A non-Hermitian system with long-range hopping under periodic driving is constructed in this work. The Hamiltonian has chiral symmetry, implying that a topological invariant can be determined. Using the non-Bloch band theory and the Floquet method, the relevant operators and topological number can be determined, thereby providing quantitative approaches for studying topological properties. For example, by calculating the non-Bloch time-evolution factor, the Floquet operator, etc., it can be found that the topological invariant is determined by changing the phase of U^+_\epsilon=0,\pi(\beta) as it moves along the generalized Brillouin zone, corresponding to the emergence of quasi-energy zero mode and π mode.

The results show that the topological structure of the static system can be significantly affected by periodic driving. The topological phase boundary of the zero mode can be changed. In the absence of periodic driving, energy spectrum does not exhibit π mode. After introducing periodic driving, a gap appears at the quasi-energy \epsilon=\pi, thereby inducing a non-trivial π-mode phase and enriching the topological phase diagram. Furthermore, the next nearest neighbor hopping has a unique effect in this system. It can induce large topological numbers. However, unlike the static system, large topological numbers only appear in specific parameter intervals under periodic driving. As the strength of the next nearest neighbor hopping increases, the large topological number phase disappears, reflecting the non-monotonic regulation characteristics of the Floquet system. In addition, introducing the phase of the next nearest neighbor hopping can change the topological phase boundary, providing new ideas for experimentally regulating topological states.

This research is of significance in the field of topological phase transitions in non-Hermitian systems. Theoretically, it reveals the synergistic effect of long-range hopping and periodic driving, and improves the theoretical framework for the cross-research of long-range and dynamic regulation in non-Hermitian systems. From an application perspective, it provides theoretical support for experimentally realizing the controllable modulation of topological states, which is helpful in promoting the development of fields such as low energy consumption electronic devices and topological quantum computing.