In this talk, we discuss the periodic homogenization of linear elliptic equations of the form $-A(x/\varepsilon):D^2 u^{\varepsilon} = f$ subject to a Dirichlet boundary condition. We  characterize good diffusion matrices $A$, i.e., those for which the sequence of solutions  converges at a rate of $\mathcal{O}(\varepsilon^2)$ in the $L^{\infty}$-norm to the solution  of the homogenized problem. Such diffusion matrices are considered “good” as the optimal rate of convergence in the generic case is only $\mathcal{O}(\varepsilon)$. First, we provide a class of good diffusion matrices, confirming a conjecture posed by Guo and Tran in 2020.  Then, we give a complete characterization of diagonal diffusion matrices in two dimensions       and a systematic study in higher dimensions. This talk is based on joint work with Xiaoqin Guo (University of Cincinnati) and Hung V. Tran (University of Wisconsin Madison).