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

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