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Given an abelian subgroup $S$ of the Pauli group on $n$ qubits with $r$ independent generators we obtain an $2^{n-r}$-dimensional subspace stabilized by $S$. By picking a basis for this space, it is easy to build a representation of the Pauli group on $k$ qubits generated by some set of matrices $\overline{X_i}, \overline{Z_i}$. However, I fail to see how to do so in a way that ensures that $\overline{X_i}, \overline{Z_i}$ are Pauli matrices in the original basis. The common way to build these logicals is to consider the quotient $\mathcal{N}(S)/S$, and note that it corresponds to the Pauli group on $k$ qubits, but I cannot quite prove it.

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