# Definition of distance d of the QECC $T(S)$

Definition of distance $$d$$:

$$d$$ of $$T(S)$$ is the weight of the smallest Pauli operator $$N$$ in $$N(S)$$ \ $$S$$. $$S$$ is the stabilizer, $$T(S)$$ is the corresponding QECC, and $$N(S)$$ are all errors commute with elements in S.

The weight of an error in QEC is the number of qubits, on which the error acts non-trivially. For example, the error $$X\otimes Y\otimes Id$$ has weight two. The weight is important, because usually we know that a code can correct errors up to some weight.