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

A Butson Hadamard matrix of order n over the kth root of unity is a square matrix H which entries are some complex kth root of unity such that HH∗ = nIn, where H∗ is the complex conjugate of H. A set of Butson Hadamard matrices of order n over the kth 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 2kth 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