  It is known that the normal three-dimensional (3D) Ising model on a cubic
lattice is dual to the Wegner's 3D $Z_2$ lattice gauge theory. Here we find an
unusual $Z_2$ lattice gauge theory which is dual to the 3D Ising model with not
only nearest-neighbor (nn) coupling, but also next-nearest-neighbor (nnn)
coupling. Our gauge theory has on each edge four $Z_2$ variables that have
product $+1$, each located on a vector perpendicular to the edge. The nn
coupling in the Ising model maps to the plaquette term in the gauge theory
where the four variables multiplied have their vectors pointing inward, while
the nnn coupling maps to the coupling between the $Z_2$ variables on nearby
vectors on each edge in the gauge theory. A Wilson loop observable in the gauge
theory depends on a framing of a loop, and maps to a surface of flipped-sign nn
and nnn couplings in the Ising model. Further numerical simulations could be
made to explore the universality at the phase transition.
