Download or read book Code Representation and Performance of Graph Based Decoding written by Junsheng Han and published by . This book was released on 2008 with total page 155 pages. Available in PDF, EPUB and Kindle. Book excerpt: Key to the success of modern error correcting codes is the effectiveness of message-passing iterative decoding (MPID). Unlike maximum-likelihood (ML) decoding, the performance of MPID depends not only on the code, but on how the code is represented. In particular, the performance of MPID is potentially improved by using a redundant representation. We focus on Tanner graphs and study combinatorial structures therein that help explain the performance disparity among different representations of the same code. Emphasis is placed on the complexity-performance tradeoff, as more and more check nodes are allowed in the graph. Our discussion applies to MPID as well as linear programming decoding (LPD), which we collectively refer to as graph-based decoding. On an erasure channel, it is well-known that the performance of MPID or LPD is determined by stopping sets. Following Schwartz and Vardy, we define the stopping redundancy as the smallest number of check nodes in a Tanner graph such that smallest size of a non-empty stopping set is equal to the minimum Hamming distance of the code. Roughly speaking, stopping redundancy measures the complexity requirement (in number of check nodes) for MPID of a redundant graph representation to achieve performance comparable to ML decoding (up to a constant factor for small channel erasure probability). General upper bounds on stopping redundancy are obtained. One of our main contribution is a new upper bound based on probabilistic analysis, which is shown to be by far the strongest. From this bound, it can be shown, for example, that for a fixed minimum distance, the stopping redundancy grows just linearly with the redundancy (codimension). Specific results on the stopping redundancy of Golay and Reed-Muller codes are also obtained. We show that the stopping redundancy of maximum distance separable (MDS) codes is bounded in between a Turan number and a single-exclusion (SE) number --- a purely combinatorial quantity that we introduce. By studying upper bounds on the SE number, new results on the stopping redundancy of MDS codes are obtained. Schwartz and Vardy conjecture that the stopping redundancy of an MDS code should only depend on its length and minimum distance. Our results provide partial confirmation, both exact and asymptotic, to this conjecture. Stopping redundancy can be large for some codes. We observe that significantly fewer checks are needed if a small number of small stopping sets are allowed. These small stopping sets can then be dealt with by ``guessing'' during the iterative decoding process. Correspondingly, the guess-g stopping redundancy is defined and it is shown that the savings in number of required check nodes are potentially significant. Another theoretically interesting question is when MPID of a Tanner graph achieves the same word error rate an ML decoder. This prompts us to define and study ML redundancy. Applicability and possible extensions of the current work to a non-erasure channel are discussed. A framework based on pseudo-codewords is considered and shown to be relevant. However, it is also observed that the polytope characterization of pseudo-codewords is not complete enough to be an accurate indicator of MPID performance. Finally, in a separate piece of work, the probability of undetected error (PUE) for over-extended Reed-Solomon codes is studied through the weight distribution bounds of the code. The resulting PUE expressions are shown to be tight in a well-defined sense.
