The paper introduces a six-dimensional movable antenna (6DMA) system to enhance wireless network capacity by adjusting the 3D positions and rotations of distributed antenna surfaces based on user spatial distribution and statistical channel information. Due to practical constraints, the 6DMA surfaces can only move in discrete positions and rotations, which limits their flexibility. The authors propose offline and online optimization algorithms to maximize the average network capacity under these constraints. The offline algorithm uses Monte Carlo simulation to approximate the average network capacity and solve a non-convex integer programming problem using the conditional gradient method. The online algorithm, based on the Fibonacci Sphere scheme, generates discrete positions and rotations without statistical channel knowledge, and optimizes them using the conditional sample mean (CSM) method. Simulation results demonstrate that the proposed 6DMA-BS can significantly improve network capacity compared to fixed-position antennas and 6DMAs with limited movability.The paper introduces a six-dimensional movable antenna (6DMA) system to enhance wireless network capacity by adjusting the 3D positions and rotations of distributed antenna surfaces based on user spatial distribution and statistical channel information. Due to practical constraints, the 6DMA surfaces can only move in discrete positions and rotations, which limits their flexibility. The authors propose offline and online optimization algorithms to maximize the average network capacity under these constraints. The offline algorithm uses Monte Carlo simulation to approximate the average network capacity and solve a non-convex integer programming problem using the conditional gradient method. The online algorithm, based on the Fibonacci Sphere scheme, generates discrete positions and rotations without statistical channel knowledge, and optimizes them using the conditional sample mean (CSM) method. Simulation results demonstrate that the proposed 6DMA-BS can significantly improve network capacity compared to fixed-position antennas and 6DMAs with limited movability.