  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 $k>2$, $k\neq4$. 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.
