New Sequences of Capacity Achieving LDPC Code Ensembles Over the Binary Erasure Channel

In this paper, new sequences (λn, ρn) of capacity achieving low-density parity-check (LDPC) code ensembles over the binary erasure channel (BEC) is introduced. These sequences include the existing sequences by Shokrollahi et al. as a special case. For a fixed code rate R, in the set of proposed sequences, Shokrollahi´s 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 δ, 0 <;; δ <;; 1 - R, however, there are infinitely many sequences in the set that are superior to Shokrollahi´s sequences in that for each of them, there exists an integer number n₀, such that for any n > n₀, the sequence (λn, ρn) requires a smaller maximum variable node degree as well as a smaller number of constituent variable node degrees to achieve a threshold within δ-neighborhood of the capacity upper bound 1 - R. Moreover, it is proven that the check-regular subset of the proposed sequences are asymptotically quasi-optimal, i.e., their decoding complexity 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.