Spatially Coupled LDPC Codes Constructed from Protographs

In this paper, we construct protograph-based spatially coupled low-density parity-check (SC-LDPC) codes by coupling together a series of L disjoint, or uncoupled, LDPC code Tanner graphs into a single coupled chain. By varying L, we obtain a flexible family of code ensembles with varying rates and frame lengths that can share the same encoding and decoding architecture for arbitrary L. We demonstrate that the resulting codes combine the best features of optimized irregular and regular codes in one design: capacity approaching iterative belief propagation (BP) decoding thresholds and linear growth of minimum distance with block length. In particular, we show that, for sufficiently large L, the BP thresholds on both the binary erasure channel (BEC) and the binary-input additive white Gaussian noise channel (AWGNC) saturate to a particular value significantly better than the BP decoding threshold and numerically indistinguishable from the optimal maximum a-posteriori (MAP) decoding threshold of the uncoupled LDPC code. When all variable nodes in the coupled chain have degree greater than two, asymptotically the error probability converges at least doubly exponentially with decoding iterations and we obtain sequences of asymptotically good LDPC codes with fast convergence rates and BP thresholds close to the Shannon limit. Further, the gap to capacity decreases as the density of the graph increases, opening up a new way to construct capacity achieving codes on memoryless binary-input symmetric-output (MBS) channels with low-complexity BP decoding.Comment: Submitted to the IEEE Transactions on Information Theor

Bias-tailored quantum LDPC codes

Bias-tailoring allows quantum error correction codes to exploit qubit noise asymmetry. Recently, it was shown that a modified form of the surface code, the XZZX code, exhibits considerably improved performance under biased noise. In this work, we demonstrate that quantum low density parity check codes can be similarly bias-tailored. We introduce a bias-tailored lifted product code construction that provides the framework to expand bias-tailoring methods beyond the family of 2D topological codes. We present examples of bias-tailored lifted product codes based on classical quasi-cyclic codes and numerically assess their performance using a belief propagation plus ordered statistics decoder. Our Monte Carlo simulations, performed under asymmetric noise, show that bias-tailored codes achieve several orders of magnitude improvement in their error suppression relative to depolarising noise.Comment: 21 Pages, 13 Figures. Comments welcome

Algebraic Design and Implementation of Protograph Codes using Non-Commuting Permutation Matrices

Random lifts of graphs, or equivalently, random permutation matrices, have been used to construct good families of codes known as protograph codes. An algebraic analog of this approach was recently presented using voltage graphs, and it was shown that many existing algebraic constructions of graph-based codes that use commuting permutation matrices may be seen as special cases of voltage graph codes. Voltage graphs are graphs that have an element of a finite group assigned to each edge, and the assignment determines a specific lift of the graph. In this paper we discuss how assignments of permutation group elements to the edges of a base graph affect the properties of the lifted graph and corresponding codes, and present a construction method of LDPC code ensembles based on noncommuting permutation matrices. We also show encoder and decoder implementations for these codes

Algebraic Design and Implementation of Protograph Codes using Non-Commuting Permutation Matrices

Random lifts of graphs, or equivalently, random permutation matrices, have been used to construct good families of codes known as protograph codes. An algebraic analog of this approach was recently presented using voltage graphs, and it was shown that many existing algebraic constructions of graph-based codes that use commuting permutation matrices may be seen as special cases of voltage graph codes. Voltage graphs are graphs that have an element of a finite group assigned to each edge, and the assignment determines a specific lift of the graph. In this paper we discuss how assignments of permutation group elements to the edges of a base graph affect the properties of the lifted graph and corresponding codes, and present a construction method of LDPC code ensembles based on noncommuting permutation matrices. We also show encoder and decoder implementations for these codes

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