A note on the existence of BH(19, 6) matrices

In this note we utilize a non-trivial block approach due to M. Petrescu to exhibit a Butson-type complex Hadamard matrix of order 19, composed of sixth roots of unity. 1 A new Butson-type complex Hadamard matrix A complex Hadamard matrix H is an n × n complex matrix with unimodular entries such that HH∗ = nIn, where ∗ denotes the Hermitian adjoint and In is the identity matrix of order n. Throughout this note we are concerned with the special case when the entries are some qth roots of unity. These matrices are called Butson-type complex Hadamard matrices and are denoted by BH(n, q) [1]. The existence of BH(n, q) matrices is wide open in general, but it is believed that real, i.e. BH(4k, 2) matrices exist for every integer number k. This is the famous Hadamard conjecture. Butson-type complex Hadamard matrices have applications in signal processing, coding theory [4] and harmonic analysis [6], among other things. They can also lead to constructions of real Hadamard matrices. This approach was demonstrated very recently in [3], where BH(n, 6) matrices were considered. For further results on BH(n, q) matrices we refer the reader to Horadam’s celebrated book [5]. The main result of this note is the following Theorem 1.1. There exists a BH(19, 6) matrix. Order n = 19 was listed as the smallest outstanding order of BH(n, 6) matrices in [2]. Addressing the existence of complex Hadamard matrices of prime orders is a notoriously difficult problem. One of the reasons for this is that the standard construction methods developed for the study of real Hadamard matrices and various