Concise encyclopedia of coding theory:
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| Language: | English |
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Boca Raton ; London ; New York
CRC Press
[2021]
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| Edition: | First edition |
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| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xxxii, 965 Seiten Illustrationen |
| ISBN: | 9781138551992 9780367709327 |
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| 245 | 1 | 0 | |a Concise encyclopedia of coding theory |c edited by W. Cary Huffman (Loyola University Chicago, USA), Jon-Lark Kim (Sogang University, Republic of Korea), Patrick Solé (University Aix-Marseille, Marseilles, France) |
| 250 | |a First edition | ||
| 264 | 1 | |a Boca Raton ; London ; New York |b CRC Press |c [2021] | |
| 264 | 4 | |c © 2021 | |
| 300 | |a xxxii, 965 Seiten |b Illustrationen | ||
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| 700 | 1 | |a Solé, Patrick |0 (DE-588)1209971208 |4 edt | |
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Contents Preface xxiii Contributors xxix I Coding Fundamentals 1 Basics of Coding Theory W. Cary Huffman, Jon-Lark Kim, and Patrick Sold 1.1 Introduction . 1.2 Finite Fields . 1.3 Codes. 1.4 Generator and Parity Check Matrices . 1.5 Orthogonality . 1.6 Distance and Weight. 1.7 Puncturing, Extending, and Shortening Codes . 1.8 Equivalence and Automorphisms. 1.9 Bounds on Codes . 1.9.1 The Sphere Packing Bound. 1.9.2 The Singleton Bound. 1.9.3 The Plotkin Bound. 1.9.4 The Griesmer Bound. 1.9.5 The Linear Programming Bound
. 1.9.6 The Gilbert Bound. 1.9.7 The Varshamov Bound. 1.9.8 Asymptotic Bounds. 1.10 Hamming Codes . 1.11 Reed-Muller Codes . 1.12 Cyclic Codes . 1.13 Golay Codes . 1.14 BCH and Reed-Solomon Codes . 1.15 Weight Distributions. 1.16 Encoding . 1.17 Decoding . 1.18 Shannon’s Theorem . 2 Cyclic Codes over Finite Fields 2.1 2.2 2.3 2.4 2.5 Cunsheng Ding Notation and Introduction . Subfield Subcodes
. Fundamental Constructionsof Cyclic Codes. The Minimum Distances ofCyclic Codes . Irreducible Cyclic Codes. 1 3 3 5 7 8 9 10 12 13 15 16 17 17 18 18 19 20 20 21 22 23 28 30 33 35 38 42 45 45 46 47 48 49 vii
viü Contents 2.6 2.7 2.8 2.9 2.10 BCH Codes and Their Properties . 2.6.1 The Minimum Distances of BCH Codes. 2.6.2 The Dimensions of BCH Codes . 2.6.3 Other Aspects of BCH Codes. Duadic Codes . Punctured Generalized Reed-Muller Codes . Another Generalization of the Punctured Binary Reed-Muller Codes . . Reversible Cyclic Codes. 3 Construction and Classification of Codes Patric R. J. Ostergdrd Introduction . Equivalence and Isomorphism . 3.2.1 Prescribing Symmetries. 3.2.2 Determining Symmetries. 3.3 Some Central Classes of Codes. 3.3.1 Perfect Codes. 3.3.2 MDS Codes. 3.3.3 Binary Error-Correcting Codes
. 3.1 3.2 4 Self-Dual Codes Stefka Bouyuklieva 4.1 Introduction . 4.2 Weight Enumerators. 4.3 Bounds for the Minimum Distance. 4.4 Construction Methods. 4.4.1 Gluing Theory. 4.4.2 Circulant Constructions . 4.4.3 Subtraction Procedure. 4.4.4 Recursive Constructions. 4.4.5 Constructions of Codes with Prescribed Automorphisms. 4.5 Enumeration and Classification. 5 Codes and Designs 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Vladimir D. Tonchev Introduction . Designs Supported by Codes . Perfect Codes and Designs . The Assmus-Mattson
Theorem. Designs from Codes Meeting the Johnson Bound. Designs and Majority Logic Decoding . Concluding Remarks. 6 Codes over Rings Steven T. Dougherty Introduction . Quaternary Codes . The Gray Map. 6.3.1 Kernels of Quaternary Codes. 6.4 Rings . 6.4.1 Codes over FrobeniusRings. 6.1 6.2 6.3 , 50 51 52 53 54 55 57 58 61 61 62 63 66 67 68 72 73 79 79 81 84 88 88 89 90 90 92 93 97 97 98 99 101 102 105 109 111 Ill 112 113 114 115 115
