Unreal $BH(n,6)$’s and Hadamard matrices

Abstract

It is well-known that a complex Hadamard matrix of order $2n$ can be used to construct a Hadamard matrix of order $4n$. Let $\gamma$ be a primitive complex cube root of unity. In this paper, we describe a method for obtaining a Hadamard matrix of order $4n$ from a Butson-type generalized Hadamard matrix $BH(n,6)$ whose entries are drawn from the set $\{\pm \gamma ,\pm \gamma ^2\}$ of non-real complex sixth roots of unity. We denote such a matrix by $BH(n,6)$. No $BH(n,6)$ can exist for odd order $n$ whose squarefree part is divisible by a prime $p\equiv 2\pmod 3$. We exhibit examples of such “unreal” $BH(n,6)$ for all orders $n<19$ not ruled out by the above condition. We obtain unreal $BH(n,6)$’s for $n=3^{k}hm$, where $k$ is a nonnegative integer, $h$ is the order of a Hadamard matrix, and $m \in \{1,7,10,13\}$.