This paper proposes a variation of 2-d cellular automata (CA) adopting a simpler structure than the normal 2-d CA and a unique neighborship characteristic – asymmetric neighborship. The randomness of 2-d CA based on asymmetric neighborship is discussed and compared with 1-d/2-d CA. The results show that they are better than 1-d CA and could compete with conventional 2-d CA under certain array setting, output method, and transition rule. Furthermore, the structures of 2-d CA based on asymmetric neighborship were evolved using some multi-objective genetic algorithm (MOGA). The evolved 2-d CA could pass DIEHARD tests with only 50 cells, which is less than the minimal number of cells (i.e. 55 cells) needed for neighbor-changing 1-d CA to pass DIEHARD. In addition, a refinement procedure to reduce the cost of 2-d CA based on asymmetric neighborship is discussed. The minimal number of cells found is 48 cells for it to pass DIEHARD. The structure of this 48-cell 2-d CA is identical to that of the evolved 10*5 2-d CA except that 2 horizontal cells in the evolved 10*5 2-d CA are removed.