Bounds on Antipodal Spherical Designs with Few Angles

A finite subset $X$ on the unit sphere $\mathbb{S}^d$ 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 $X$ for small $s$. Motivated by the method developed by Nozaki and Suda, for each even integer $s\in[\frac{t+5}{2}, t+1]$ with $t\geq 3$, we improve the best known upper bound of Delsarte, Goethals and Seidel. 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 improve the previous estimate on the size of real ETFs and Levenstein-equality packings. This in turn gives an upper bound on $|X|$ when $s=3,\ t=3$ and $s=4,\ t=5$, respectively.

SICs: Some Explanations

The problem of constructing maximal equiangular tight frames or SICs was raised by Zauner in 1998. Four years ago it was realized that the problem is closely connected to a major open problem in number theory. We discuss why such a connection was perhaps to be expected, and give a simplified sketch of some developments that have taken place in the past 4 years. The aim, so far unfulfilled, is to prove existence of SICs in an infinite sequence of dimensions.