Classifying cocyclic Butson Hadamard matrices

TL;DR

This paper classifies all cocyclic Butson Hadamard matrices of certain orders over roots of unity for odd primes, providing a comprehensive list and computational methods for these matrices up to order 100.

Contribution

It provides the first complete classification of cocyclic Butson Hadamard matrices for orders with $np \,\leq\, 100$, including non-existence results and computational tools.

Findings

Complete list of cocyclic BH(n,p) matrices for specified orders.

Non-existence results for certain matrix orders.

Development of computational methods for 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.