Download or read book Low Density Parity Check Codes Based on Finite Geometries written by National Aeronautics and Space Adm Nasa and published by . This book was released on 2018-09-15 with total page 36 pages. Available in PDF, EPUB and Kindle. Book excerpt: Low density parity check (LDPC) codes with iterative decoding based on belief propagation achieve astonishing error performance close to Shannon limit. No algebraic or geometric method for constructing these codes has been reported and they are largely generated by computer search. As a result, encoding of long LDPC codes is in general very complex. This paper presents two classes of high rate LDPC codes whose constructions are based on finite Euclidean and projective geometries, respectively. These classes of codes a.re cyclic and have good constraint parameters and minimum distances. Cyclic structure adows the use of linear feedback shift registers for encoding. These finite geometry LDPC codes achieve very good error performance with either soft-decision iterative decoding based on belief propagation or Gallager's hard-decision bit flipping algorithm. These codes can be punctured or extended to obtain other good LDPC codes. A generalization of these codes is also presented.Kou, Yu and Lin, Shu and Fossorier, MarcGoddard Space Flight CenterEUCLIDEAN GEOMETRY; ALGORITHMS; DECODING; PARITY; ALGEBRA; INFORMATION THEORY; PROJECTIVE GEOMETRY; TWO DIMENSIONAL MODELS; COMPUTERIZED SIMULATION; ERRORS; BLOCK DIAGRAMS...

Download or read book Finite Geometry Low Density Parity Check Codes written by Yu Kou and published by . This book was released on 2001 with total page 366 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Aspects of Low density Parity check Codes Based on Finite Geometries written by Elvira Chkoulepova and published by . This book was released on 2004 with total page 61 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Construction and Decoding of Codes on Finite Fields and Finite Geometries written by Li Zhang and published by . This book was released on 2010 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In this doctoral dissertation, two constructions of binary low-density parity-check (LDPC) codes with quasi-cyclic (QC) structures are presented. A general construction of RC-constrained arrays of circulant permutation matrices is introduced, then two specific construction methods based on Latin squares and cyclic subgroups are presented. Array masking is also proposed to improve the waterfall-region performance of the QC-LDPC codes. Also, by analyzing the parity check matrices of these codes, combinatorial expressions for their ranks and dimensions are derived. Experimental results show that, with iterative decoding algorithms, the constructed codes perform very well over both the additive white Gaussian noise (AWGN) and the binary erasure channels (BEC). Also presented in this dissertation are constructions of QC-LDPC codes based on two special classes of balanced incomplete block designs (BIBDs) derived by Bose. Codes are constructed for both the AWGN channel and the binary burst erasure channel (BBEC). Experimental results show that the codes constructed perform well not only over these two types of channels but also over the BEC. Finally, a two stage iterative decoding is presented to decode a class of cyclic Euclidean geometry codes. By exploiting the inherent geometry structure of the codes and avoiding the degrading effect of short cycles, the proposed algorithm provides good decoding performance of the codes.

Download or read book LDPC Codes from Semipartial Geometries written by James Rhys Harwood Hutton and published by . This book was released on 2011 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: A binary low-density parity-check (LDPC) code is a linear block code that is defined by a sparse parity-check matrix H, that is H has a low density of 1's. LDPC codes were originally presented by Gallager in his doctoral dissertation [9], but largely overlooked for the next 35 years. A notable exception was [29], in which Tanner introduced a graphical representation for LDPC codes, now known as Tanner graphs. However, interest in these codes has greatly increased since 1996 with the publication of [22] and other papers, since it has been realised that LDPC codes are capable of achieving near-optimal performance when decoded using iterative decoding algorithms. LDPC codes can be constructed randomly by using a computer algorithm to generate a suitable matrix H. However, it is also possible to construct LDPC codes explicitly using various incidence structures in discrete mathematics. For example, LDPC codes can be constructed based on the points and lines of finite geometries: there are many examples in the literature (see for example [18, 28]). These constructed codes can possess certain advantages over randomly-generated codes. For example they may provide more efficient encoding algorithms than randomly-generated codes. Furthermore it can be easier to understand and determine the properties of such codes because of the underlying structure. LDPC codes have been constructed based on incidence structures known as partial geometries [16]. The aim of this research is to provide examples of new codes constructed based on structures known as semipartial geometries (SPGs), which are generalisations of partial geometries. Since the commencement of this thesis [19] was published, which showed that codes could be constructed from semipartial geometries and provided some examples and basic results. By necessity this thesis contains a number of results from that paper. However, it should be noted that the scope of [19] is fairly limited and that the overlap between the current thesis and [19] is consequently small. [19] also contains a number of errors, some of which have been noted and corrected in this thesis.

