I'd like to check Knill-Laflamme conditions given a set of possible errors $\mathcal{E} \subset \mathbb{C}^{2^x \times 2^n}$ and a parity check matrix, i.e. find $c_{ab}$ in $\left\langle\psi_i\left|E_a^{\dagger} E_b\right| \psi_j\right\rangle=c_{a b} \delta_{i j}$
My question is: how to get logical codewords $\{ | \psi \rangle_i \}$ (a base of the subspace that defines the code $\mathcal{C} \subset \mathcal{H}_2^{\otimes n}$) from the parity check matrix? In a classical case space of codewords would just be kernel of parity check matrix but I believe that kernel of a quantum parity check matrix is the set of logical operators.
I suppose one could extract stabilizer generators from rows of parity check matrix and then find a logical zero using \begin{equation} \left|0_L\right\rangle=\frac{1}{2^n}\left(\prod_{i=1}^n\left(\mathbb{I}+g_i\right)\right)|0\rangle \end{equation}.
And then we're left with getting $ \left|1_L\right\rangle = X_L\left|0_L\right\rangle$. So we only need $X_L$. How exactly can we find it from kernel of parity check matrix?
Is there no easier way to check KL conditions given a list of errors and a parity check matrix? Would it make things easier if we only considered CSS codes?
