  We study the rank of the random $n\times m$ 0/1 matrix ${\bf A}_{n,m;k}$
where each column is chosen independently from the set $\Omega_{n,k}$ of 0/1
vectors with exactly $k$ 1's. Here 0/1 are the elements of the field $GF_2$. We
obtain an asymptotically correct estimate for the rank in terms of $c,n,k$,
assuming that $m=cn$.
  In addition, we assign i.i.d. $U[0,1]$ weights $X_{{\bf c}},{\bf
c}\in\Omega_{n,k}$ and let the weight of a set of columns $C$ be
$X(C)=\sum_{{\bf c}\in C}X_{{\bf c}}$. Let a basis be a set of
$n-1_{k\text{even}}$ linearly independent columns. We obtain an asymptotically
correct estimate for the minimum weight of a basis. This generalises the
well-known result for $k=2$ viz. that the expected length of a minimum weight
spanning tree tends to $\zeta(3)$.
