Bounds on antipodal spherical designs with few angles

TL;DR

This paper establishes new bounds on the maximum size of antipodal spherical designs with few angles, improving existing estimates for specific cases related to equiangular tight frames and Levenstein-equality packings.

Contribution

It provides improved bounds on the size of antipodal spherical designs with few angles, especially for cases s=3,t=3 and s=4,t=5, linking to ETFs and Levenstein packings.

Findings

Improved bounds for even s in [ (t+5)/2, t+1 ] for t ≥ 3.

New estimates for the size of real ETFs.

Enhanced bounds for Levenstein-equality packings.

Abstract

A finite subset $X$ on the unit sphere $\mathbb{S}^{d-1}$ is called an $s$-distance set with strength $t$ if its angle set $A(X):=\{\langle \mathbf{x},\mathbf{y}\rangle : \mathbf{x},\mathbf{y}\in X,\mathbf{x}\neq\mathbf{y} \}$ has size $s$, and $X$ is a spherical $t$-design but not a spherical $(t+1)$-design. In this paper, we consider to estimate the maximum size of such antipodal set for small $s$. First, we improve the known bound on $|X|$ for each even integer $s\in[\frac{t+5}{2}, t+1]$ when $t\geq 3$. We next focus on two special cases: $s=3,\ t=3$ and $s=4,\ t=5$. Estimating the size of $X$ for these two cases is equivalent to estimating the size of real equiangular tight frames (ETFs) and Levenstein-equality packings, respectively. We first improve the previous estimate on the size of real ETFs and Levenstein-equality packings. This in turn gives a bound on $|X|$ when $s=3,\ t=3$ and $s=4,\ t=5$, respectively.