Let $M\in\{-\ell,\dots,-1,0,+1,\dots,+\ell\}^{n\times n}$ be a matrix of rank $r$ where $\ell\geq1$ such that there is a permutation matrix in $\{0,1\}^{m\times m}$ of order $2\ell$.
Fix a permutation matrix $T$ of order $2\ell$.
Consider the matrix $f(M)\in\{0,+1\}^{mn\times mn}$ where $0$ in $M$ is replaced by $m\times m$ all $0$ matrix, $a$ in $M$ is replaced by $T^{2a}$ and $-a$ is replaced by $T^{-2a-1}$ at every $a\in\{1,2,\dots,\ell\}$.
Example at $m=2$:
$0$ is replaced by $\begin{bmatrix}0&0\\0&0\end{bmatrix}$, $+1$ is replaced by $\begin{bmatrix}1&0\\0&1\end{bmatrix}$ and $-1$ is replaced by $\begin{bmatrix}0&1\\1&0\end{bmatrix}$.
- How is the the real rank of the new matrix related to $m,\ell$ and $r$? Is there a precise bound?
- Is the $\mathbb F_2$ rank of the new matrix related to $m,\ell$ and $r$? Is there a precise bound?
