  The essential insight of quantum error correction was that quantum
information can be protected by suitably encoding this quantum information
across multiple independently erred quantum systems. Recently it was realized
that, since the most general method for encoding quantum information is to
encode it into a subsystem, there exists a novel form of quantum error
correction beyond the traditional quantum error correcting subspace codes.
These new quantum error correcting subsystem codes differ from subspace codes
in that their quantum correcting routines can be considerably simpler than
related subspace codes. Here we present a class of quantum error correcting
subsystem codes constructed from two classical linear codes. These codes are
the subsystem versions of the quantum error correcting subspace codes which are
generalizations of Shor's original quantum error correcting subspace codes. For
every Shor-type code, the codes we present give a considerable savings in the
number of stabilizer measurements needed in their error recovery routines.
