Consider $m$ random 0-1 vectors of length $n$. Let $L$ be the lattice spanned by them. What is the value of $m$ (as a function of $n$) for which it is true with positive probability that $L=Z^n$? More generally, let $V(L)$ be the rank of $Z^n/L$ (The volume of $L$). What is the behavior of $V(L)$ as a function of $n$ and $m$?