# Reference for partial Hadamard matrices

Definition. An $m\times n$ matrix is said to be a partial Hadamard matrix (let's say PHM) if its entries are chosen from $\lbrace -1, 1 \rbrace$ such that the dot product of each pair of row vectors is $0$.

Details. This of course relates to the famous Hadamard conjecture which looks at $m=n$, but I am interested in the weaker $m<n$ case. Specifically, I am wondering about how many distinct $m\times n$ PHM exist given some, say $2$, of the row vectors, up to row permutation - as well as their construction.

Questions. What are some good resources on results known about PHM? Are there iconic papers in the field?

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I imagine doing a web search on "partial Hadamard" will go far towards answering your question. Further checking resources such as the handbook of combinatorial designs should provide decent references. – The Masked Avenger Jul 25 '13 at 6:03
I searched before posting and Google yielded little outside sources making superficial observations. I'll check out that handbook, thanks. – Seraj Jul 25 '13 at 6:58
Did your web search turn up this paper? dx.doi.org/10.1007/s12095-010-0033-z I'm not sure it addresses exactly what you're looking for, but it's certainly not superficial. – Mark Meckes Jul 25 '13 at 13:29