Group distance magic labeling of the Cartesian product of two directed cycles

Let $\overrightarrow{G}$ be a finite simple directed graph with $n$ vertices, and let $\mathrm{\Gamma }$ be a finite abelian group of order $n$. A $\mathrm{\Gamma }$-distance magic labeling is a bijection $\phi :V(\overrightarrow{G})⟶\mathrm{\Gamma }$ for which there exists $c\in \mathrm{\Gamma }$ such that $\sum _{y\in N^{+}(x)}\phi (y)-\sum _{y\in N^{-}(x)}\phi (y)=c$ for any $x\in V(\overrightarrow{G})$, where $N^{+}(x)$ and $N^{-}(x)$ denote the set of the head and the tail of $x$, respectively. In this paper, we obtain a necessary and sufficient condition for that there exists a $\mathrm{\Gamma }$-distance magic labeling for the Cartesian products of two directed cycles.