Suppose, $G$ is a $k \times n$ binary matrix with $\operatorname{rank}(G) = k$. The first $k$ columns of $G$ are linearly independent and the next $n-k$ columns are linear combinations of the first $k$ columns.
$G$ matrix can be a generator matrix of a linear code where the fist $k$ columns are the $k \times k$ identity matrix and the next $n-k$ column contain the parity bits and these columns are linear combinations if the first $k$ columns.
I am choosing $u$ number of columns randomly from $G$ to form the matrix $G_u$, where $u<=k$.
I want to find the probability that $\operatorname{rank}(G_u) = u$ for any random sub-matrix.
Really appreciate any suggestions to solve this problem.
Thanks in advance.