Low-Complexity LDPC Codes with Near-Optimum Performance over the BEC

Recent works showed how low-density parity-check (LDPC) erasure correcting codes, under maximum likelihood (ML) decoding, are capable of tightly approaching the performance of an ideal maximum-distance-separable code on the binary erasure channel. Such result is achievable down to low error rates, even for small and moderate block sizes, while keeping the decoding complexity low, thanks to a class of decoding algorithms which exploits the sparseness of the parity-check matrix to reduce the complexity of Gaussian elimination (GE). In this paper the main concepts underlying ML decoding of LDPC codes are recalled. A performance analysis among various LDPC code classes is then carried out, including a comparison with fixed-rate Raptor codes. The results confirm that a judicious LDPC code design allows achieving achieving a near-optimum performance on the erasure channel, with very low error floors. Furthermore, it is shown that LDPC and Raptor codes, under ML decoding, provide almost identical performance in terms of decoding failure probability vs. overhead.