I suspect the following to be true and well known. I am looking for a reference.
About the notations: $d$ is the dimension, $S^{d-1}$ is the unit euclidean sphere in $\mathbb{R}^d$, $d(x,y)$ is the distance between $x$ and $y$.
There exist constants $c_1$ and $c_2$ depending on $d$ such that for any $n$ we can choose $u_1,\ldots,u_n\in S^{d-1}$ with the following properties:
- $d(u_i,u_j)> c_1 n^{\frac{-1}{(d-1)}}$ (the balls of centers $u_i$ and radius $\frac{c_1}2 n^{\frac{-1}{(d-1)}}$ don't overlap)
- $\forall u\in S^{d-1}, \min_i d(u,u_i)< c_2 n^{\frac{-1}{(d-1)}}$ (the balls of centers $u_i$ and radius $c_2 n^{\frac{-1}{(d-1)}}$ cover the sphere)