On Universal Properties of Capacity-Approaching LDPC Code Ensembles

Abstract

This paper is focused on the derivation of some universal properties of capacity-approaching lowdensity parity-check (LDPC) code ensembles whose transmission takes place over memoryless binary-input output-symmetric (MBIOS) channels. Properties of the degree distributions, graphical complexity and the number of fundamental cycles in the bipartite graphs are considered via the derivation of informationtheoretic bounds. These bounds are expressed in terms of the target block/ bit error probability and the gap (in rate) to capacity. Most of the bounds are general for any decoding algorithm, and some others are proved under belief propagation (BP) decoding. Proving these bounds under a certain decoding algorithm, validates them automatically also under any sub-optimal decoding algorithm. A proper modification of these bounds makes them universal for the set of all MBIOS channels which exhibit a given capacity. Bounds on the degree distributions and graphical complexity apply to finite-length LDPC codes and to the asymptotic case of an infinite block length. The bounds are compared with capacity-approaching LDPC code ensembles under BP decoding, and they are shown to be informative and are easy to calculate. Finally, some interesting open problems are considered.

Bounds on the number of iterations for turbo-like code ensembles over the binary erasure channel

“…The results in Table I are based on lower bounds and some achievability results which are related to the graphical complexity of various families of code ensembles defined on graphs (see [8], [9], [11], [12]); the results related to the number of iterations are based on the lower bounds introduced here (for rigorous proofs, see [13] families of code ensembles (including LDPC codes, systematic and non-systematic IRA codes, and ARA codes), the number of iterations which are required to achieve a fixed bit erasure probability scales at least like the inverse of the gap between the channel capacity and the design rate of the ensemble. This conclusion holds provided that the fraction of degree-2 variable nodes in the Tanner graph does not vanish as the gap to capacity vanishes (where under mild conditions, this property is satisfied for sequences of capacity-achieving LDPC code ensembles, see [10,Lemma 5]). The behavior of these lower bounds matches well with the experimental results and the conjectures of Khandekar and McEliece [4].…”

Replacing the Soft-Decision FEC Limit Paradigm in the Design of Optical Communication Systems

“…9.5]. The universality property of LDPC codes for binaryinput memoryless channels was initially discussed in [32], [37], later studied in, e.g., [38], [39], and recently for spatially-coupled LDPC codes in [40]. …”

Design of Finite-Length Generalized LDPC Codes

“…In addition, SCNs have more powerful error correcting capability than SPC codes, therefore GLDPC codes generally show better error floor behaviors than LDPC codes. One drawback of GLDPC codes is the so-called "rate loss under iterative decoding", which refers to the fact that GLDPC codes suffer from a certain degree of decoding threshold degradation under iterative decoding [15,16,36]. The rate loss issue is more prominent on sGLDPC codes, which can be addressed by (1) using hGLDPC codes, (2) generalizing GLDPC codes to DGLDPC codes, or (3) efficient puncturing, which will be discussed in Chapter 7.…”

“…In Fig. 4.4, a blowup of the lower right corner of the H 2 matrix of an EE-hGLDPC code using n = [..., 16,8,8,8,4,4,4] is shown: the red dots correspond to "1"s in H 2 matrix. It can be observed that the distribution of red dots is fairly irregular.…”

“…We designed (960, 448) EE-sGLDPC codes with various n's. [64,32,16,8,4,2,1,1] and [32,32,16,16,16,8,4,4]. The maximum number of iterations is set to I max = 50. changes from those with longer encoding time, due to the error correcting capability of Hamming codes.…”