For any two (complex) Hadamard matrices $A\in\mathcal{H}_K$ and $B\in\mathcal{H}_M$ their tensor product $A\otimes B\in\mathcal{H}_{KM}$. A more general construction by Diţă [56] allows to use apart of a given Hadamard matrix $A$ of size $K$ an entire set of $K$ (possibly different) Hadamard matrices $\{B_1, ... , B_K\}$ of size $M$. Then the matrix $$H=\left[\begin{array}{cccc} A_{11}B_1 & A_{12}E_2B_2 & ... & A_{1K}E_KB_K\\ \vdots & \vdots & & \vdots\\ A_{K1}B_1 & A_{K2}E_2B_2 & ... & A_{KK}E_KB_K \end{array}\right]$$ of size $N = K M$ is Hadamard. We may introduce additional free phases by using $m=(K - 1)$ diagonal unitary matrices $E_k$. The total number $f$ of free phases reads $f=(M - 1)(K - 1) + a + b_1 + ... + b_K$ where $a$ and $b_i$ denote the numbers of free phases used in the definitions of $A$ and $B_i$ respectively.
Other constructions.