  We introduce a stabilizer formalism for the general quantum error correction
framework called operator algebra quantum error correction (OAQEC), which
generalizes Gottesman's formulation for traditional quantum error correcting
codes (QEC) and Poulin's for operator quantum error correction and subsystem
codes (OQEC). The construction generates hybrid classical-quantum stabilizer
codes and we formulate a theorem that fully characterizes the Pauli errors that
are correctable for a given code, generalizing the fundamental theorems for the
QEC and OQEC stabilizer formalisms. We discover hybrid versions of the
Bacon-Shor subsystem codes motivated by the formalism, and we apply the theorem
to derive a result that gives the distance of such codes. We show how some
recent hybrid subspace code constructions are captured by the formalism, and we
also indicate how it extends to qudits.
