My understanding is that there is no general closed-form solution to the task of maximally separating $k$ points on an $n$-sphere ($S^n$).
But I wonder if this problem has known solutions when $k$ and $n$ satisfy some conditions.
I am wondering specifically whether we can evenly distribute $k\in\{2^{i},2^{i+1},2^{i+2},\ldots \}$ on $S^{2^i-1}\subset \mathbb{R}^{2^i}$.
In other words, consider the space $\mathbb{R}^n$ for $n$ a power of $2$. Do we have closed form solutions to evenly distribute $2n$ points on the sphere of that space ($S^{n-1}$)? what about $4n$ points? $8n$? $2^jn$?
Apart from the trivial solutions for $n=2$, my intuition tells me we can always maximally separate $k = 2n$ points in $S^{n-1}$ (e.g. choose $e_i$ and $-e_i$ for all coordinates)
Edit:
By "maximally separating $k$ points on an $n$-sphere" I would try to formalise like this:
$$\arg max_{S\subset S^n, |S|=k}\min\{\|s_i-s_j\|:\forall s_i, s_j \in S\} $$
where as usual
$$S^n=\{p\in\mathbb{R}^{n+1}:\|p\|=1\}$$
Any help is appreciated