I have a $k \times n$ matrix $G$ over ${\mathbb F_2}$ that's full rank. This can always be put in systematic form : $G \sim [I_k \mid P]$ 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 $G \approx [I_k\mid P] : P^t P = I_{n-k}$? Thanks for any suggestions.
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$\begingroup$ "Put into" by what operations? $\endgroup$– Robert IsraelCommented Aug 28, 2016 at 18:37
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$\begingroup$ I'm allowing these two types of operations : the first is multiply by invertible matrix $U$ : $ G \to U G = [I_k \mid P]$ and the second is a column permutations of $UG$ to make $P$ orthogonal (if possible). $\endgroup$– unknownCommented Aug 28, 2016 at 19:41
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$\begingroup$ If a column permutation of $P$ is orthogonal, so is $P$. $\endgroup$– Robert IsraelCommented Aug 28, 2016 at 21:17
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1$\begingroup$ yes, but I'm allowing any combination of $(U,V)$ in any order; here $V$ is a permutation of order $n$. So it's equivalence under a larger group containing $GL_k({\mathbb F}_2)$ and $S_n$ as subgroups. (general linear group and symmetric group). Two matrices equivalent under these operations generate equivalent codes (hence the coding theory tag). $\endgroup$– unknownCommented Aug 28, 2016 at 21:54
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