Download or read book Construction of Structured Low density Parity check Codes written by Lei Chen and published by . This book was released on 2005 with total page 340 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Error Correction Coding and Decoding written by Martin Tomlinson and published by Springer. This book was released on 2017-02-21 with total page 527 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book discusses both the theory and practical applications of self-correcting data, commonly known as error-correcting codes. The applications included demonstrate the importance of these codes in a wide range of everyday technologies, from smartphones to secure communications and transactions. Written in a readily understandable style, the book presents the authors’ twenty-five years of research organized into five parts: Part I is concerned with the theoretical performance attainable by using error correcting codes to achieve communications efficiency in digital communications systems. Part II explores the construction of error-correcting codes and explains the different families of codes and how they are designed. Techniques are described for producing the very best codes. Part III addresses the analysis of low-density parity-check (LDPC) codes, primarily to calculate their stopping sets and low-weight codeword spectrum which determines the performance of th ese codes. Part IV deals with decoders designed to realize optimum performance. Part V describes applications which include combined error correction and detection, public key cryptography using Goppa codes, correcting errors in passwords and watermarking. This book is a valuable resource for anyone interested in error-correcting codes and their applications, ranging from non-experts to professionals at the forefront of research in their field. This book is open access under a CC BY 4.0 license.

Download or read book Applied Algebra Algebraic Algorithms and Error Correcting Codes written by Marc Fossorier and published by Springer. This book was released on 2003-08-03 with total page 275 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 15th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-15, held in Toulouse, France, in May 2003.The 25 revised full papers presented together with 2 invited papers were carefully reviewed and selected from 40 submissions. Among the subjects addressed are block codes; algebra and codes: rings, fields, and AG codes; cryptography; sequences; decoding algorithms; and algebra: constructions in algebra, Galois groups, differential algebra, and polynomials.

Download or read book Codes on Finite Geometries written by Heng Tang and published by . This book was released on 2002 with total page 374 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Channel Codes written by William Ryan and published by Cambridge University Press. This book was released on 2009-09-17 with total page 709 pages. Available in PDF, EPUB and Kindle. Book excerpt: Channel coding lies at the heart of digital communication and data storage, and this detailed introduction describes the core theory as well as decoding algorithms, implementation details, and performance analyses. In this book, Professors Ryan and Lin provide clear information on modern channel codes, including turbo and low-density parity-check (LDPC) codes. They also present detailed coverage of BCH codes, Reed-Solomon codes, convolutional codes, finite geometry codes, and product codes, providing a one-stop resource for both classical and modern coding techniques. Assuming no prior knowledge in the field of channel coding, the opening chapters begin with basic theory to introduce newcomers to the subject. Later chapters then extend to advanced topics such as code ensemble performance analyses and algebraic code design. 250 varied and stimulating end-of-chapter problems are also included to test and enhance learning, making this an essential resource for students and practitioners alike.

Download or read book LDPC Code Designs Constructions and Unification written by Juane Li and published by Cambridge University Press. This book was released on 2017 with total page 259 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book, leading authorities unify algebraic- and graph-based LDPC code designs and constructions into a single theoretical framework.

Download or read book Algebraic Low density Parity check Codes written by Qiuju Diao and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The ever-growing needs for cheaper, faster, and more reliable communication systems have forced many researchers to seek means to attain the ultimate limits on reliable communications. Low densityparity-check (LDPC) codes are currently the most promising coding technique to achieve the Shannon capacities for a wide range of channels. Many LDPC codes have been chosen as the standard codes for various next generations of communication systems and they are appearing in recent data storage products. More applications are expected to come.Major methods for constructing LDPC codes can be divided into two general categories, graphtheoretic-based methods (using computer search) and algebraic methods. Each type of constructions has its advantages and disadvantages in terms overall error performance, encoding and decoding implementations. In general, algebraically constructed LDPC codes have lower error-floor and their decoding using iterative message-passing algorithms converges at a much faster rate than the LDPC codes constructed using a graph theoretic-based method. Furthermore, it is much easier to constructalgebraic LDPC codes with large minimum distances.This research project is set up to investigate several important aspects of algebraic LDPC codes for the purpose of achieving overall good error performance required for future high-speed communication systems and high-density data storage systems. The subjects to be investigated include: (1) new constructions of algebraic LDPC codes based on finite geometries; (2) analysis of structural properties of algebraic LDPC codes, especially the trapping set structure that determines how lowthe error probability of a given LDPC code can achieve; (3) construction of algebraic LDPC codes and design coding techniques for correcting combinations of random errors and erasures that occursimultaneously in many physical communication and storage channels; and (4) analysis and construction of algebraic LDPC codes in transform domain.Research effort has resulted in important findings in all four proposed research subjects which may have a significant impact on future generations of communication and storage systems andadvance the state-of-the-art of channel coding theory.

