We introduce a generic approach, called FAID diversity, for improving the error correction capability of regular low-density parity check codes, beyond the belief propagation performance. The method relies on operating a set of finite alphabet iterative decoders (FAID). The message-passing update rules are interpreted as discrete dynamical systems, and are judiciously chosen to ensure that decoders have different dynamics on a specific finite-length code. An algorithm is proposed which uses random jumps in the iterative message passing trajectories, such that the system is not trapped in periodic attractors. We show by simulations that the FAID diversity approach with random jumps has the potential of approaching the performance of maximum-likelihood decoding for finite-length regular, column-weight three codes.