  Recently a framework for assisted quantum error correction was proposed in
which a specific type of error is allowed to occur on auxiliary qubits, which
is in contrast to standard entanglement assistance that requires noiseless
auxiliary qubits. However, while the framework maintains the ability to import
any binary or quaternary linear code without sacrificing active error
correction power, it requires the code designer to turn a parity-check matrix
of the underlying classical code into an equivalent one in standard form. This
means that classical coding theoretic techniques that require parity-check
matrices to be in specific form may not fully be exploitable. Another issue of
the recently proposed scheme is that the error correction capabilities for bit
errors and phase errors are generally equal, which is not ideal for asymmetric
error models. This paper addresses these two problems. We generalize the
framework in such a way that any parity-check matrix of any binary or
quaternary linear code can be exploited. Our generalization also allows for
importing a pair of distinct linear codes so that error correction capabilities
become suitably asymmetric.
