Construction of Butson matrices using Fourier matrices as input

Butson matrices are square orthogonal matrices, denoted by $BH(m,n)$, whose entries are the complex mth roots of unity and satisfy the condition\\ BH(m,n)·BH(m,n)^*=nIₙ, where BH(m,n)^* is the conjugate transpose of $BH(m,n)$ and Iₙ is the identity matrix. In this work, we propose constructions for $BH(m,(n-1)n)$ then BH(m,(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. Moreover, we present some results about the construction of Hadamard matrices.