A real orthogonal design in $n$ variables is an $m \times n$ matrix with entries from the set $\pm x_1,\pm x_2,\cdots,\pm x_n$ that satisfies : $$ A A^T = (x_1^2 + x_2^2 + \cdots x_n^2) I_m $$ Normally the matrix is taken to be square so that $m=n$; in this case it is known that such a design exists iff $n=2,4$, or $8$. My question is : what is the largest possible $m$ for a given $n$? By experimentation, if $n$ is a power of two and $n>8$, (which is the case I'm most interested in), I can always find matrices with $m=(n/2)+1$ rows; is this the best possible?
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1$\begingroup$ Have a look at Hurwitz's theorem $\endgroup$ – Boris Bukh Oct 28 '16 at 2:30