  The two-dimensional q-state Potts model is subjected to a Z_q symmetric
disorder that allows for the existence of a Nishimori line. At q=2, this model
coincides with the +/- J random-bond Ising model. For q>2, apart from the usual
pure and zero-temperature fixed points, the ferro/paramagnetic phase boundary
is controlled by two critical fixed points: a weak disorder point, whose
universality class is that of the ferromagnetic bond-disordered Potts model,
and a strong disorder point which generalizes the usual Nishimori point. We
numerically study the case q=3, tracing out the phase diagram and precisely
determining the critical exponents. The universality class of the Nishimori
point is inconsistent with percolation on Potts clusters.
