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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? 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

Many of the key results on partial Hadamard matrices are due to De Launey, going back to his 2000 paper On the asymptotic existence of partial complex Hadamard matrices and related combinatorial objects. [Discrete Applied Mathematics 102, 37–45, (2000)]. You can find references in one of his most recent papers, A Fourier-analytic Approach to Counting Partial Hadamard Matrices (which is the one mentioned by Mark Meckes). A slightly more recent reference is Searching for partial Hadamard matrices.

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