Orderly generation of Butson Hadamard matrices

In this paper Butson-type complex Hadamard matrices $\mathrm{B}\mathrm{H}(n,q)$ of order $n$ and complexity $q$ are classified for small parameters by computer-aided methods. Our main results include the enumeration of $\mathrm{B}\mathrm{H}(21,3)$, $\mathrm{B}\mathrm{H}(16,4)$, and $\mathrm{B}\mathrm{H}(14,6)$ matrices. There are exactly $72$, 1786763, and $167776$ such matrices, up to monomial equivalence. Additionally, we show an example of a $\mathrm{B}\mathrm{H}(14,10)$ matrix for the first time, and show the nonexistence of $\mathrm{B}\mathrm{H}(8,15)$, $\mathrm{B}\mathrm{H}(11,q)$ for $q\in \{10,12,14,15\}$, and $\mathrm{B}\mathrm{H}(13,10)$ matrices.