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?

• 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. Jul 25, 2013 at 6:03
• I searched before posting and Google yielded little outside sources making superficial observations. I'll check out that handbook, thanks. Jul 25, 2013 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. Jul 25, 2013 at 13:29

One great paper that I recently became aware of on this topic is Integral Rational Completions of Combinatorial Matrices" by Eric Verheiden in the Journal of Combinatorial Theory Series A (1978). This paper studies extendebility of partial Hadamard matrices to Hadamard matrices. In particular, it is proven that any partial $$n-r \times n$$ Hadamard matrix can be extended to an $$n \times n$$ Hadamard matrix if $$r \leq 7$$.
It is not possible to improve this result as there are many examples of $$n-8 \times n$$ partial Hadamard matrices that cannot be extended to a Hadamard matrix, see $$$$A counter example to Beder's conjectures about Hadamard matrices" in Journal of Statistical Planning and Inference (2009) by Bulutoglu and Kaziska.