  We construct Pauli topological subsystem codes characterized by arbitrary
two-dimensional Abelian anyon theories--this includes anyon theories with
degenerate braiding relations and those without a gapped boundary to the
vacuum. Our work both extends the classification of two-dimensional Pauli
topological subsystem codes to systems of composite-dimensional qudits and
establishes that the classification is at least as rich as that of Abelian
anyon theories. We exemplify the construction with topological subsystem codes
defined on four-dimensional qudits based on the $\mathbb{Z}_4^{(1)}$ anyon
theory with degenerate braiding relations and the chiral semion theory--both of
which cannot be captured by topological stabilizer codes. The construction
proceeds by "gauging out" certain anyon types of a topological stabilizer code.
This amounts to defining a gauge group generated by the stabilizer group of the
topological stabilizer code and a set of anyonic string operators for the anyon
types that are gauged out. The resulting topological subsystem code is
characterized by an anyon theory containing a proper subset of the anyons of
the topological stabilizer code. We thereby show that every Abelian anyon
theory is a subtheory of a stack of toric codes and a certain family of twisted
quantum doubles that generalize the double semion anyon theory. We further
prove a number of general statements about the logical operators of translation
invariant topological subsystem codes and define their associated anyon
theories in terms of higher-form symmetries.
