Studies of cellular automata (CA) based random number generators (RNGs) have focused mainly upon symmetrically connected networks with neighborhood sizes of three or five. Popular field programmable gate array configurations feature a four-input (i.e., 16-row) lookup table. Full utilization of the four-input lookup table leads to the potential for asymmetrically connected cellular automata networks with a neighborhood size of four. 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.