  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 $A(X):=\{\langle
\mathbf{x},\mathbf{y}\rangle : \mathbf{x},\mathbf{y}\in
X,\mathbf{x}\neq\mathbf{y} \}$ 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 for small $s$. First, we improve the known
bound on $|X|$ for each even integer $s\in[\frac{t+5}{2}, t+1]$ when $t\geq 3$.
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 $|X|$ when $s=3,\
t=3$ and $s=4,\ t=5$, respectively.
