Using a materials genome approach on the basis of the density functional theory, we have formulated a new class of inorganic electrolytes for fast diffusion of Li+ ions, through fine-tuning of lattice chemistry of anti-perovskite structures. Systematic modelling has been carried out to elaborate the structural stability and ion transportation characteristics in Li3AX based cubic anti-perovskite, through alloying on the chalcogen lattice site (A) and alternative occupancy of the halogen site (X). In addition to identifying effective ways for reduction of diffusion barriers for Li+ ions in anti-perovskite phases via suitable designation of lattice occupancy, the current theoretical study leads to discovery and synthesis of a new phase with a double-anti-perovskite structure, Li6OSI2 (or Li3O0.5S0.5I). Such a new compound is of fairly low activation barrier for Li+ diffusion, together with a wide energy band gap to hinder conduction of electrons.