Read-Rezayi fractional quantum Hall states are among the prime candidates for realizing non-Abelian anyons which, in principle, can be used for topological quantum computation. We present a prescription for efficiently finding braids which can be used to carry out a universal set of quantum gates on encoded qubits based on anyons of the Read-Rezayi states with k > 2, k not equal 4. This work extends previous results which only applied to the case k = 3 (Fibonacci) and clarifies why, in that case, gate constructions are simpler than for a generic Read-Rezayi state.