Bounds on antipodal spherical designs with few angles

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$\mathbf{y}$ }$hassize$s$,and$X$isaspherical$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 $\mathrm{\mid }X\mathrm{\mid }$ for each even integer $s\in [\frac{t+5}{2},t+1]$ when $t\ge 3$. 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 $\mathrm{\mid }X\mathrm{\mid }$ when $s=3,
t=3$and$s=4,\ t=5$, respectively.