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.
A new proof of the Larman–Rogers upper bound for the chromatic number of the Euclidean space