  A new method for low-complexity near-maximum-likelihood (ML) decoding of
low-density parity-check (LDPC) codes over the additive white Gaussian noise
channel is presented. The proposed method termed belief-propagation--list
erasure decoding (BP-LED) is based on erasing carefully chosen unreliable bits
performed in case of BP decoding failure. A strategy of introducing erasures
into the received vector and a new erasure decoding algorithm are proposed. The
new erasure decoding algorithm, called list erasure decoding, combines ML
decoding over the BEC with list decoding applied if the ML decoder fails to
find a unique solution. The asymptotic exponent of the average list size for
random regular LDPC codes from the Gallager ensemble is analyzed. Furthermore,
a few examples of regular and irregular quasi-cyclic LDPC codes of short and
moderate lengths are studied by simulations and their performance is compared
with the upper bound on the LDPC ensemble-average performance and the upper
bound on the average performance of random linear codes under ML decoding. A
comparison with the BP decoding performance of the WiMAX standard codes and
performance of the near-ML BEAST decoding are presented. The new algorithm is
applied to decoding a short nonbinary LDPC code over the extension of the
binary Galois field. The obtained simulation results are compared to the upper
bound on the ensemble-average performance of the binary image of regular
nonbinary LDPC codes.
