  In this paper, we study the fourth-order Schr\"{o}dinger equation
\begin{equation*}
  i \partial_t u + {\Delta}^2 u - \gamma \Delta u = \pm |u|^{s-1}u
\end{equation*} on the lattice $\mathbb{Z}^d$ with dimensions $d=1,2$ and
parameter $\gamma \in \mathbb{R}$. In order to establish sharp dispersive
estimates, we consider the fundamental solution as an oscillatory integral and
analyze the Newton polyhedron of its phase function. Furthermore, we prove
Strichartz estimates which yield the existence of global solutions to nonlinear
equations with small data.
