Perfect codes in quintic Cayley graphs on abelian groups

A subset $C$ of the vertex set of a graph $\mathrm{\Gamma }$ is called a perfect code of $\mathrm{\Gamma }$ if every vertex of $\mathrm{\Gamma }$ is at distance no more than one to exactly one vertex in $C$. In this paper, we classify all connected quintic Cayley graphs on abelian groups that admit a perfect code, and determine completely all perfect codes of such graphs.