Bounds on antipodal spherical designs with few angles

A finite subset X on the unit sphere S^d is called an s-distance set with strength t if its angle set A(X):= {〈x, y〉: x, y ∈ X, x ≠ 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 ∈ [^t+5 2, t + 1] with t ≥ 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.