I have a $k \times n$ matrix $G$ over ${\mathbb F_2}$ that's full rank. This can always be put in systematic form : $G \sim [I_k | P]$ where $I_k$ is a $k \times k$ identity matrix and $P$ is a $k \times (n-k)$ matrix. I'd like to impose this additional orthogonality condition on $P : P^t P = I_{n-k}$. I allow permutations of the columns of $G$ as additional operations. With these additional operations, when can I have $G \approx [I_k| P] : P^t P = I_{n-k}$? Thanks for any suggestions.