Given the vector space $\mathbb{R}^n$, endowed with the standard inner (dot) product $\langle\cdot,\cdot\rangle:\mathbb{R}^n\times \mathbb{R}^n\to\mathbb{R}$, the problem of **almost-orthogonal sets** asks, for each $\epsilon> 0$, for the construction (or at the very least, bounds on the cardinality) of a **maximal** subset $S$ of $U_n$, the set of unit vectors of $\mathbb{R}^n$, such that

$\forall x,y\in S: x\ne y\implies |\langle x,y\rangle|\le\epsilon$.

For known results, see Matt Cheung's Major Qualifying Project monograph at this link.

**Question**: What are known results, if we restrict ourselves to some subset of $U_n$? Specifically, I am interested in the subset $\lbrace\frac{1}{\sqrt{n}},-\frac{1}{\sqrt{n}}\rbrace^n$ consisting of the vertices of the scaled-down unit cube inscribed inside the unit ball centered at $0$.