Minors of a random binary matroid

Let A be an n × m matrix over GF ₂ where each column consists of k ones, and let M be an arbitrary fixed binary matroid. The matroid growth rate theorem implies that there is a constant C_M such that m ≥ C_Mn² implies that the binary matroid induced by A contains M as a minor. We prove that if the columns of A = A _n,m,k are chosen randomly, then there are constants k_M,L_M such that k ≥ k_M and m ≥ L_Mn implies that A contains M as a minor with high probability .