On spherical designs obtained from $Q$-polynomial association schemes

TL;DR

This paper characterizes when the embedding of a $Q$-polynomial association scheme forms a spherical $t$-design, linking it to Krein numbers, and establishes bounds on their strengths as spherical designs.

Contribution

It provides a characterization of spherical $t$-designs arising from $Q$-polynomial schemes using Krein numbers and bounds their strengths.

Findings

Embedding images form spherical $t$-designs characterized by Krein numbers.

Strengths of $P$- and $Q$-polynomial schemes as spherical designs are bounded by a constant.

Theoretical link between association schemes and spherical design properties.

Abstract

We characterize that the image of the embedding of the $Q$-polynomial association scheme into eigenspace by primitive idempotent $E_1$ is a spherical $t$-design in terms of the Krein numbers. And we show that the strengths of $P$- and $Q$-polynomial schemes as spherical designs are bounded by constant.