This paper proposes a 6D movable antenna (6DMA)-aided base station (BS) system that enhances wireless network capacity by optimizing the 3D positions and 3D rotations of multiple 6DMA surfaces, subject to discrete movement constraints. The 6DMA surfaces can be adjusted in 3D positions and rotations based on the users' spatial distribution and statistical channel information. The system considers practical cases with and without statistical channel knowledge, and proposes offline and online optimization algorithms using Monte Carlo and conditional sample mean (CSM) methods. Simulation results show that the 6DMA-BS significantly improves network capacity compared to conventional BSs with fixed-position antennas (FPAs) and 6DMAs with limited movability, even under discrete position/rotation constraints. The key contributions include extending the 6DMA-BS system model to discrete position/rotation adjustment, formulating a new optimization problem for maximizing average network capacity, and proposing offline and online algorithms for solving the problem with and without statistical channel knowledge. The offline algorithm uses Monte Carlo simulation to approximate network capacity, while the online algorithm uses CSM to optimize positions and rotations based on measured sum-rate values. The results demonstrate that the 6DMA-BS can significantly enhance wireless network capacity by exploiting the spatial distribution characteristics of users.This paper proposes a 6D movable antenna (6DMA)-aided base station (BS) system that enhances wireless network capacity by optimizing the 3D positions and 3D rotations of multiple 6DMA surfaces, subject to discrete movement constraints. The 6DMA surfaces can be adjusted in 3D positions and rotations based on the users' spatial distribution and statistical channel information. The system considers practical cases with and without statistical channel knowledge, and proposes offline and online optimization algorithms using Monte Carlo and conditional sample mean (CSM) methods. Simulation results show that the 6DMA-BS significantly improves network capacity compared to conventional BSs with fixed-position antennas (FPAs) and 6DMAs with limited movability, even under discrete position/rotation constraints. The key contributions include extending the 6DMA-BS system model to discrete position/rotation adjustment, formulating a new optimization problem for maximizing average network capacity, and proposing offline and online algorithms for solving the problem with and without statistical channel knowledge. The offline algorithm uses Monte Carlo simulation to approximate network capacity, while the online algorithm uses CSM to optimize positions and rotations based on measured sum-rate values. The results demonstrate that the 6DMA-BS can significantly enhance wireless network capacity by exploiting the spatial distribution characteristics of users.