What is the discrete gauge symmetry of the minimal supersymmetric standard model

We systematically study the extension of the supersymmetric standard model (SSM) by an anomaly-free discrete gauge symmetry ${\mathbit{Z}}_{N}$. We extend the work of Ib\'a\~nez and Ross with $N=2$, 3 to arbitrary values of $N$. As new fundamental symmetries, we find four ${\mathbit{Z}}_{6}$, nine ${\mathbit{Z}}_{9}$, and nine ${\mathbit{Z}}_{18}$. We then place three phenomenological demands upon the low-energy effective SSM: (i) the presence of the $\ensuremath{\mu}$ term in the superpotential, (ii) baryon-number conservation up to dimension-five operators, and (iii) the presence of the seesaw neutrino mass term $L{H}_{u}L{H}_{u}$. We are then left with only two anomaly-free discrete gauge symmetries: baryon triality, ${\mathbit{B}}_{3}$, and a new ${\mathbit{Z}}_{6}$, which we call proton hexality, ${\mathbit{P}}_{6}$. Unlike ${\mathbit{B}}_{3}$, ${\mathbit{P}}_{6}$ prohibits the dimension-four lepton-number violating operators. This we propose as the discrete gauge symmetry of the minimal SSM, instead of $R$ parity.