  Domination-type parameters are difficult to manage in Cartesian product
graphs and there is usually no general relationship between the parameter in
both factors and in the product graph. This is the situation of the domination
number, the Roman domination number or the $2$-domination number, among others.
Contrary to what happens with the domination number and the Roman domination
number, the $2$-domination number remains unknown in cylinders, that is, the
Cartesian product of a cycle and a path and in this paper, we will compute this
parameter in the cylinders with small cycles. We will develop two algorithms
involving the $(\min,+)$ matrix product that will allow us to compute the
desired values of $\gamma_2(C_n\Box P_m)$, with $3\leq n\leq 15$ and $m\geq 2$.
We will also pose a conjecture about the general formulae for the
$2$-domination number in this graph class.
