The rectangular rank profile of a complex Hadamard matrix $H$ of order $N$ is the following ordered set [126], [37] $$\mathcal{R}(H) = \Big\{\big\{\left(r_i(j,k), m_i(j,k)\right) : i\in I(j,k)\big\}_{j\times k} : 2\leqslant j,k\leqslant n-2\Big\}$$ where $I(j,k)$ is an index set for every $2\leqslant j,k\leqslant n-2$, and $r_i(j,k)$ and $m_i(j,k)$ are the possible values of the rank of the $j\times k$ submatrices of the matrix $H$ and their multiplicities, respectively.
For example $$ \mathcal{R}(F_2\otimes F_2)=\Big\{\big\{(1,12),(2,24)\big\}_{2\times 2}\Big\} $$ and $$ \mathcal{R}(F_4)=\Big\{\big\{(1,4),(2,32)\big\}_{2\times 2}\Big\}. $$
Rectangular rank profile can be used to distinguish a complex Hadamard matrix from its transpose with respect to equivalence relation.
See also Haagerup-set and matrix fingerprint.