Protograph-based Low-density Parity-check Codes Research Articles

Fully Absorbing Set Enumerators for Protograph-Based LDPC Code Ensembles

The average finite-length (elementary) fully absorbing set enumerators for binary protograph-based low-density parity-check (LDPC) code ensembles are derived. The method relies on generating functions, and it provides an efficient mean to obtain the normalized logarithmic asymptotic distribution of (elementary) fully absorbing sets. Using these results, the asymptotic distribution of (elementary) fully absorbing sets is evaluated for some protograph-based LDPC code ensembles. Numerical results show how the distributions can be used to estimate the error floor performance of protograph LDPC codes.

Design of Protograph-Based LDPC Codes for Two-User Gaussian Multiple Access Channel

In this letter, we study the performance of protograph-based low-density parity-check (LDPC) codes for two-user Gaussian multiple access channel (GMAC). Considering the characteristic of a joint protograph of two users, we first derive a joint protograph-based extrinsic information transfer (JP-EXIT) chart to analyze the iterative user decoding behavior of two-user GMAC. Guided by the JP-EXIT, we further propose a new design scheme and construct protograph base matrices of different code rates for two users, respectively. Both theoretical analyses and simulation results show that the proposed protograph-based LDPC codes of low decoding threshold outperform the conventional irregular LDPC codes for both equal-power and unequal-power GMAC with different code rates.

New Unequal Error Protection Strategy for Image Transmission Based on Bilayer-Lengthened PLDPC Code in Half-Duplex Relay System

To reduce the waste of energy in communications, unequal error protection (UEP) is used to provide asymmetric protection for messages with different levels of importance. This paper proposes new efficient strategies of UEP based on bilayer protograph-based low-density parity check (PLDPC) codes in decoding-and-forward (DF) relay systems. In particular, we jointly utilize source coding and channel coding to design UEP strategies and then save transmission energy. According to the different levels of importance of discrete cosine transform (DCT) coefficients of image and variance statistical characteristics of image sub-blocks, bilayer-lengthened PLDPC codes are exploited to protect the transmitted image information with different importance levels at the half-duplex relay system. In the end, the simulation result shows that the proposed UEP schemes achieve excellent performance gains compared to conventional equal error protection (EEP) scheme. Additionally, the complexity analysis of the UEP strategies is given.

Performance Analysis and Optimization of Spatially Coupled Protograph-Based Low-Density Parity-Check Codes for Two-Dimensional Magnetic Recording Systems

As a class of convolutional-based low-density parity-check (LDPC) codes, spatially coupled protograph (SC-P) codes not only can exhibit outstanding error performance but also can realize low-complexity implementation. Motivated by the prominent superiorities, SC-P codes have been intensively investigated in various communication environments. However, no research works regarding SC-P codes in the magnetic recording (MR) scenarios have been done so far. To explore such an issue mentioned previously, this article carries out a comprehensive study on the SC-P codes in the two-dimensional magnetic recording (TDMR) systems. Specifically, a turbo-like extrinsic information transfer (EXIT) algorithm is conceived for the SC-P-coded TDMR systems. With the help of such an algorithm, we evaluate the convergence performance (i.e., decoding thresholds) for conventional SC-P codes over TDMR channels with turbo equalization and illustrate that such codes should be re-constructed to attain better performance. Therefore, we develop an optimization scheme to generate a novel class of SC-P codes, called enhanced SC-P codes, which can exhibit promising decoding thresholds without losing the linear-minimum-distance superiority. The convergence-performance analyses and simulated results reveal that such enhanced SC-P codes can obtain more desirable coding gains in contrast to the conventional SC-P codes and the state-of-the-art protograph codes for the whole signal-to-noise-ratio (SNR) range under study. The excellent performance and simple implementation make the enhanced SC-P codes a very competitive error-correction solution for the next-generation TDMR devices.

