Consider the subset of $\mathbb{R}^n$ consisting of the vectors with only 1 or -1 as its entries, that is $\{1, -1\}^n$. I am trying to find how many orthogonal (in the Euclidean sense) vectors can be found as a function of $n$.
I know that when $n$ is odd, there exists no pair of orthogonal vectors, and when $n$ is a power of 2, there are $n$ of them (and we can find them by computing the Hadamard matrix of order $n$).
However, when $n$ falls in another category, I'm not sure how to proceed.