High-performance random number generators (RNGs) can be economically implemented in popular field programmable gate arrays without the need for arithmetic circuitry by employing cellular automata (CA) with a neighborhood size of four and an asymmetrical, non-local neighborhood connection scheme. Each cell (i.e., RNG bit) requires only a single 4-input lookup table and a single flip-flop. From each of various 1-d, 2-d, and 3-d networks with periodic boundary conditions, the 1000 highest entropy CA RNGs were selected from the set of 65,536 possible uniform (all CA truth tables the same) implementations. Each set of 1000 high-entropy CA was then submitted to Marsaglia?s DIEHARD suite of random number tests. A number of 64-bit, neighbor-of-four CA-based RNGs have been discovered that pass all tests in DIEHARD without resorting to either site spacing or time spacing to improve the RNG quality.