I am really puzzled when it comes to the construction of a stabilizer code. By that, I mean how to come up with the subgroup $S$ and the matrix generators $M= \{M_1, M_2, \cdots, M_{n-k}\}$? In classical error correcting codes, there is a generator matrix $G$ where each row is linearly independent and from there, you can find the codewords.
Though stabilizer code deals with operators rather the state itself, what is the process?
I found this that might be relatable: Given $n-k$ stabiliser generators, how can we find an additional $k$ commuting generators?
