Given real matrix $M\in\{0,1\}^{n\times n}$ of rank $r$. How many $M_i\in\{0,1\}^{n\times n}$ of rank $s$ does one need to write $M'=\sum_{i=1}^ta_iM_i$ for some $a_i\in\Bbb R$ where maximum absolute value entry of $M'-M$ is $\frac{1}{4}$? We know $t=n^2$ suffices. However can we do better (say in a number as a function of $r,s$)?
For instance consider $M$ to be block diagonal matrix with $2$ blocks of $1$. $r=2$ here and $M$ can indeed be given as sum of $2$ rank $s=1$ matrices.