  One of the main problems in quantum information systems is the presence of
errors due to noise, and for this reason quantum error-correcting codes (QECCs)
play a key role. While most of the known codes are designed for correcting
generic errors, i.e., errors represented by arbitrary combinations of Pauli X ,
Y and Z operators, in this paper we investigate the design of stabilizer QECC
able to correct a given number eg of generic Pauli errors, plus eZ Pauli errors
of a specified type, e.g., Z errors. These codes can be of interest when the
quantum channel is asymmetric in that some types of error occur more frequently
than others. We first derive a generalized quantum Hamming bound for such
codes, then propose a design methodology based on syndrome assignments. For
example, we found a [[9,1]] quantum error-correcting code able to correct up to
one generic qubit error plus one Z error in arbitrary positions. This,
according to the generalized quantum Hamming bound, is the shortest code with
the specified error correction capability. Finally, we evaluate analytically
the performance of the new codes over asymmetric channels.
