Given positive integers $k$, $m$, $n$, let $A$ be an $m \times n$ matrix over $GF(2)$ constructed as follows. Let $X_1, \ldots, X_m$ be independent random subsets of $\{1,\ldots,n\}$ with cardinality $k$, and take $A_{i,j} = 1$ if $j \in X_i$, $0$ otherwise. What can be said about the probability $P(k,m,n)$ that $A$ has rank $m$ over $GF(2)$?

In particular, I suspect that there is some $t(k) > 1$ such that as $m, n \to \infty$ with $n/m < t(k) - \epsilon$, $P(k,m,n) \to 0$, while for $n/m > t(k) + \epsilon$, $P(k,m,n)$ is bounded below. Is this true?