Relation between Hamming distances of columns and rows
The condition on the columns implies that $2t &lt; n$. Suppose that the condition on the rows is not satisfied. The rows therefore form a code of minimal distance at least $n-t &gt; n/2$ of size $n$. The Plotkin bound implies that unless $n-t$ is extremely close to $n/2$, such a code cannot exist. If $t$ is very close to $n/2$, however, the columns form a code of minimal distance close to $n$, which is again ruled out by the Plotkin bound (except perhaps for very small $n$).

I'll let you fill up the details and check for which $n$ this argument works.