  A perfect code in a graph is an independent set of the graph such that every
vertex outside the set is adjacent to exactly one vertex in the set. A
circulant graph is a Cayley graph of a cyclic group. In this paper we study
perfect codes in circulant graphs of degree $p^l - 1$, where $p$ is a prime and
$l \ge 1$. We obtain a necessary and sufficient condition for such a circulant
graph to admit perfect codes, give a construction of all such circulant graphs
which admit perfect codes, and prove a lower bound on the number of distinct
perfect codes in such a circulant graph. This extends known results for the
case $l=1$ and provides insight on the general problem on the existence and
structure of perfect codes in circulant graphs.