Download or read book Cryptography and Coding written by Michael Walker and published by Springer. This book was released on 2003-07-31 with total page 323 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Optimizing and Decoding LDPC Codes with Graph based Techniques written by Amir H. Djahanshahi and published by . This book was released on 2010 with total page 117 pages. Available in PDF, EPUB and Kindle. Book excerpt: Low-density parity-check (LDPC) codes have been known for their outstanding error-correction capabilities. With low-complexity decoding algorithms and a near capacity performance, these codes are among the most promising forward error correction schemes. LDPC decoding algorithms are generally sub-optimal and their performance not only depends on the codes, but also on many other factors, such as the code representation. In particular, a given non-binary code can be associated with a number of different field or ring image codes. Additionally, each LDPC code can be described with many different Tanner graphs. Each of these different images and graphs can possibly lead to a different performance when used with iterative decoding algorithms. Consequently, in this dissertation we try to find better representations, i.e., graphs and images, for LDPC codes. We take the first step by analyzing LDPC codes over multiple-input single-output (MISO) channels. In an n_T by 1 MISO system with a modulation of alphabet size 2^M, each group of n_T transmitted symbols are combined and produce one received symbol at the receiver. As a result, we consider the LDPC-coded MISO system as an LDPC code over a 2^{M n_T}-ary alphabet. We introduce a modified Tanner graph to represent MISO-LDPC systems and merge the MISO symbol detection and binary LDPC decoding steps into a single message passing decoding algorithm. We present an efficient implementation for belief propagation decoding that significantly reduces the decoding complexity. With numerical simulations, we show that belief propagation decoding over modified graphs outperforms the conventional decoding algorithm for short length LDPC codes over unknown channels. Subsequently, we continue by studying images of non-binary LDPC codes. The high complexity of belief propagation decoding has been proven to be a detrimental factor for these codes. Thereby, we suggest employing lower complexity decoding algorithms over image codes instead. We introduce three classes of binary image codes for a given non-binary code, namely: basic, mixed, and extended binary image codes. We establish upper and lower bounds on the minimum distance of these binary image codes, and present two techniques to find binary image codes with better performance under belief propagation decoding algorithm. In particular, we present a greedy algorithm to find optimized binary image codes. We then proceed by investigation of the ring image codes. Specifically, we introduce matrix-ring-image codes for a given non-binary code. We derive a belief propagation decoding algorithm for these codes, and with numerical simulations, we demonstrate that the low-complexity belief propagation decoding of optimized image codes has a performance very close to the high complexity BP decoding of the original non-binary code. Finally, in a separate study, we investigate the performance of iterative decoders over binary erasure channels. In particular, we present a novel approach to evaluate the inherent unequal error protection properties of irregular LDPC codes over binary erasure channels. Exploiting the finite length scaling methodology, that has been used to study the average bit error rate of finite-length LDPC codes, we introduce a scaling approach to approximate the bit erasure rates in the waterfall region of variable nodes with different degrees. Comparing the bit erasure rates obtained from Monte Carlo simulation with the proposed scaling approximations, we demonstrate that the scaling approach provides a close approximation for a wide range of code lengths. In view of the complexity associated with the numerical evaluation of the scaling approximation, we also derive simpler upper and lower bounds and demonstrate through numerical simulations that these bounds are very close to the scaling approximation.
Download or read book Graph Representation Learning written by William L. William L. Hamilton and published by Springer Nature. This book was released on 2022-06-01 with total page 141 pages. Available in PDF, EPUB and Kindle. Book excerpt: Graph-structured data is ubiquitous throughout the natural and social sciences, from telecommunication networks to quantum chemistry. Building relational inductive biases into deep learning architectures is crucial for creating systems that can learn, reason, and generalize from this kind of data. Recent years have seen a surge in research on graph representation learning, including techniques for deep graph embeddings, generalizations of convolutional neural networks to graph-structured data, and neural message-passing approaches inspired by belief propagation. These advances in graph representation learning have led to new state-of-the-art results in numerous domains, including chemical synthesis, 3D vision, recommender systems, question answering, and social network analysis. This book provides a synthesis and overview of graph representation learning. It begins with a discussion of the goals of graph representation learning as well as key methodological foundations in graph theory and network analysis. Following this, the book introduces and reviews methods for learning node embeddings, including random-walk-based methods and applications to knowledge graphs. It then provides a technical synthesis and introduction to the highly successful graph neural network (GNN) formalism, which has become a dominant and fast-growing paradigm for deep learning with graph data. The book concludes with a synthesis of recent advancements in deep generative models for graphs—a nascent but quickly growing subset of graph representation learning.
Download or read book Turbo like Codes written by Aliazam Abbasfar and published by Springer. This book was released on 2010-10-19 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces turbo error correcting concept in a simple language, including a general theory and the algorithms for decoding turbo-like code. It presents a unified framework for the design and analysis of turbo codes and LDPC codes and their decoding algorithms. A major focus is on high speed turbo decoding, which targets applications with data rates of several hundred million bits per second (Mbps).
