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We propose families of protocols for magic-state distillation—important components of fault-tolerance schemes—for systems of odd prime dimension. Our protocols utilize quantum Reed-Muller codes with transversal non-Clifford gates. We find that, in higher dimensions, small and effective codes can be used that have no direct analogue in qubit (two-dimensional) systems. We present several concrete protocols, including schemes for three-dimensional (qutrit) and five-dimensional (ququint) systems. The five-dimensional protocol is, by many measures, the best magic-state-distillation scheme yet discovered. It excels both in terms of error threshold with respect to depolarizing noise () and the efficiency measure known as yield, where, for a large region of parameters, it outperforms its qubit counterpart by many orders of magnitude.