Minors of a random binary matroid

Let $\mathbf{A}={\mathbf{A}}_{n,m,k}$ be a random $n\times m$ matrix over ${\mathbf{G}\mathbf{F}}_2$ wher each column consists of $k$ randomly chosen ones. Let $M$ be an arbirary fixed binary matroid. We show that if $m\mathrm{/}n$ and $k$ are sufficiently large then as $n\to \mathrm{\infty }$ the binary matroid induced by {\bf A} contains $M$ as a minor.