Matrices $M_1,M_2,\dots,M_{k-1},M_k\in\{0,1\}^{n\times n}$ with real ranks $r_1,r_2,\dots,r_{k-1},r_k$ respectively.
What is the rank of the matrix $M=M_1\oplus M_2\oplus\dots\oplus M_{k-1}\oplus M_k\in\Bbb Z^{n\times n}$ (not $\Bbb F_2^{n\times n}$) which is defined by:
$M_{ij}=-1\iff\sum_{\ell=1}^k M_{\ell,ij}=1$
$M_{ij}=+1\iff\sum_{\ell=1}^k M_{\ell,ij}\neq1$?
Note $M_1+M_2+\dots+M_{k-1}+M_k$ has rank at most $r_1+r_2+\dots+r_{k-1}+r_k$.
I just want to know if it is upper bound by $poly(r_1+r_2+\dots+r_{k-1}+r_k)$.
