  We introduce a new paradigm for finite precision iterative decoding on
low-density parity-check codes over the Binary Symmetric channel. The messages
take values from a finite alphabet, and unlike traditional quantized decoders
which are quantized versions of the Belief propagation (BP) decoder, the
proposed finite alphabet iterative decoders (FAIDs) do not propagate quantized
probabilities or log-likelihoods and the variable node update functions do not
mimic the BP decoder. Rather, the update functions are maps designed using the
knowledge of potentially harmful subgraphs that could be present in a given
code, thereby rendering these decoders capable of outperforming the BP in the
error floor region. On certain column-weight-three codes of practical interest,
we show that there exist 3-bit precision FAIDs that surpass the BP decoder in
the error floor. Hence, FAIDs are able to achieve a superior performance at
much lower complexity. We also provide a methodology for the selection of FAIDs
that is not code-specific, but gives a set of candidate FAIDs containing
potentially good decoders in the error floor region for any column-weight-three
code. We validate the code generality of our methodology by providing
particularly good three-bit precision FAIDs for a variety of codes with
different rates and lengths.
