  We study subsystem codes whose gauge group has local generators in the 2D
geometry. It is shown that there exists a family of such codes defined on
lattices of size LxL with the number of logical qubits k and the minimum
distance d both proportional to L. The gauge group of these codes involves only
two-qubit generators of type XX and ZZ coupling nearest neighbor qubits (and
some auxiliary one-qubit generators). Our proof is not constructive as it
relies on a certain version of the Gilbert-Varshamov bound for classical codes.
Along the way we introduce and study properties of generalized Bacon-Shor codes
which might be of independent interest. Secondly, we prove that any 2D
subsystem [n,k,d] code with spatially local generators obeys upper bounds
kd=O(n) and d^2=O(n). The analogous upper bound proved recently for 2D
stabilizer codes is kd^2=O(n). Our results thus demonstrate that subsystem
codes can be more powerful than stabilizer codes under the spatial locality
constraint.
