Antipodality of spherical designs with odd harmonic indices

TL;DR

This paper establishes the minimal size of certain spherical designs with specific harmonic indices, demonstrating a key property related to antipodality, and extends the result to interval designs.

Contribution

It determines the smallest size of non-antipodal spherical designs with odd harmonic indices and proves an analogous result for interval designs.

Findings

Smallest size of non-antipodal spherical designs is 2m+1 for harmonic indices {1,3,...,2m-1}

Results apply to interval designs, extending the understanding of design size constraints

Provides a characterization of antipodality in spherical designs with odd harmonic indices

Abstract

We determine the smallest size of a non-antipodal spherical design with harmonic indices $\{1,3,\dots,2m-1\}$ to be $2m+1$, where $m$ is a positive integer. This is achieved by proving an analogous result for interval designs.