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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?

M. Al Jumaily
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    Rather than talking about stabilizer codes in general, you might find it more accessible to read about CSS codes. These are explicitly constructed from classical codes and, once you've done that, easily translate into the matrix generators. – DaftWullie Nov 26 '19 at 08:45

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