ML and Near-ML Decoding of LDPC Codes Over the BEC: Bounds and Decoding Algorithms

The performance of maximum-likelihood (ML) decoding on the binary erasure channel for finite-length low-density paritycheck (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.