Given a Hadamard matrix of size $n$, I want to know what is the minimum number of unique sign patterns in any $\frac n2$ columns (or rows). I count a sign pattern and its negation to be the same.
My guess is exactly $\frac n2$. It is true for the Sylvester construction. Does it remain true in general when their columns and rows are permuted? If yes, any clue on how to prove it formally will be helpful.