Symplectic representation of a stabilizer code

A stabilizer group with $r$ independent generators maps injectively to an r-dimensional isotropic subspace of the binary symplectic space of Pauli labels. For generator rows $(A\mid B)$, commutation is $A{B}^T+B{A}^T=0$ over ${\mathbb{F}}_2$. Its stabilizer subspace has dimension $2^{n-r}$. The labels of the centralizer of a stabilizer group form $L^{\perp }$; labels in $L^{\perp }\setminus L$ act as nontrivial logical Pauli operators.