<abstract><p>In this paper, we consider the following periodic discrete nonlinear Schrödinger equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} Lu_{n}-\omega u_{n} = g_{n}(u_{n}), \qquad n = (n_{1}, n_{2}, ..., n_{m})\in \mathbb{Z}^{m}, \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ \omega\notin \sigma(L) $(the spectrum of $ L $) and $ g_{n}(s) $ is super or asymptotically linear as $ |s|\to\infty $. Under weaker conditions on $ g_{n} $, the existence of ground state solitons is proved via the generalized linking theorem developed by Li and Szulkin and concentration-compactness principle. Our result sharply extends and improves some existing ones in the literature.</p></abstract>