Is there a theorem about when a (sparse) 0-1 matrix $A = (a_{ij})_{\substack{1\leq i\leq n\\ 1\leq j\leq m}}$ has full rank? More concretely, I know the row sums of $A$, $$ r_i = \sum\limits_{j=1}^m a_{ij}$$ and how many of the $r_i$ are equal (depending on $m$ and $n$. Furthermore, about the column sums $$c_j = \sum\limits_{i=1}^n a_{ij} $$ I know lower and upper bounds, and how many of the $c_j$ are equal, I also can estimate in both directions (depending on $m$ and $n$ again. The easiest case (constant row sums and constant column sums) is known to me, but too weak for what I want.
I appreciate any help!