An $m\times n$ matrix with entries $\pm 1$ is said to be partial Hadamard if any two rows are orthogonal. See Reference for partial Hadamard matrices. Given $n\equiv 0\,(\mathrm{mod}\,4)$, what is the minimum value of $m$ for which there exists an $m\times n$ partial Hadamard matrix that cannot be extended to an $(m+1)\times n$ partial Hadamard matrix?

• I think m=4, n=12. Take 3 times H_4. Minimality of m and n should be easy for you to prove. – The Masked Avenger Mar 16 '14 at 16:01
• Sorry. I just now realized n=(2k+1)2^t was a parameter. t+2 then. – The Masked Avenger Mar 16 '14 at 16:08

In particular they explicitly found $m=13$ when $n=32$, which is odd like $m$.