  Binary message-passing decoders for low-density parity-check (LDPC) codes are
studied by using extrinsic information transfer (EXIT) charts. The channel
delivers hard or soft decisions and the variable node decoder performs all
computations in the L-value domain. A hard decision channel results in the
well-know Gallager B algorithm, and increasing the output alphabet from hard
decisions to two bits yields a gain of more than 1.0 dB in the required signal
to noise ratio when using optimized codes. The code optimization requires
adapting the mixing property of EXIT functions to the case of binary
message-passing decoders. Finally, it is shown that errors on cycles consisting
only of degree two and three variable nodes cannot be corrected and a necessary
and sufficient condition for the existence of a cycle-free subgraph is derived.
