Given an integer matrix $Minmathbb Z^{ntimes n}$ of real rank $k$ what is the minimum and maximum number of rank $1$ matrices $B_1$ to $B_t$ we require so that $Mequivsum_{i=1}^tB_ibmod 2$?

If $mathbb F_2$ rank of $M$ is $k’$ then we need $t=k’$. I am asking the best gap between $k$ and $k’$.

If $k’=n$ when is $k<n$?
I guess the converse $k'<n$ and $k=n$ does not happen.