  Let $G$ be a finite abelian group and let $\exp(G)$ denote the least common
multiple of the orders of all elements of $G$. A $BH(G,h)$ matrix is a
$G$-invariant $|G|\times |G|$ matrix $H$ whose entries are complex $h$th roots
of unity such that $HH^*=|G|I_{|G|}$. In this paper, we study the relation
between $G$ and $h$ so that a $BH(G,h)$ matrix exists. We will only focus on
$BH(\mathbb{Z}_n,h)$ matrices and $BH(G,2p^b)$ matrices, where $p$ is an odd
prime. By our results, there are $2687$ open cases left for the existence of
$BH(\mathbb{Z}_n,h)$ matrices in which $1\leq n,h \leq 100$. In the last
section, we show that $BH(\mathbb{Z}_n,h)$ matrices can be used to construct a
new family of perfect polyphase arrays.