Download or read book Fundamentals of Codes Graphs and Iterative Decoding written by Stephen B. Wicker and published by Springer Science & Business Media. This book was released on 2006-04-18 with total page 241 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fundamentals of Codes, Graphs, and Iterative Decoding is an explanation of how to introduce local connectivity, and how to exploit simple structural descriptions. Chapter 1 provides an overview of Shannon theory and the basic tools of complexity theory, communication theory, and bounds on code construction. Chapters 2 - 4 provide an overview of "classical" error control coding, with an introduction to abstract algebra, and block and convolutional codes. Chapters 5 - 9 then proceed to systematically develop the key research results of the 1990s and early 2000s with an introduction to graph theory, followed by chapters on algorithms on graphs, turbo error control, low density parity check codes, and low density generator codes.
Download or read book Constrained Coding and Soft Iterative Decoding written by John L. Fan and published by Springer Science & Business Media. This book was released on 2001-08-31 with total page 284 pages. Available in PDF, EPUB and Kindle. Book excerpt: Constrained Coding and Soft Iterative Decoding is the first work to combine the issues of constrained coding and soft iterative decoding (e.g., turbo and LDPC codes) from a unified point of view. Since constrained coding is widely used in magnetic and optical storage, it is necessary to use some special techniques (modified concatenation scheme or bit insertion) in order to apply soft iterative decoding. Recent breakthroughs in the design and decoding of error-control codes (ECCs) show significant potential for improving the performance of many communications systems. ECCs such as turbo codes and low-density parity check (LDPC) codes can be represented by graphs and decoded by passing probabilistic (a.k.a. `soft') messages along the edges of the graph. This message-passing algorithm yields powerful decoders whose performance can approach the theoretical limits on capacity. This exposition uses `normal graphs,' introduced by Forney, which extend in a natural manner to block diagram representations of the system and provide a simple unified framework for the decoding of ECCs, constrained codes, and channels with memory. Soft iterative decoding is illustrated by the application of turbo codes and LDPC codes to magnetic recording channels. For magnetic and optical storage, an issue arises in the use of constrained coding, which places restrictions on the sequences that can be transmitted through the channel; the use of constrained coding in combination with soft ECC decoders is addressed by the modified concatenation scheme also known as `reverse concatenation.' Moreover, a soft constraint decoder yields additional coding gain from the redundancy in the constraint, which may be of practical interest in the case of optical storage. In addition, this monograph presents several other research results (including the design of sliding-block lossless compression codes, and the decoding of array codes as LDPC codes). Constrained Coding and Soft Iterative Decoding will prove useful to students, researchers and professional engineers who are interested in understanding this new soft iterative decoding paradigm and applying it in communications and storage systems.
Download or read book Decoding Linear Codes Via Optimization and Graph based Techniques written by Mohammad H. Taghavi and published by . This book was released on 2008 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt: Low-density parity-check (LDPC) codes have made it possible to communicate at information rates very close to the Shannon capacity by combining sparsity with quasi-randomness, which enables the use of low-complexity iterative message-passing (IMP) decoders. So far, most systematic studies of IMP decoders have focused on evaluating the average performance of random ensembles of LDPC codes with infinite length. However, the statistical nature of IMP algorithms does not seem very suitable for rigorous analysis the decoding of individual finite-length codes. The need for finite-length studies are most critical in applications such as data storage, where the required decoding error rate is too low to be verifiable by simulation. As an alternative to IMP algorithms, linear programming (LP) decoding is based on relaxing the optimal decoding into a linear optimization. The geometric nature of this approach makes it more amenable to deterministic finite-length analysis than IMP decoding. On the other hand, LP decoding is computationally more complex than IMP decoding, due to both the large number of constraints in the relaxed problem, and the inefficiency of using general-purpose LP solvers. In this dissertation, we study several aspects of LP decoding, starting by some steps toward reducing its complexity. We introduce an adaptive implementation of LP decoding, where the relaxed problem is replaced by a sequence of subproblems of much smaller size, resulting in a complexity reduction by orders of magnitude. This is followed by a sparse implementation of an interior-point LP solver which exploits the structure of the decoding problem. We further propose a cutting-plane approach to improve the error-correcting capability of LP decoding. Along the way, several properties are proved for LP decoding and its proposed variations. We continue by investigating the application of an optimization-based approach to decoding linear codes in the presence of intersymbol interference (ISI). By relaxing the optimal detection problem into a linear program, we derive a new graphical representation for the ISI channel, which can be used for combined equalization and decoding by LP or IMP decoders. Finally, in a separate piece of work, we study the effect of nonlinearities on the multiuser capacity of optical fibers.