On the Tanner Graph Cycle Distribution of Random LDPC, Random Protograph-Based LDPC, and Random Quasi-Cyclic LDPC Code Ensembles

In this paper, we study the cycle distribution of random low-density parity-check (LDPC) codes, randomly constructed protograph-based LDPC codes, and random quasi-cyclic (QC) LDPC codes. We prove that for a random bipartite graph, with a given (irregular) degree distribution, the distributions of cycles of different length tend to independent Poisson distributions, as the size of the graph tends to infinity. We derive asymptotic upper and lower bounds on the expected values of the Poisson distributions that are independent of the size of the graph, and only depend on the degree distribution and the cycle length. For a random lift of a bi-regular protograph, we prove that the asymptotic cycle distributions are essentially the same as those of random bipartite graphs as long as the degree distributions are identical. For random QC-LDPC codes, however, we show that the cycle distribution can be quite different from the other two categories. In particular, depending on the protograph and the value of $c$ , the expected number of cycles of length $c$ , in this case, can be either $\Theta (N)$ or $\Theta (1)$ , where $N$ is the lifting degree (code length). We also provide numerical results that match our theoretical derivations. Our results provide a theoretical foundation for emperical results that were reported in the literature but were not well-justified. They can also be used for the analysis and design of LDPC codes and associated algorithms that are based on cycles.

Designing Protograph-Based LDPC Codes for Iterative Receivers on ${M}$ -ary DCSK Systems

In this brief, protograph-based low density parity check (P-LDPC) codes, which have simple structure and excellent error-correction capability, are designed for iterative receivers (IRs) on M-ary differential chaos shift keying (M-ary DCSK) systems. Due to the non-periodic and random characteristics of chaos carriers and their interaction with soft information, the conventional optimal P-LDPC codes over Bi-AWGN channels in the tandem mode may not perform well in M-ary DCSK systems with IR. With the help of a finite-length extrinsic information transfer (EXIT) algorithm, the convergence of IR is analyzed, by which a new design principle is proposed to construct new P-LDPC codes for the M-ary DCSK systems. Both EXIT analysis and simulated results demonstrate the superiority of the proposed P-LDPC codes.

Nonbinary Protograph-Based LDPC Codes Based on Additive Group of Finite Field

This letter proposes a flexible construction method of nonbinary protograph-based low-density parity-check codes based on finite field additive group operation from node perspective. First, nonzero element labels of nodes are optimized by minimum distance of corresponding binary image of rows of the parity-check matrix. Then, quasi-girth calculating method is proposed to jointly optimize node permutation label so as to improve global code’s minimum distance upper bound. Simulation results illustrate the excellent performance of the proposed codes with adequately selected node’s value label and permuting label.

Protograph-Based LDPC Code Design for Shaped Bit-Metric Decoding

A protograph-based low-density parity-check (LDPC) code design technique for bandwidth-efficient coded modulation with probabilistic shaping is presented. The approach jointly optimizes the LDPC code node degrees and the mapping of the coded bits to the bit-interleaved coded modulation (BICM) bit-channels. For BICM with uniform inputs and for BICM with probabilistic shaping, binary-input symmetric-output surrogate channels for the code design are used. The constructed codes for uniform inputs perform as good as the multi-edge type codes of Zhang and Kschischang (2013). For 8-ASK and 64-ASK with probabilistic shaping, codes of rates 2/3 and 5/6 with blocklength 64800 are designed, which operate within 0.63 and 0.69 dB of $\frac{1}{2}\,\log_2(1+\text{SNR})$ for a target frame error rate of $10^{-3}$ at spectral efficiencies of 1.38 and 4.25 bits/channel use, respectively.

Re-jagged AR4JA LDPC codes and their applications

This work investigated the performance of a newly developed protograph-based low-density parity-check code (LDPC) families belonging to accumulate-repeat-4-jagged-accumulate codes. The proposed code has hierarchical irregularity and quasi-cyclic property that retains it to be suitable for fast encoder and decoder implementation. The code used in this work is a rate-compatible linearly punctured code. The modified iterative decoding algorithm as mentioned in the literature is used along with a newly proposed encoder and simulation has been done for proposed LDPC coded orthogonal frequency division multiplexing (OFDM) link over wireless and optical wireless channels with hard- and soft-decision decoding. The code relates to new methods for designing rate-compatible families of protographs of different rates, all having the linear minimum distance property showing small performance degradation compared with the threshold. The parity-check matrix is composed of small identical modules to reduce the routing complexity of the decoder. The main essence of this code lies in the fact that LDPC codes of arbitrarily large size can be built by expanding from individual protographs or rate-compatible protograph families.

