Mathematical Theory and Applications of Error Correcting Codes

Mathematical Theory and Applications of Error-Correcting Codes book presents those concepts of mathematics that continue to reveal essentials in making reliable the data exchange between systems and their parts. For such reason, its applications range over many areas, including physical science, engineering, and medicine. In particular, it treats of hard and soft-decision decoding, of cyclic and quasi-cyclic codes, of Reed-Solomon codes, of low density parity check codes, and quantum error-correcting codes. Cover Title Page Copyright DECLARATION ABOUT THE EDITOR TABLE OF CONTENTS List of Contributors List of Abbreviations Preface Section 1 Introduction to Error-Correcting Codes Chapter 1 Error-correcting Codes and Neural Networks Abstract Introduction And Summary Codes And Good Codes Neural Encodings Of Stimulus Spaces Acknowledgements References Section 2 Hard and Soft Decision Decoding Chapter 2 Sum of the Magnitude for Hard Decision Decoding Algorithm Based on Loop Update Detection Abstract Introduction Basic Definitions Algorithm Description Complexity Analysis Simulation Results and Statistical Analysis Conclusions Acknowledgments Conflicts Of Interest References Chapter 3 Soft-Decision Low-Complexity Chase Decoders for the RS (255,239) Code Abstract Introduction Rs Decoders Low-Complexity Chase Decoder Decoder Architecture Implementation Results Conclusions References Chapter 4 Low-energy Error Correction of NAND Flash Memory through Soft-decision Decoding Abstract Introduction Energy Consumption Of Multi-Bit Data Read In Nand Flash Memory Soft-Decision Error Correcting Performance In Nand Flash Memory Hardware Performance Of (68254, 65536) Ldpc Decoder Low-Energy Error Correction Scheme For Nand Flash Memory Concluding Remarks Acknowledgements References Chapter 5 Performance of Soft Viterbi Decoder enhanced with Non-Transmittable Codewords for Storage Media Abstract Public Interest Statement Introduction Binary Convolutional Encoding And Decoding Enhanced Soft Viterbi Algorithm Developed Model Result And Discussion Conclusion Funding References Section 3 Linear Codes: Cyclic and Constacyclic Codes Chapter 6 The Structure of One Weight Linear and Cyclic Codes Over Zr2 x (Z2 + uZ2)S Abstract Introduction Preliminaries The Structure of One Weight Z2Z2[u]-Linear Codes One Weight Z2Z2[u]-Cyclic Codes Examples Of One Weight Z2Z2[u]-Cyclic Codes Conclusion Acknowledgement References Chapter 7 (1 + u)-Constacyclic Codes over Z 4 + uZ 4 Abstract Background (1 + U)-Constacyclic Codes Over Z4 + UZ 4 Gray Images Of (1 + U)-Constacyclic Codes Over R Conclusion Acknowledgements Competing Interests References Section 4 Introduction to Error-Correcting Codes Chapter 8 Projection Decoding of Some Binary Optimal Linear Codes of Lengths 36 and 40 Abstract Introduction Projection of Binary Linear Codes Construction of Binary Optimal [36, 19, 8] And [40, 22, 8] Codes Projection Decoding Examples Conclusions Funding Conflicts of Interest References Chapter 9 Reed-Solomon Turbo Product Codes for Optical Communications: From Code Optimization to Decoder Design Abstract Introduction Reed-Solomon Product Codes Turbo Decoding of RS Product Codes RS Product Code Design For Optical Communications Full-Parallel Turbo Decoding Architecture Dedicated to Product Codes Complexity and Throughput Analysis of The Full-Parallel Reed-Solomon Turbo Decoders Implementation of an RS Turbo Decoder For Ultra High Throughput Communication Conclusion Acknowledgments References Chapter 10 Enhancing BER Performance Limit of BCH and RS Codes Using Multipath Diversity Abstract Introduction Related Work Forward Error Correction Multipath Propagation Methodology Column Weight Multipath Combiner (Cwmc) Results Conclusions Acknowledgements Conflicts Of Interest Abbreviations References Section 5 Quasi Cyclic Codes Chapter 11 Quasi-Cyclic Codes via Unfolded Cyclic Codes and Their Reversibility Abstract Section I Introduction Section II Preliminaries Section III Generator Polynomial Matrix Of F.(C) Section IV Reversibility of Unfolded Cyclic Codes Section V Numerical Examples Section VI Conclusion References Chapter 12 Skew Cyclic and Quasi-Cyclic Codes of Arbitrary Length over Galois Rings Abstract Preliminaries Skew Cyclic Codes Skew Quasi-Cyclic Codes Examples References Section 6 Low Density Parity Check Codes Chapter 13 On the use of Ordered Statistics Decoders For Low-Density Parity-Check Codes In Space Telecommand Links Abstract Introduction Ldpc Codes For Space Telecommand Links Decoding Algorithms Error Rate Versus Complexity Tradeoff Evaluation Impact Of Limited Memory Conclusions Endnote Appendix Acknowledgements Competing Interests References Chapter 14 Optimization of LDPC Codes over the Underwater Acoustic Channel Abstract Introduction Underwater Acoustic Channel Model Iterative DFE-LDPC Structure and Its Exit Charts Optimization Of LDPC Codes Conclusions Conflict Of Interests Acknowledgments References Chapter 15 Augmented Decoders for LDPC Codes Abstract Introduction Brief Overview LDPC Codes Method Numerical Results Conclusions Abbreviations References Chapter 16 Adaptive Rate-Compatible Non-Binary LDPC Coding Scheme for the B5G Mobile System Abstract Introduction The Construction of RC-NB-LDPC Codes Channel Clustering Based on The K-Means++ Algorithm Design And Performance Simulation Of The Adaptive RC-NB-LDPC Coding Scheme Conclusions Conflicts of Interest References Chapter 17 Improved Symbol Value Selection for Symbol Flipping-based Non-binary LDPC Decoding Abstract Introduction System Model Non-Binary LDPC Decoding Proposed Symbol Value Selection Algorithm Computational Complexity Simulation Results and Discussion Conclusions Acknowledgments References Chapter 18 DNA Barcoding through Quaternary LDPC Codes Abstract Introduction Results Discussion Methods Acknowledgments Funding Statement References Chapter 19 High Speed and Adaptable Error Correction for Megabit/s Rate Quantum Key Distribution Abstract Introduction Results Discussion Methods Acknowledgements References Section 7 Introduction to Error-Correcting Codes Chapter 20 Duality of Quantum and Classical Error Correction Codes: Design Principles and Examples Abstract Section I Section II Quantum Decoherence Section III Quantum Coding Theory Section IV Section V Stabilizer Formalism Section VI Quantum-To-Classical Isomorphism Section VII Taxonomy Of Stabilizer Codes Section VIII Design Examples Section IX Conclusion & Design Guidelines References Chapter 21 Dynamic Concatenation of Quantum Error Correction in Integrated Quantum Computing Architecture Abstract Introduction Results Discussion Acknowledgements References Index Back Cover