  Let $\gamma(C_m\Box C_n)$ be the domination number of the Cartesian product
of directed cycles $C_m$ and $C_n$ for $m,n\geq2$. Shaheen [] and Liu and al.[
], [ ] determined the value of $\gamma(C_m\Box C_n)$ when $m \leq 6$ and when
both $m$ and $n$ $\equiv 0$ $(mod\: 3)$. In this article we give, in general,
the value of $\gamma(C_m\Box C_n)$ when $m\equiv 2$ $(mod\: 3)$ and improve the
known lower bound for most of the remaining cases. We also disprove the
conjectured formula for the case $m$ $\equiv 0$ $(mod\: 3)$ appearing in
\cite{}
