Design of Non-Binary Quasi-Cyclic LDPC Codes by ACE Optimization

An algorithm for constructing Tanner graphs of non-binary irregular quasi-cyclic LDPC codes is introduced. It employs a new method for selection of edge labels allowing control over the code's non-binary ACE spectrum and resulting in low error-floor. The efficiency of the algorithm is demonstrated by generating good codes of short to moderate length over small fields, outperforming codes generated by the known methods.Comment: Accepted to 2013 IEEE Information Theory Worksho

Design of Finite-Length Irregular Protograph Codes with Low Error Floors over the Binary-Input AWGN Channel Using Cyclic Liftings

We propose a technique to design finite-length irregular low-density parity-check (LDPC) codes over the binary-input additive white Gaussian noise (AWGN) channel with good performance in both the waterfall and the error floor region. The design process starts from a protograph which embodies a desirable degree distribution. This protograph is then lifted cyclically to a certain block length of interest. The lift is designed carefully to satisfy a certain approximate cycle extrinsic message degree (ACE) spectrum. The target ACE spectrum is one with extremal properties, implying a good error floor performance for the designed code. The proposed construction results in quasi-cyclic codes which are attractive in practice due to simple encoder and decoder implementation. Simulation results are provided to demonstrate the effectiveness of the proposed construction in comparison with similar existing constructions.Comment: Submitted to IEEE Trans. Communication

On generalized LDPC codes for ultra reliable communication

Ultra reliable low latency communication (URLLC) is an important feature in future mobile communication systems, as they will require high data rates, large system capacity and massive device connectivity [11]. To meet such stringent requirements, many error-correction codes (ECC)s are being investigated; turbo codes, low density parity check (LDPC) codes, polar codes and convolutional codes [70, 92, 38], among many others. In this work, we present generalized low density parity check (GLDPC) codes as a promising candidate for URLLC. Our proposal is based on a novel class of GLDPC code ensembles, for which new analysis tools are proposed. We analyze the trade-o_ between coding rate and asymptotic performance of a class of GLDPC codes constructed by including a certain fraction of generalized constraint (GC) nodes in the graph. To incorporate both bounded distance (BD) and maximum likelihood (ML) decoding at GC nodes into our analysis without resorting to multi-edge type of degree distribution (DD)s, we propose the probabilistic peeling decoding (P-PD) algorithm, which models the decoding step at every GC node as an instance of a Bernoulli random variable with a successful decoding probability that depends on both the GC block code as well as its decoding algorithm. The P-PD asymptotic performance over the BEC can be efficiently predicted using standard techniques for LDPC codes such as Density evolution (DE) or the differential equation method. We demonstrate that the simulated P-PD performance accurately predicts the actual performance of the GLPDC code under ML decoding at GC nodes. We illustrate our analysis for GLDPC code ensembles with regular and irregular DDs. This design methodology is applied to construct practical codes for URLLC. To this end, we incorporate to our analysis the use of quasi-cyclic (QC) structures, to mitigate the code error floor and facilitate the code very large scale integration (VLSI) implementation. Furthermore, for the additive white Gaussian noise (AWGN) channel, we analyze the complexity and performance of the message passing decoder with various update rules (including standard full-precision sum product and min-sum algorithms) and quantization schemes. The block error rate (BLER) performance of the proposed GLDPC codes, combined with a complementary outer code, is shown to outperform a variety of state-of-the-art codes, for URLLC, including LDPC codes, polar codes, turbo codes and convolutional codes, at similar complexity rates.Programa Oficial de Doctorado en Multimedia y ComunicacionesPresidente: Juan José Murillo Fuentes.- Secretario: Matilde Pilar Sánchez Fernández.- Vocal: Javier Valls Coquilla

Analysis of Quasi-Cyclic LDPC codes under ML decoding over the erasure channel

In this paper, we show that Quasi-Cyclic LDPC codes can efficiently accommodate the hybrid iterative/ML decoding over the binary erasure channel. We demonstrate that the quasi-cyclic structure of the parity-check matrix can be advantageously used in order to significantly reduce the complexity of the ML decoding. This is achieved by a simple row/column permutation that transforms a QC matrix into a pseudo-band form. Based on this approach, we propose a class of QC-LDPC codes with almost ideal error correction performance under the ML decoding, while the required number of row/symbol operations scales as $k\sqrt{k}$, where $k$ is the number of source symbols.Comment: 6 pages, ISITA1

