Classifying cocyclic Butson Hadamard matrices

Abstract

We classify all the cocyclic Butson Hadamard matrices $\mathrm{BH}(n,p)$ of order $n$ over the $p$th roots of unity for an odd prime $p$ and $np\leq 100$. That is, we compile a list of matrices such that any cocyclic $\mathrm{BH}(n,p)$ for these $n$, $p$ is equivalent to exactly one element in the list. Our approach encompasses non-existence results and computational machinery for Butson and generalized Hadamard matrices that are of independent interest.