  We consider the problem of large data scattering for the defocusing cubic
nonlinear Schr\"odinger equation on $\mathbb{R}^2$ $\times$ $\mathbb{T}^2$.
This equation is critical both at the level of energy and mass. The key
ingredients contain a global-in-time Stricharz estimate, resonant system
approximation and profile decomposition. Assuming the large data scattering for
the 2d cubic resonant system, we can prove the large data scattering for this
problem.