Near-capacity fixed-rate and rateless channel code constructions

Fixed-rate and rateless channel code constructions are designed for satisfying conflicting design tradeoffs, leading to codes that benefit from practical implementations, whilst offering a good bit error ratio (BER) and block error ratio (BLER) performance. More explicitly, two novel low-density parity-check code (LDPC) constructions are proposed; the first construction constitutes a family of quasi-cyclic protograph LDPC codes, which has a Vandermonde-like parity-check matrix (PCM). The second construction constitutes a specific class of protograph LDPC codes, which are termed as multilevel structured (MLS) LDPC codes. These codes possess a PCM construction that allows the coexistence of both pseudo-randomness as well as a structure requiring a reduced memory. More importantly, it is also demonstrated that these benefits accrue without any compromise in the attainable BER/BLER performance. We also present the novel concept of separating multiple users by means of user-specific channel codes, which is referred to as channel code division multiple access (CCDMA), and provide an example based on MLS LDPC codes. In particular, we circumvent the difficulty of having potentially high memory requirements, while ensuring that each user’s bits in the CCDMA system are equally protected. With regards to rateless channel coding, we propose a novel family of codes, which we refer to as reconfigurable rateless codes, that are capable of not only varying their code-rate but also to adaptively modify their encoding/decoding strategy according to the near-instantaneous channel conditions. We demonstrate that the proposed reconfigurable rateless codes are capable of shaping their own degree distribution according to the nearinstantaneous requirements imposed by the channel, but without any explicit channel knowledge at the transmitter. Additionally, a generalised transmit preprocessing aided closed-loop downlink multiple-input multiple-output (MIMO) system is presented, in which both the channel coding components as well as the linear transmit precoder exploit the knowledge of the channel state information (CSI). More explicitly, we embed a rateless code in a MIMO transmit preprocessing scheme, in order to attain near-capacity performance across a wide range of channel signal-to-ratios (SNRs), rather than only at a specific SNR. The performance of our scheme is further enhanced with the aid of a technique, referred to as pilot symbol assisted rateless (PSAR) coding, whereby a predetermined fraction of pilot bits is appropriately interspersed with the original information bits at the channel coding stage, instead of multiplexing pilots at the modulation stage, as in classic pilot symbol assisted modulation (PSAM). We subsequently demonstrate that the PSAR code-aided transmit preprocessing scheme succeeds in gleaning more information from the inserted pilots than the classic PSAM technique, because the pilot bits are not only useful for sounding the channel at the receiver but also beneficial for significantly reducing the computational complexity of the rateless channel decoder

Hierarchical and High-Girth QC LDPC Codes

We present a general approach to designing capacity-approaching high-girth low-density parity-check (LDPC) codes that are friendly to hardware implementation. Our methodology starts by defining a new class of "hierarchical" quasi-cyclic (HQC) LDPC codes that generalizes the structure of quasi-cyclic (QC) LDPC codes. Whereas the parity check matrices of QC LDPC codes are composed of circulant sub-matrices, those of HQC LDPC codes are composed of a hierarchy of circulant sub-matrices that are in turn constructed from circulant sub-matrices, and so on, through some number of levels. We show how to map any class of codes defined using a protograph into a family of HQC LDPC codes. Next, we present a girth-maximizing algorithm that optimizes the degrees of freedom within the family of codes to yield a high-girth HQC LDPC code. Finally, we discuss how certain characteristics of a code protograph will lead to inevitable short cycles, and show that these short cycles can be eliminated using a "squashing" procedure that results in a high-girth QC LDPC code, although not a hierarchical one. We illustrate our approach with designed examples of girth-10 QC LDPC codes obtained from protographs of one-sided spatially-coupled codes.Comment: Submitted to IEEE Transactions on Information THeor

