Fix numbers $m, n, k\in {\Bbb Z}_+$ and $r\in {\Bbb R}_+$.

What non-trivial estimates exist for the probability that a random $m\times n$ matrix, with integer entries and with all its rows of Euclidean norm less than or equal to $r$, will have rank $k$?

I'm particularly interested in results asymptotic in the variable $r$. (I've worked out the coefficient for $m=n=2, k=1$, but my method doesn't seem to generalize.)