their energy efficiency.

Findings

Bounds are close to lower energy bounds.

Spherical designs are nearly energy optimal.

Method uses Hermite interpolation in linear programming.

Abstract

We derive upper bounds for the potential energy of spherical designs of cardinality close to the Delsarte-Goethals-Seidel bound. These bounds are obtained by linear programming with the use of the Hermite interpolating polynomial of the potential function in suitable nodes. Numerical computations show that the results are quite close to certain lower energy bounds confirming that spherical designs are, in a sense, energy efficient. \end{abstract}