Analysis and design of binary message passing decoders

Binary message passing decoders for low-density parity-check codes are studied by using extrinsic information transfer charts. The channel delivers hard or soft decisions and the variable node decoder performs all computations in the log-likelihood ratio (L-value) domain. A hard decision results in the Gallager B algorithm and examples show that increasing the channel output alphabet to two bits gains more than 1.0 dB in signal to noise ratio when using optimized codes. 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.