Download or read book Fundamentals of Classical and Modern Error Correcting Codes written by Shu Lin and published by Cambridge University Press. This book was released on 2021-12-09 with total page 843 pages. Available in PDF, EPUB and Kindle. Book excerpt: An accessible textbook that uses step-by-step explanations, relatively easy mathematics and numerous examples to aid student understanding.

Download or read book Classes of High performance Quantum LDPC Codes from Finite Projective Geometries written by Jacob M. Farinholt and published by . This book was released on 2012 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Due to their fast decoding algorithms, quantum generalizations of low-density parity check, or LDPC, codes have been investigated as a solution to the problem of decoherence in fragile quantum states [1, 2]. However, the additional twisted inner product requirements of quantum stabilizer codes force four-cycles and eliminate the possibility of randomly generated quantum LDPC codes. Moreover, the classes of quantum LDPC codes discovered thus far generally have unknown or small minimum distance, or a fixed rate (see [3, 4] and references therin). This paper presents several new classes of quantum LDPC codes constructed from finite projective planes. These codes have rates that increase with the block length n and minimum weights proportional to n1=2. For the sake of completeness, we include an introduction to classical error correction and LDPC codes, and provide a review of quantum communication, quantum stabilizer codes, and finite projective geometry.

Download or read book Construction of Structured LDPC Codes Based on Geometry Decomposition Masking and Superposition written by Jun Xu and published by . This book was released on 2003 with total page 356 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book QC LDPC Code Based Cryptography written by Marco Baldi and published by Springer Science & Business. This book was released on 2014-05-02 with total page 128 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book describes the fundamentals of cryptographic primitives based on quasi-cyclic low-density parity-check (QC-LDPC) codes, with a special focus on the use of these codes in public-key cryptosystems derived from the McEliece and Niederreiter schemes. In the first part of the book, the main characteristics of QC-LDPC codes are reviewed, and several techniques for their design are presented, while tools for assessing the error correction performance of these codes are also described. Some families of QC-LDPC codes that are best suited for use in cryptography are also presented. The second part of the book focuses on the McEliece and Niederreiter cryptosystems, both in their original forms and in some subsequent variants. The applicability of QC-LDPC codes in these frameworks is investigated by means of theoretical analyses and numerical tools, in order to assess their benefits and drawbacks in terms of system efficiency and security. Several examples of QC-LDPC code-based public key cryptosystems are presented, and their advantages over classical solutions are highlighted. The possibility of also using QC-LDPC codes in symmetric encryption schemes and digital signature algorithms is also briefly examined.

Download or read book VLSI Architectures for Modern Error Correcting Codes written by Xinmiao Zhang and published by CRC Press. This book was released on 2017-12-19 with total page 410 pages. Available in PDF, EPUB and Kindle. Book excerpt: Error-correcting codes are ubiquitous. They are adopted in almost every modern digital communication and storage system, such as wireless communications, optical communications, Flash memories, computer hard drives, sensor networks, and deep-space probing. New-generation and emerging applications demand codes with better error-correcting capability. On the other hand, the design and implementation of those high-gain error-correcting codes pose many challenges. They usually involve complex mathematical computations, and mapping them directly to hardware often leads to very high complexity. VLSI Architectures for Modern Error-Correcting Codes serves as a bridge connecting advancements in coding theory to practical hardware implementations. Instead of focusing on circuit-level design techniques, the book highlights integrated algorithmic and architectural transformations that lead to great improvements on throughput, silicon area requirement, and/or power consumption in the hardware implementation. The goal of this book is to provide a comprehensive and systematic review of available techniques and architectures, so that they can be easily followed by system and hardware designers to develop en/decoder implementations that meet error-correcting performance and cost requirements. This book can be also used as a reference for graduate-level courses on VLSI design and error-correcting coding. Particular emphases are placed on hard- and soft-decision Reed-Solomon (RS) and Bose-Chaudhuri-Hocquenghem (BCH) codes, and binary and non-binary low-density parity-check (LDPC) codes. These codes are among the best candidates for modern and emerging applications due to their good error-correcting performance and lower implementation complexity compared to other codes. To help explain the computations and en/decoder architectures, many examples and case studies are included. More importantly, discussions are provided on the advantages and drawbacks of different implementation approaches and architectures.