Bounds for the sum of distances of spherical sets of small size

TL;DR

This paper establishes bounds on the total pairwise distances in spherical codes of varying sizes, using recent energy bounds, and explores their asymptotic behavior with examples.

Contribution

It introduces new bounds for the sum of distances in spherical codes of small size, extending previous energy bounds to this context.

Findings

Bounds are tight for certain code configurations.

Asymptotic behavior of bounds is characterized.

Examples show codes closely follow the upper bounds.

Abstract

We derive upper and lower bounds on the sum of distances of a spherical code of size $N$ in $n$ dimensions when $N\sim n^\alpha, 0<\alpha\le 2.$ The bounds are derived by specializing recent general, universal bounds on energy of spherical sets. We discuss asymptotic behavior of our bounds along with several examples of codes whose sum of distances closely follows the upper bound.