New Protograph-Based Construction of GLDPC Codes for Binary Erasure Channel and LDPC Codes for Block Fading Channel

학위논문(박사) -- 서울대학교대학원 : 공과대학 전기·정보공학부, 2022.2. 노종선 교수님.이 학위 논문에서는 다음 두 가지의 연구가 이루어졌다: i) 이진 소실 채널에서 새로운 구조의 프로토그래프 기반 generalized low-density parity-check (GLDPC) 부호의 설계 방법 ii) 블록 페이딩 채널을 위한 프로토그래프 기반의 LDPC 부호 설계. 첫 번째로, 이진 소실 채널에서 새롭게 제안된 부분적 도핑 기법을 이용한 프로토그래프 기반의 GLDPC 부호가 제안되었다. 기존의 프로토그래프 기반의 GLDPC 부호의 경우 프로토그래프 영역에서 single parity-check (SPC) 노드를 generalized constraint (GC) 노드로 치환(도핑)하는 형태로 부호가 설계되어 여러 변수 노드 걸쳐 GC 노드가 연결되는 형태를 가진다. 반면, 제안된 부분적 도핑 기법은 한 개의 변수 노드에 GC 노드를 연결하도록 만들 수 있다. 바꿔 말하면, 제안된 부분적 도핑 기법은 더 세밀한 도핑이 가능해서 결과적으로 부호 설계에 있어 높은 자유도를 가지고 더 세련된 부호 최적화가 가능하다. 본 학위 논문에서는 부분적 도핑과 PEXIT 분석을 이용하여 partially doped GLDPC (PD-GLDPC) 부호를 설계하고 최적화 하였다. 더불어, PD-GLDPC 부호의 일반적 최소 거리를 가지는 조건을 제시하였고 이를 이 론적으로 증명하였다. 결과적으로, 제안된 PD-GLDPC 부호는 현존하는 GLDPC 부호의 성능보다 유의미하게 워터플 성능이 좋았고 동시에 오류 마루가 없었다. 마지막으로, 최적화된 PD-GLDPC 부호는 현존하는 최신 블록 LDPC 부호들에 근접한 성능을 가짐을 보여주었다. 두 번째로, 블록 페이딩 (BF) 채널에서 resolvable block design (RBD)를 이용한 프로토그래프 LDPC 부호 설계가 이루어졌다. 제안된 부호의 성능을 확인하기 위한 비트 오류율의 상한을 감마 진화라는 제안된 기법을 이용해 유도하였다. 또한, 시뮬레이션을 통해 유도된 오류율 상한과 부호의 프레임 오류율이 높은 SNR 영역에서 채널 outage 확률에 근접함을 알 수 있다.In this dissertation, two main contributions are given as: i) new construction methods for protograph-based generalized low-density parity-check (GLDPC) codes for the binary erasure channel using partial doping technique and ii) new design of protograph-based low-density parity-check (LDPC) codes for the block fading channel using resolvable block design. First, a new code design technique, called partial doping, for protograph-based GLDPC codes is proposed. While the conventional construction method of protograph-based GLDPC codes is to replace some single parity-check (SPC) nodes with generalized constraint (GC) nodes applying to multiple connected variable nodes (VNs) in the protograph, the proposed technique of partial doping can select any number of partial VNs in the protograph to be protected by GC nodes. In other words, the partial doping technique enables finer tuning of doping, which gives higher degrees of freedom in the code design and enables a sophisticated code optimization. The proposed partially doped GLDPC (PD-GLDPC) codes are constructed using the partial doping technique and optimized by the protograph extrinsic information transfer (PEXIT) analysis. In addition, the condition guaranteeing the linear minimum distance growth of the PD-GLDPC codes is proposed and analytically proven so that the PD-GLDPC code ensembles satisfying this condition have the typical minimum distance. Consequently, the proposed PD-GLDPC codes outperform the conventional GLDPC codes with a notable improvement in the waterfall performance and without the error floor phenomenon. Finally, the PD-GLDPC codes are shown to achieve a competitive performance compared to the existing state-of-the-art block LDPC codes. Second, the protograph-based LDPC codes constructed from resolvable balanced incomplete block design (RBIBD) are designed and proposed for block fading (BF) channel. In order to analyze the performance of the proposed LDPC codes, the upper bounds on bit error rate (BER) using the novel method called gamma evolution are derived. In addition, the numerical analysis shows that the upper bound and the frame error rate (FER) of the proposed LDPC