  We introduce new reliability definitions for bit and check nodes. Maximizing
global reliability, which is the sum reliability of all bit nodes, is shown to
be equivalent to minimizing a decoding metric which is closely related to the
maximum likelihood decoding metric. We then propose novel bit-flipping (BF)
decoding algorithms that take into account the check node reliability. Both
hard-decision (HD) and soft-decision (SD) versions are considered. The former
performs better than the conventional BF algorithm and, in most cases, suffers
less than 1 dB performance loss when compared with some well known SD BF
decoders. For one particular code it even outperforms those SD BF decoders. The
performance of the SD version is superior to that of SD BF decoders and is
comparable to or even better than that of the sum-product algorithm (SPA). The
latter is achieved with a complexity much less than that required by the SPA.