Download or read book On Linear Programming Based Decoding of Graph based Codes written by Idan Goldenberg and published by . This book was released on 2013 with total page 127 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Cryptography and Coding written by and published by . This book was released on 1999 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Tractability written by Lucas Bordeaux and published by Cambridge University Press. This book was released on 2014-02-06 with total page 401 pages. Available in PDF, EPUB and Kindle. Book excerpt: Classical computer science textbooks tell us that some problems are 'hard'. Yet many areas, from machine learning and computer vision to theorem proving and software verification, have defined their own set of tools for effectively solving complex problems. Tractability provides an overview of these different techniques, and of the fundamental concepts and properties used to tame intractability. This book will help you understand what to do when facing a hard computational problem. Can the problem be modelled by convex, or submodular functions? Will the instances arising in practice be of low treewidth, or exhibit another specific graph structure that makes them easy? Is it acceptable to use scalable, but approximate algorithms? A wide range of approaches is presented through self-contained chapters written by authoritative researchers on each topic. As a reference on a core problem in computer science, this book will appeal to theoreticians and practitioners alike.
Download or read book Channel Coding Theory Algorithms and Applications written by and published by Academic Press. This book was released on 2014-07-29 with total page 687 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives a review of the principles, methods and techniques of important and emerging research topics and technologies in Channel Coding, including theory, algorithms, and applications. Edited by leading people in the field who, through their reputation, have been able to commission experts to write on a particular topic. With this reference source you will: Quickly grasp a new area of research Understand the underlying principles of a topic and its applications Ascertain how a topic relates to other areas and learn of the research issues yet to be resolved Quick tutorial reviews of important and emerging topics of research in Channel Coding Presents core principles in Channel Coding theory and shows their applications Reference content on core principles, technologies, algorithms and applications Comprehensive references to journal articles and other literature on which to build further, more specific and detailed knowledge
Download or read book Studies on Graph based Coding Systems written by Jing Sun and published by . This book was released on 2004 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: To make full use of the valuable radio spectrum, one of the targets of communications system design is to convey as much information as possible through the spectrum (the channel) allocated for the purpose. For a given channel, the amount of information that can be passed through it is upper bounded by the well-known Shannon channel capacity. The invention of turbo codes in 1993 was a key step in the 50-year effort to design good coding schemes achieving the Shannon capacity. Since then, other coding schemes with similar performance, such as Low Density Parity Check (LDPC) codes and turbo product codes, have been re-discovered or invented. The common characteristics of these codes are that they all can be represented by a large (pseudo- )random graph, and iteratively decoded. In this dissertation, we treat three topics in the design and analysis of the two most important graph-based coding schemes: turbo codes and LDPC codes. Together with two component convolutional codes, an interleaver is a key component of a turbo code. We introduce a class of deterministic interleavers for turbo codes based on permutation polynomials over Z (sub)N . It is observed that the performance of a turbo code using these permutation polynomial-based interleavers is usually dominated by a subset of input weight 2m error events. Due to the structure of these interleavers, we derive a simple method to find the weight spectrum of those error events. Therefore good permutation polynomials can be searched for a given component code to achieve better performance. LDPC codes can be constructed using an interleaver. In a previous work, the use of maximum length linear congruential sequences (MLLCS) has been proposed for the construction of interleavers for regular LDPC codes with data node degree 3. Since the smallest loop size (girth) is a key characteristic of the graph of the LDPC code, a sufficient condition on the parameters of the MLLCS to generate a graph with girth larger than 4 is given. We extend the sufficient condition to general irregular LDPC codes and also provide sufficient conditions to guarantee even larger girth. It is observed that the error floor of LDPC code (bit error performance at high signal-to-noise ratios) is usually caused by trapping sets, which are sets of data nodes that cannot be corrected by the iterative decoder. We develop an approximated linear system model for the iterative decoding process in a trapping set. Then the probability that the trapping set can be corrected can be estimated by observing the response of the linear system. Using the idea from the analysis of the linear system, the iterative decoder for regular LDPC codes can be slightly modified to greatly decrease the error floor.
