If you sample $n$ vectors each with $m$ entries, with each entry chosen from the set $\{-1, 1\}$, how can you calculate the expected maximum absolute value of the inner product between all pairs of vectors? That is let us call the vectors $v_i$ and let $X_{n,m} = \max_{i \ne j} |\langle v_i,v_j \rangle|$. I would like $\mathbb{E}(X_{n,m})$.

We can assume both $n$ and $m$ are large. We can also assume that $n^c \leq m \leq n^{d}$ for constant $c,d > 0$.