codes approach the channel outage probability in a finite signal-to-noise ratio (SNR) region.1 INTRODUCTION 1 1.1 Background 1 1.2 Overview of Dissertation 3 2 Overview of LDPC Codes 5 2.1 LDPC Codes 5 2.2 Decoding of LDPC Codes in the BEC 7 2.3 Analysis tool for LDPC Codes 8 2.3.1 Density Evolution 8 2.4 Protograph-Based LDPC Codes 9 3 Construction of Protograph-Based Partially Doped Generalized LDPC Codes 11 3.1 Code Structure of Protograph-Based GLDPC Ensembles 14 3.1.1 Construction of Protograph Doped GLDPC Codes 14 3.1.2 PEXIT Analysis and Decoding Process of Protograph Doped GLDPC Codes 15 3.2 The Proposed PD-GLDPC Codes 18 3.2.1 Construction Method of PD-GLDPC Codes 18 3.2.2 PEXIT Analysis of PD-GLDPC Codes 22 3.2.3 Condition for the Existence of the Typical Minimum Distance of the PD-GLDPC Code Ensemble 23 3.2.4 Comparison between Proposed PD-GLDPC Codes and Protograph Doped GLDPC Codes 25 3.3 Optimization of PD-GLDPC Codes 26 3.3.1 Construction of PD-GLDPC Codes from Regular Protographs 26 3.3.2 Differential Evolution-Based Code Construction from the Degree Distribution of Random LDPC Code Ensembles 28 3.3.3 Optimization of PD-GLDPC Codes Using Protograph Differential Evolution 32 3.4 Numerical Results and Analysis 36 3.4.1 Simulation Result for Optimized PD-GLDPC Code from Regular and Irregular Random LDPC Code Ensembles 36 3.4.2 Simulation Result for PD-GLDPC Code from Optimized Protograph 43 3.5 Proof of Theorem 1: The Constraint for the Existence of the Typical Minimum Distance of the Proposed Protograph-Based PD-GLDPC Codes 45 4 Design of Protograph-Based LDPC Code Using Resolvable Block Design for Block Fading Channel 52 4.1 Problem Formulation 53 4.1.1 BF Channel Model 53 4.1.2 Performance Metrics of BF Channel 54 4.1.3 Protograph-Based LDPC Codes and QC LDPC Codes 57 4.2 New Construction of Protograph-Based LDPC Codes from Resolvable Block Designs 57 4.2.1 Resolvable Block Designs 57 4.2.2 Construction of the Proposed Protograph-Based LDPC Codes 59 4.2.3 Theoretical Analysis of the Proposed Protograph-Based LDPC Code from RBD 61 4.2.4 Numerical Analysis of the Proposed Protograph-Based LDPC Code Codes for BF Channel 65 4.2.5 BER Comparison with Analytical Results from Gamma Evolution 65 4.2.6 FER Comparison with Channel Outage Probability 67 5 Conclusions 69 Abstract (In Korean) 78박

Deriving Good LDPC Convolutional Codes from LDPC Block Codes

Low-density parity-check (LDPC) convolutional codes are capable of achieving excellent performance with low encoding and decoding complexity. In this paper we discuss several graph-cover-based methods for deriving families of time-invariant and time-varying LDPC convolutional codes from LDPC block codes and show how earlier proposed LDPC convolutional code constructions can be presented within this framework. Some of the constructed convolutional codes significantly outperform the underlying LDPC block codes. We investigate some possible reasons for this "convolutional gain," and we also discuss the --- mostly moderate --- decoder cost increase that is incurred by going from LDPC block to LDPC convolutional codes.Comment: Submitted to IEEE Transactions on Information Theory, April 2010; revised August 2010, revised November 2010 (essentially final version). (Besides many small changes, the first and second revised versions contain corrected entries in Tables I and II.