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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 $1\\leq n,h \\leq 100$. 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