  The performance of maximum-likelihood (ML) decoding on the binary erasure
channel for finite-length low-density parity-check (LDPC) codes from two random
ensembles is studied. The theoretical average spectrum of the Gallager ensemble
is computed by using a recurrent procedure and compared to the empirically
found average spectrum for the same ensemble as well as to the empirical
average spectrum of the Richardson-Urbanke ensemble and spectra of selected
codes from both ensembles. Distance properties of the random codes from the
Gallager ensemble are discussed. A tightened union-type upper bound on the ML
decoding error probability based on the precise coefficients of the average
spectrum is presented. A new upper bound on the ML decoding performance of LDPC
codes from the Gallager ensemble based on computing the rank of submatrices of
the code parity-check matrix is derived. A new low-complexity near-ML decoding
algorithm for quasi-cyclic LDPC codes is proposed and simulated. Its
performance is compared to the upper bounds on the ML decoding performance.
