  In this paper we study the topological and metric directional entropy of
$\mathbb{Z}^2$-actions by generated additive cellular automata (CA hereafter),
defined by a local rule $f[l, r]$, $l, r\in \mathbb{Z}$, $l\leq r$, i.e. the
maps $T_{f[l, r]}: \mathbb{Z}^\mathbb{Z}_{m} \to \mathbb{Z}^\mathbb{Z}_{m}$
which are given by $T_{f[l, r]}(x) =(y_n)_ {-\infty}^{\infty}$, $y_{n} =
f(x_{n+l}, ..., x_{n+r}) = \sum_{i=l}^r\lambda_{i}x_{i+n}(mod m)$, $x=(x_n)_
{n=-\infty}^{\infty}\in \mathbb{Z}^\mathbb{Z}_{m}$, and $f:
\mathbb{Z}_{m}^{r-l+1}\to \mathbb{Z}_{m}$, over the ring $\mathbb{Z}_m (m \geq
2)$, and the shift map acting on compact metric space
$\mathbb{Z}^\mathbb{Z}_{m}$, where $m$ $(m \geq2)$ is a positive integer. Our
main aim is to give an algorithm for computing the topological directional
entropy of the $\mathbb{Z}^2$-actions generated by the additive CA and the
shift map. Thus, we ask to give a closed formula for the topological
directional entropy of $\mathbb{Z}^2$-action generated by the pair $(T_{f[l,
r]}, \sigma)$ in the direction $\theta$ that can be efficiently and rightly
computed by means of the coefficients of the local rule f as similar to [Theor.
Comput. Sci. 290 (2003) 1629-1646]. We generalize the results obtained by Ak\i
n [The topological entropy of invertible cellular automata, J. Comput. Appl.
Math. 213 (2) (2008) 501-508] to the topological entropy of any invertible
linear CA.
