The answer is actually yes ( $k^2$ and $2k$ instead of $10000$ and $200$ ) Let $X$ be set of all rows which have exactly one digit $0$ in first $k$ places ( section $A$ ) and exactly one in last $k$ places ( section $B$ ) ( which the number of these rows is $k^2$. ) And also let $Y$ be set of all rows which have at least $2$ zero digits in one of sections $A,B$ and $Z$ the set of rows with one zero digit. Now if the blowed away sheets in rows of set $X$ be its zero digits , for each row in set $Y$ two zero sheets from one section and also for $Z$ members their zero sheet and one of the $1$ digits , then one can easily see that all rows of set $X$ get uniquely determined and we're done.