The rapid advancement of additive manufacturing technologies enables a pixelated or voxelated structure consisting of multiple materials in 2D or 3D, respectively, called digital materials. It maximizes design flexibility without constraints in geometry and material, realizing unprecedented physical properties and functionalities that cannot be realized by conventional manufacturing processes. However, the enormous design space of digital materials has become a significant challenge for maximizing the availability of digital materials. In this study, we developed a novel deep generative model with discrete representations of the latent space for designing digital materials. The proposed model, inspired by the discrete nature of digital materials, retains reconstruction and prediction accuracies with one-third of the data usage compared to the conventional generative model. The physical insight of discrete representations of latent space is rigorously interpreted, proving that certain discrete representations are strongly related to the mechanical behavior of digital materials, such as auxeticity. It was also confirmed that the proposed model is an excellent tool for generating functionally graded structures as well as the unseen auxetic structures.