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In this work, we propose constructions for $BH(m,(n-1)n)$ then $BH(m,(\\frac{n}{2}-1)n)$, when $n$ and $m$ are even numbers, using the existing $BH(m,n)$. For each case, we provide two construction methods: one uses a single input Butson matrix, and another uses two input Butson matrices. 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In this work, we propose constructions for $BH(m,(n-1)n)$ then $BH(m,(\\frac{n}{2}-1)n)$, when $n$ and $m$ are even numbers, using the existing $BH(m,n)$. For each case, we provide two construction methods: one uses a single input Butson matrix, and another uses two input Butson matrices. 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