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?