  Combinatorial gauge symmetry is a principle that allows us to construct
lattice gauge theories with two key and distinguishing properties: a) only one-
and two-body interactions are needed; and b) the symmetry is exact rather than
emergent in an effective or perturbative limit. The ground state exhibits
topological order for a range of parameters. This paper is a generalization of
the construction to any finite Abelian group. In addition to the general
mathematical construction, we present a physical implementation in
superconducting wire arrays, which offers a route to the experimental
realization of lattice gauge theories with static Hamiltonians.
