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

Abstract

In this paper, new sequences <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$(\lambda ^{n},\rho ^{n})$</tex> </formula> 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 <etal xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> as a special case. For a fixed code rate <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$R$</tex></formula> , in the set of proposed sequences, Shokrollahi's sequences are superior to the rest of the set in that for any given value of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$n$</tex></formula> , their threshold is closer to the capacity upper bound <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$1- R$</tex></formula> . For any given <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\delta $</tex></formula> , <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$0 &lt; \delta &lt; 1-R$</tex></formula> , 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 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n_{0}$</tex> </formula> , such that for any <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n &gt; n_{0}$</tex></formula> , the sequence <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$(\lambda ^{n},\rho ^{n})$</tex></formula> requires a smaller maximum variable node degree as well as a smaller number of constituent variable node degrees to achieve a threshold within <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\delta $</tex></formula> -neighborhood of the capacity upper bound <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$1-R$</tex></formula> . 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.