This Letter presents a search for a heavy neutral particle decaying into an opposite-sign different-flavor dilepton pair, $e^\pm \mu^\mp$, $e^\pm \tau^\mp$, or $\mu^\pm \tau^\mp$ using 20.3 fb$^{-1}$ of $pp$ collision data at $\sqrt{s} = 8$ TeV collected by the ATLAS detector at the LHC. The numbers of observed candidate events are compatible with the Standard Model expectations. Limits are set on the cross section of new phenomena in two scenarios: the production of $\tilde{\nu}_{\tau}$ in $R$-parity-violating supersymmetric models and the production of a lepton-flavor-violating $Z'$ vector
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