Existence and nonexistence of spherical $5$-designs of minimal type

TL;DR

This paper explores the existence and structure of spherical 5-designs of minimal type, establishing conditions for their existence, linking them to equiangular tight frames, and proving nonexistence in many dimensions.

Contribution

It characterizes tight spherical 5-designs of minimal type via Q-polynomial structures, links them to ETFs, and proves their nonexistence in infinitely many dimensions.

Findings

Tight spherical 5-designs of minimal type correspond to specific Q-polynomial configurations.

Half of the derived code from such designs forms an ETF with particular parameters.

Such designs cannot exist in dimensions satisfying certain arithmetic conditions, including 119 and 527.

Abstract

This paper investigates the existence and properties of spherical $5$-designs of minimal type. We focus on two cases: tight spherical $5$-designs and antipodal spherical $4$-distance $5$-designs. We prove that a tight spherical $5$-design is of minimal type if and only if it possesses a specific $Q$-polynomial coherent configuration structure. For tight spherical $5$-designs in $\mathbb{R}^d$ of minimal type, we demonstrate that half of the derived code forms an equiangular tight frames (ETF) with parameters $(d-1, \frac{(d-1)(d+1)}{3})$. This provides a sufficient condition for constructing such ETFs from maximal ETFs with parameters $(d, \frac{d(d+1)}{2})$. Moreover, we establish that tight spherical $5$-designs of minimal type cannot exist if the dimension $d$ satisfies a certain arithmetic condition, which holds for infinitely many values of $d$, including $d=119$ and $527$. For antipodal spherical $4$-distance $5$-designs, we utilize valency theory to derive necessary conditions for certain special types of antipodal spherical $4$-distance $5$-designs to be of minimal type.