Rationality of the inner products of spherical $s$-distance $t$-designs for $t \geq 2s-2$, $s \geq 3$

TL;DR

This paper proves that the inner products of certain spherical designs and codes are rational, with the exception of the icosahedron, revealing a fundamental property of these geometric configurations.

Contribution

It establishes the rationality of inner products for spherical $s$-distance $t$-designs with $t \,\geq\, 2s-2$ and $s \,\geq\, 3$, extending understanding of their algebraic structure.

Findings

Inner products of spherical $s$-distance $t$-designs are rational for $t \,\geq\, 2s-2$, $s \,\geq\, 3$

All sharp configurations have rational inner products

Spherical codes attaining the Levenshtein bound have rational inner products

Abstract

We prove that the inner products of spherical $s$-distance $t$-designs with $t \geq 2s-2$ (Delsarte codes) and $s \geq 3$ are rational with the only exception being the icosahedron. In other formulations, we prove that all sharp configurations have rational inner products and all spherical codes which attain the Levenshtein bound, have rational inner products, except for the icosahedron.