The effect of cycles on binary message-passing decoding of LDPC codes

We study the error-floor behavior of binary message-passing decoders for low-density parity-check (LDPC) codes. We find that the stability condition is independent of the quality of the channel messages (i.e. soft or hard decision). Furthermore, we identify a structure in the graph where the involved variable nodes cannot be corrected by the binary message-passing decoder. This leads to an error floor for regular LDPC codes with variable node degree smaller than four. An additional constraint on the degree distribution is derived which allows to avoid this structure, leading to optimized LDPC codes without error floors.