  Ashikhmin and Litsyn showed that all binary stabilizer codes - pure or impure
- of sufficiently large length obey the quantum Hamming bound, ruling out the
possibility that impure codes of large length can outperform pure codes with
respect to sphere packing. In contrast we show that impure subsystem codes do
not obey the quantum Hamming bound for pure subsystem codes, not even
asymptotically. We show that there exist arbitrarily long Bacon-Shor codes that
violate the quantum Hamming bound.