해양 위성 통신을 위한 프로토그래프 기반 블록 저밀도 패리티 검사 부호 설계

본 논문에서는 기존의 해양 위성 방송 통신을 위해 사용되어온 2세대 위성 디지털 방송 표준 (Digital Video Broadcasting Satellite Second Generation : DVB-S2) 의 저밀도 패리티 검사 (Low Density Parity Check : LDPC) 부호 성능 개선과 부호화 및 복호화 복잡도를 줄일 수 있는 프로토그래프 (protograph) 기반 블록 저밀도 패리티 검사 부호를 제안한다. 모의 실험을 통하여 기존의 DVB-S2의 저밀도 패리티 검사 부호 대비 우수한 비트 오류확률 및 프레임오류확률 성능을 갖는 것을 보이며, 또한 부호화 및 복호화 계산 복잡도 분석 및 비교를 통하여 제안하는 부호가 효율적인 부호화 및 복호화 구조를 가짐을 보인다. In this paper, the protograph-based block low density parity check (LDPC) code, which improves the performance and reduces the encoder/decoder complexity than the conventional Digital Video Broadcasting Satellite Second Generation (DVB-S2) LDPC code used for the marine satellite communication, is proposed. The computer simulation results verify that the proposed protograph-based LDPC code has the better performance in both the bit error rate (BER) and the frame error rate (FER) than the conventional DVB-S2 LDPC code. Furthermore, by analyzing the encoding and decoding computational complexity, we show that the protograph-based block LDPC code has the efficient encoder/decoder structure.

Non-Binary Protograph-Based LDPC Codes: Enumerators, Analysis, and Designs

This paper provides a comprehensive analysis of nonbinary low-density parity check (LDPC) codes built out of protographs. We consider both random and constrained edge-weight labeling, and refer to the former as the unconstrained nonbinary protograph-based LDPC codes (U-NBPB codes) and to the latter as the constrained nonbinary protograph-based LDPC codes (C-NBPB codes). Equipped with combinatorial definitions extended to the nonbinary domain, ensemble enumerators of codewords, trapping sets, stopping sets, and pseudocodewords are calculated. The exact enumerators are presented in the finite-length regime, and the corresponding growth rates are calculated in the asymptotic regime. An EXIT chart tool for computing the iterative decoding thresholds of protograph-based LDPC codes is presented, followed by several examples of finite-length U-NBPB and C-NBPB codes with high performance. Throughout this paper, we provide accompanying examples, which demonstrate the advantage of nonbinary protograph-based LDPC codes over their binary counterparts and over random constructions. The results presented in this paper advance the analytical toolbox of nonbinary graph-based codes.

Bounds on the Minimum Distance of Punctured Quasi-Cyclic LDPC Codes

Recent work by Divsalar has shown that properly designed protograph-based low-density parity-check codes typically have minimum (Hamming) distance linearly increasing with block length. This fact rests on ensemble arguments over all possible expansions of the base protograph. However, when implementation complexity is considered, the expansions are frequently selected from a smaller class of structured expansions. For example, protograph expansion by cyclically shifting connections generates a quasi-cyclic (QC) code. Other recent work by Smarandache and Vontobel has provided upper bounds on the minimum distance of QC codes. In this paper, we generalize these bounds to punctured QC codes and then show how to tighten these for certain classes of codes. We then evaluate these upper bounds for the family of protograph codes known as AR4JA codes that have been recommended for use in deep space communications in a standard established by the Consultative Committee for Space Data Systems. At block lengths larger than 4400 bits, these upper bounds fall well below the ensemble lower bounds.

Nonbinary Hybrid LDPC Codes

In this paper, a new class of low-density parity-check (LDPC) codes, named hybrid LDPC codes, is introduced. Hybrid LDPC codes are characterized by an irregular connectivity profile and heterogeneous orders of the symbols in the codeword. It is shown in particular that the class of hybrid LDPC codes can be asymptotically characterized and optimized using density evolution (DE) framework, and a technique to maximize the minimum distance of the code is presented. Numerical assessment of hybrid LDPC code performances is provided, by comparing them to protograph-based and multiedge-type (MET) LDPC codes. Hybrid LDPC codes are shown to allow to achieve an interesting tradeoff between good error-floor performance and good waterfall region with nonbinary coding techniques.