Systematic design of low-density parity-check code ensembles for binary erasure channels

We propose a systematic method to design irregular low-density parity-check (LDPC) codes for binary erasure channels (BEC). Compared to the existing methods, which are based on the application of asymptotic analysis tools such as density evolution or Extrinsic Information Transfer (EXIT) charts in an optimization process, the proposed method is much simpler and faster. Through a number of examples, we demonstrate that the codes designed by the proposed method perform very closely to the best codes designed by optimization. An important property of the proposed designs is the flexibility to select the number of constituent variable node degrees P. The proposed designs include existing systematic designs as a special case with P = N - 1, where N is the maximum variable node degree. Compared to the existing systematic designs, for a given rate and a given ¿ > 0, the designed ensembles can have a threshold in ¿-neighborhood of the capacity upper bound with smaller values of P and N. They can also achieve the capacity of the BEC as N, and correspondingly P and the maximum check node degree tend to infinity.