Strong quantum nonlocality with genuine entanglement in an $N$-qutrit system

In this paper, we construct genuinely multipartite entangled bases in $(\mathbb{C}^{3})^{\otimes N}$ for $N\geq3$, where every state is one-uniform state. By modifying this construction, we successfully obtain strongly nonlocal orthogonal genuinely entangled sets and strongly nonlocal orthogonal genuinely entangled bases, which provide an answer to the open problem raised by Halder $et~al.$ [\href{https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.122.040403} {Phy. Rev. Lett. \textbf{122}, 040403 (2019)}]. The strongly nonlocal orthogonal genuine entangled set we constructed in $(\mathbb{C}^{3})^{\otimes N}$ contains much fewer quantum states than all known ones. When $N>3$, our result answers the open question given by Wang $et~al$. [\href{https://journals.aps.org/pra/abstract/10.1103/PhysRevA.104.012424} {Phys. Rev. A \textbf{104}, 012424 (2021)}].