  A partial Butson matrix is a matrix $H\in M_{M\times N}(\mathbb Z_q)$ having
its rows pairwise orthogonal, where $\mathbb Z_q\subset\mathbb C^\times$ is the
group of $q$-th roots of unity. We investigate here the counting problem for
these matrices in the "thin" regime, where $M=2,3,\ldots$ is small, and where
$N\to\infty$ (subject to the condition $N\in p\mathbb N$ when $q=p^k>2$). The
proofs are inspired from the de Launey-Levin and Richmond-Shallit counting
results.
