On equivalence classes of Butson Hadamard Matrices $BH(4,2k)$

Source abstract

A Butson Hadamard matrix of order $n$ over the $k^{th}$ root of unity is a square matrix $H$ which entries are some complex $k^{th}$ root of unity such that $H H^{*} = nI_{n}$, where $H^{*}$ is the complex conjugate of $H$. A set of Butson Hadamard matrices of order $n$ over the $k^{th}$ root of unity is denoted by $BH(n,k)$. It is well-known that a Butson Hadamard matrices is a generalization of a Hadamard matrix. In this paper, we give some properties of Butson Hadamard matrices of order $4$ which implies to the upper and the lower bounds of the number of its equivalence classes. We also showed that the entries of Butson Hadamard matrices of order $4$ is $2k$-th root of unity for some integer $k$. Furthermore, we describe the equivalence classes of Butson Hadamard matrices of order 4 by constructing the representative of the class. Received: 12 May 2022 | Accepted: 14 November 2022