Download or read book Design and Analysis of Graph based Codes Using Algebraic Lifts and Decoding Networks written by Allison Beemer and published by . This book was released on 2018 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Error-correcting codes seek to address the problem of transmitting information efficiently and reliably across noisy channels. Among the most competitive codes developed in the last 70 years are low-density parity-check (LDPC) codes, a class of codes whose structure may be represented by sparse bipartite graphs. In addition to having the potential to be capacity-approaching, LDPC codes offer the significant practical advantage of low-complexity graph-based decoding algorithms. Graphical substructures called trapping sets, absorbing sets, and stopping sets characterize failure of these algorithms at high signal-to-noise ratios.
Download or read book Cryptography and Coding written by Matthew G. Parker and published by Springer. This book was released on 2009-12-02 with total page 505 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 12th IMA International Conference on Cryptography and Coding, held in Cirencester, UK in December 2009. The 26 revised full papers presented together with 3 invited contributions were carefully reviewed and selected from 53 submissions. The papers are organized in topical sections on coding theory, symmetric cryptography, security protocols, asymmetric cryptography, Boolean functions and side channels and implementations.
Download or read book Design Techniques for Graph based Error correcting Codes and Their Applications written by Ching Fu Lan and published by . This book was released on 2006 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In Shannon's seminal paper, "A Mathematical Theory of Communication", he defined "Channel Capacity" which predicted the ultimate performance that transmission systems can achieve and suggested that capacity is achievable by error-correcting (channel) coding. The main idea of error-correcting codes is to add redundancy to the information to be transmitted so that the receiver can explore the correlation between transmitted information and redundancy and correct or detect errors caused by channels afterward. The discovery of turbo codes and rediscovery of Low Density Parity Check codes (LDPC) have revived the research in channel coding with novel ideas and techniques on code concatenation, iterative decoding, graph-based construction and design based on density evolution. This dissertation focuses on the design aspect of graph-based channel codes such as LDPC and Irregular Repeat Accumulate (IRA) codes via density evolution, and use the technique (density evolution) to design IRA codes for scalable image/video communication and LDPC codes for distributed source coding, which can be considered as a channel coding problem. The first part of the dissertation includes design and analysis of rate-compatible IRA codes for scalable image transmission systems. This part presents the analysis with density evolution the effect of puncturing applied to IRA codes and the asymptotic analysis of the performance of the systems. In the second part of the dissertation, we consider designing source-optimized IRA codes. The idea is to take advantage of the capability of Unequal Error Protection (UEP) of IRA codes against errors because of their irregularities. In video and image transmission systems, the performance is measured by Peak Signal to Noise Ratio (PSNR). We propose an approach to design IRA codes optimized for such a criterion. In the third part of the dissertation, we investigate Slepian-Wolf coding problem using LDPC codes. The problems to be addressed include coding problem involving multiple sources and non-binary sources, and coding using multi-level codes and nonbinary codes.
Download or read book Decoding Error correcting Codes Via Linear Programming written by Jon Feldman and published by . This book was released on 2003 with total page 151 pages. Available in PDF, EPUB and Kindle. Book excerpt: (Cont.) Our decoder is particularly attractive for analysis of these codes because the standard message-passing algorithms used for decoding are often difficult to analyze. For turbo codes, we give a relaxation very close to min-cost flow, and show that the success of the decoder depends on the costs in a certain residual graph. For the case of rate-1/2 repeat-accumulate codes (a certain type of turbo code), we give an inverse polynomial upper bound on the probability of decoding failure. For LDPC codes (or any binary linear code), we give a relaxation based on the factor graph representation of the code. We introduce the concept of fractional distance, which is a function of the relaxation, and show that LP decoding always corrects a number of errors up to half the fractional distance. We show that the fractional distance is exponential in the girth of the factor graph. Furthermore, we give an efficient algorithm to compute this fractional distance. We provide experiments showing that the performance of our decoders are comparable to the standard message-passing decoders. We also give new provably convergent message-passing decoders based on linear programming duality that have the ML certificate property.
