Bounds on antipodal spherical designs with few angles (Q2049613)

Summary: A finite subset \(X\) on the unit sphere \(\mathbb{S}^d\) is called an \(s\)-distance set with strength \(t\) if its angle set \(A(X):=\{\langle x,y\rangle : x,y\in X, x\neq y\}\) 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 \textit{H. Nozaki} and \textit{S. Suda} [SIAM J. Discrete Math. 25, No. 4, 1699--1713 (2011; Zbl 1237.05217)], for each even integer \(s\in[\frac{t+5}{2}, t+1]\) with \(t\geq 3\), we improve the best known upper bound of \textit{P. Delsarte} et al. [Geom. Dedicata 6, 363--388 (1977; Zbl 0376.05015)]. 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.