  Quantum error correction requires the use of error syndromes derived from
measurements that may be unreliable. Recently, quantum data-syndrome (QDS)
codes have been proposed as a possible approach to protect against both data
and syndrome errors, in which a set of linearly dependent stabilizer
measurements are performed to increase redundancy. Motivated by wanting to
reduce the total number of measurements performed, we introduce QDS subsystem
codes, and show that they can outperform similar QDS stabilizer codes derived
from them. We also give a construction of single-error-correcting QDS
stabilizer codes from impure stabilizer codes, and show that any such code must
satisfy a variant of the quantum Hamming bound for QDS codes. Finally, we use
this bound to prove a new bound that applies to impure, but not pure,
stabilizer codes that may be of independent interest.
