New sequences of capacity achieving LDPC code ensembles over the binary erasure channel

In this paper, we introduce new sequences (lambdaⁿ, rhoⁿ) of capacity achieving low-density parity-check (LDPC) code ensembles over the binary erasure channel (BEC). These sequences include the existing sequences by Shokrollahi as a special case. For a fixed code rate R, in the set of proposed sequences, Shokrollahipsilas sequences are superior to the rest of the set in that for any given value of n, their threshold is closer to the capacity upper bound 1 - R. For any given delta, 0 < delta < 1 - R, however, there are infinitely many sequences in the set that are superior to Shokrollahipsilas sequences in that for each of them, there exists an integer number n₀, such that for any n > n₀, the sequence (lambdaⁿ, rhoⁿ) requires a smaller maximum variable node degree as well as a smaller number of constituent variable node degrees to achieve a threshold within delta-neighborhood of the capacity upper bound 1 - R. Moreover, we prove that the check-regular subset of the proposed sequences are asymptotically quasi-optimal, i.e., their decoding complexity per iteration increases only logarithmically with the relative increase of the threshold. A stronger result on asymptotic optimality of some of the proposed sequences is also established.