  Quaternary unit Hadamard (QUH) matrices were introduced by Fender, Kharagani
and Suda along with a method to construct them at prime power orders. We
present a novel construction of real Hadamard matrices from QUH matrices. Our
construction recovers the result by Mukhopadhyay on the existence of real
Hadamard matrices of order $q^n+q^{n-1}$ for each prime power $q\equiv 3\mod
4$. Furthermore we provide nonexistence conditions for QUH matrices.
