The asymptotic error floor of LDPC ensembles under BP decoding

We consider communication over binary memoryless symmetric channels using random elements from irregular low density parity check code (LDPC) ensembles, and belief propagation (BP) decoding. Under the assumption that the corresponding Tanner graph includes a non-vanishing fraction of degree 2 variable nodes, we determine the large blocklength behavior of the bit error rate for noise levels below threshold. More precisely, we show that the BP bit error rate is, asymptotically, F/n, where n is the blocklength and F an explicit constant. Surprisingly, this is the same behavior found for maximum likelihood (ML) decoding, implying that the BP and ML error floor are asymptotically equal.