We study the \(\pm J\) random-plaquette \(Z_2\) gauge model (RPGM) in three spatial dimensions, a three-dimensional analog of the two-dimensional \(\pm J\) random-bond Ising model (RBIM). The model is a pure \(Z_2\) gauge theory in which randomly chosen plaquettes (occurring with concentration \(p\)) have coupling with the “wrong sign” so that magnetic flux is energetically favored on these plaquettes. Excitations of the model are one-dimensional “flux tubes” that terminate at “magnetic monopoles” located inside lattice cubes that contain an odd number of wrong-sign plaquettes. Electric confinement can be driven by thermal fluctuations of the flux tubes, by the quenched background of magnetic monopoles, or by a combination of the two. Like the RBIM, the RPGM has enhanced symmetry along a “Nishimori line” in the \(p-T\) plane (where \(T\) is the temperature). The critical concentration \(p_c\) of wrong-sign plaquettes at the confinement-Higgs phase transition along the Nishimori line can be identified with the accuracy threshold for robust storage of quantum information using topological error-correcting codes: if qubit phase errors, qubit bit-flip errors, and errors in the measurement of local check operators all occur at rates below \(p_c\), then encoded quantum information can be protected perfectly from damage in the limit of a large code block. Through Monte Carlo simulations, we measure \(p_{c0}\), the critical concentration along the \(T=0\) axis (a lower bound on \(p_c\)), finding \(p_{c0}=.0293\pm.0002\). We also measure the critical concentration of antiferromagnetic bonds in the two-dimensional RBIM on the \(T=0\) axis, finding \(p_{c0}=.1031\pm 0001\). Our value of \(p_{c0}\) is incompatible with the value of \(p_c=.1093\pm.0002\) found in earlier numerical studies of the RBIM, in disagreement with the conjecture that the phase boundary of the RBIM is vertical (parallel to the \(T\) axis) below the Nishimori line. The model can be generalized to a rank-\(r\) antisymmetric tensor field in \(d\) dimensions, in the presence of quenched disorder.