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

Abstract

In this paper, new sequences$(\lambda ^{n},\rho ^{n})$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 Shokrollahias 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$\delta $,$0 < \delta < 1-R$, 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$n_{0}$, such that for any$n > n_{0}$, the sequence$(\lambda ^{n},\rho ^{n})$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, 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.