Uniqueness of Butson Hadamard matrices of small degrees

Abstract

For positive integers $m$ and $n$, we denote by $\mathrm{BH}(m,n)$ the set of all $H\in M_{n\times n}(\mathbb{C})$ such that $HH^\ast=nI_n$ and each entry of $H$ is an $m$-th root of unity where $H^\ast$ is the adjoint matrix of $H$ and $I_n$ is the identity matrix. For $H_1,H_2\in \mathrm{BH}(m,n)$ we say that $H_1$ is \textit{equivalent} to $H_2$ if $H_1=PH_2 Q$ for some monomial matrices $P, Q$ whose nonzero entries are $m$-th roots of unity. In this paper we classify $\mathrm{BH}(17,17)$ up to equivalence by computer search.