We present a search for nine lepton-number-violating and three lepton-flavor-violating neutral charm decays of the type $D^0rightarrow h^{prime -} h^{-}ell^{prime +} ell^{+}$ and $D^0rightarrow h^{prime -} h^{+}ell^{primepm} ell^{mp}$, where $h$ and $h^{prime}$ represent a $K$ or $pi$ meson and $ell$ and $ell^{prime}$ an electron or muon. The analysis is based on $468$ fb$^{-1}$ of $e^+e^-$ annihilation data collected at or close to the $Y(4S)$ resonance with the BaBar detector at the SLAC National Accelerator Laboratory. No significant signal is observed for any of the twelve modes and we establish 90% confidence level upper limits on the branching fractions in the range $(1.0 - 30.6)times 10^{-7}$. The limits are between one and three orders of magnitude times more stringent than previous measurements.
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