  A subset $C$ of the vertex set of a graph $\Gamma$ is called a perfect code
of $\Gamma$ if every vertex of $\Gamma$ is at distance no more than one to
exactly one vertex in $C$. Let $A$ be a finite abelian group and $T$ a
square-free subset of $A$. The Cayley sum graph of $A$ with respect to the
connection set $T$ is a simple graph with $A$ as its vertex set, and two
vertices $x$ and $y$ are adjacent whenever $x+y\in T$. A subgroup of $A$ is
said to be a subgroup perfect code of $A$ if the subgroup is a perfect code of
some Cayley sum graph of $A$. In this paper, we give some necessary and
sufficient conditions for a subset of $A$ to be a perfect code of a given
Cayley sum graph of $A$. We also characterize all subgroup perfect codes of
$A$.
