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 $ \Gamma $ be a finite abelian group of order $ n $. A $ \Gamma $-distance magic labeling is a bijection $ \varphi:V(\overrightarrow{G})\longrightarrow \Gamma $ for which there exists $ c\in\Gamma $ such that $ \sum_{y\in N^+(x)}\varphi(y)-\sum_{y\in N^-(x)}\varphi(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 $ \Gamma $-distance magic labeling for the Cartesian products of two directed cycles.