I am interested in answering the following question:
Question
For a given $n$, does there exist a $2n \times 2n$ matrix with entries in $\{1, 0, -1\}$ having orthogonal rows and columns with exactly $n$ zeroes in each row and column?
Conjectures
For $n=2^k$, $k\ge0$ such a matrix always exists.
For $n=3$ such a matrix does not exist.
For $n=5$ such a matrix exists, for example: $$ \left( \begin{array}{cccccccccc} 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & -1 & 0 & -1 & -1 & -1 & 0 \\ 1 & 0 & 0 & -1 & 0 & 0 & -1 & 1 & 1 & 0 \\ 0 & -1 & -1 & 1 & 1 & 0 & -1 & 0 & 0 & 0 \\ 1 & 0 & -1 & 0 & 0 & -1 & 1 & 0 & 0 & -1 \\ 1 & -1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & -1 & 1 & -1 & 0 & -1 & 0 & 1 \\ 0 & -1 & 1 & 0 & 0 & 0 & 0 & -1 & 1 & -1 \\ 0 & -1 & 1 & 0 & 0 & -1 & 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & -1 & 1 & 1 & 0 & 0 & -1 & -1 \\ \end{array} \right) $$
For $n=7$ such a matrix does not exist.