Contents 6.5 6.6 6.7 6.8 6.4.2 Families of Rings. 6.4.3 The Chinese RemainderTheorem. The MacWilliams Identities. Generating Matrices. The Singleton Bound and MDRCodes . Conclusion . 7 Quasi-Cyclic Codes Cem Güneri, San Ling, and Buket Özkaya Introduction . Algebraic Structure . Decomposition of Quasi-Cyclic Codes . 7.3.1 The Chinese Remainder Theorem and Concatenated Decomposi tions of QC Codes. 132 7.3.2 Applications. 7.3.2.1 Trace Representation. 7.3.2.2 Self-Dual and Complementary Dual QC Codes. 7.4 Minimum Distance Bounds . 7.4.1 The Jensen Bound
. 7.4.2 The Lally Bound. 7.4.3 Spectral Bounds. 7.4.3.1 Cyclic Codes and Distance Bounds From Their Zeros . . 7.4.3.2 Spectral Theory of QC Codes. 7.4.3.3 Spectral Bounds for QC Codes . 7.5 Asymptotics . 7.5.1 Good Self-Dual QC Codes Exist. 7.5.2 Complementary Dual QC Codes Are Good. 7.6 Connection to Convolutional Codes . 7.1 7.2 7.3 8 Introduction to Skew-Polynomial Rings and Skew-Cyclic Codes 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 Heide Gluesing-Luerssen Introduction . Basic Properties of Skew-Polynomial Rings . Skew Polynomials and Linearized Polynomials . Evaluation of Skew Polynomials and Roots . Algebraic Sets and Wedderburn Polynomials . A Circulant Approach Toward Cyclic Block Codes. Algebraic
Theory of Skew-Cyclic Codes with General Modulus . Skew-Constacyclic Codes and Their Duals . The Minimum Distance of Skew-Cyclic Codes . 9 Additive Cyclic Codes Jiirgen Bierbrauer, Stefano Marcugini, and Fernanda Pambianco 9.1 Introduction . 9.2 Basic Notions. 9.3 Code Equivalenceand Cyclicity . 9.4 Additive Codes Which AreCyclic in the Permutation Sense. 9.4.1 The Linear Case m = l. 9.4.2 The General Case m l. 9.4.3 Equivalence. 9.4.4 Duality and Quantum Codes. ix 117 120 121 123 126 127 129 129 130 132 134 134 135 137 137 137 138 138 140 141 143 143 147 148 151 151 152 157 158 162 166 169 174 176 181 181 182 182 184 185 187 190 191
Contents X 9.5 Additive Codes Which Are Cyclic in the Monomial Sense . 9.5.1 The Linear Case m = 1. 9.5.2 The General Case m 1. 10 Convolutional Codes Julia Lieb, Raquel Pinto, and Joachim Rosenthal 10.1 Introduction . 10.2 Foundational Aspects of Convolutional Codes. 10.2.1 Definition of Convolutional Codes via Generator and Parity Check Matrices. 198 10.2.2 Distances of Convolutional Codes. 10.3 Constructions of Codes with Optimal Distance. 10.3.1 Constructions of MDS Convolutional Codes. 10.3.2 Constructions of MDP Convolutional Codes. 10.4 Connections to Systems Theory . 10.5 Decoding of Convolutional Codes. 10.5.1 Decoding over the Erasure Channel. 10.5.1.1 The Case 5 = 0. 10.5.1.2 The General Case. 10.5.2 The Viterbi Decoding Algorithm. 10.6 Two-
Dimensional Convolutional Codes . 10.6.1 Definition of 2D Convolutional Codes via Generator and Parity Check Matrices. 218 10.6.2 ISO Representations . 10.7 Connections to Symbolic Dynamics . 11 Rank-Metric Codes Elisa Gorla 11.1 Definitions, Isometries, and Equivalenceof Codes. 11.2 The Notion of Support in the Rank Metric . 11.3 MRD Codes and Optimal Anticodes. 11.4 Duality and the MacWilliams Identities. 11.5 Generalized Weights. 11.6 q-Polymatroids and Code Invariants . 12 Linear Programming Bounds Peter Boyvalenkov and Danyo Danev 12.1 Preliminaries - Krawtchouk Polynomials, Codes, andDesigns. 12.2 General Linear Programming Theorems. 12.3 Universal Bounds . 12.4 Linear Programming on Sn-1. 12.5 Linear Programming in Other
Coding Theory Problems. 13 Semidefinite Programming Bounds for Error-Correcting Codes Frank Vallentin 13.1 Introduction . 13.2 Conic Programming . 13.2.1 Conic Programming andits Duality Theory. 13.2.2 Linear Programming. 13.2.3 Semidefinite Programming. 13.3 Independent Sets in Graphs. 193 194 195 197 197 198 203 207 207 208 211 214 214 214 215 217 218 221 223 227 227 230 232 236 239 245 251 251 253 255 261 265 267 267 268 268 270 270 272
xi Contents 13.4 13.5 II 13.3.1 Independence Number and Codes. 13.3.2 Semidefinite Programming Bounds for the Independence Number Symmetry Reduction and Matrix *-Algebras . 13.4.1 Symmetry Reduction of Semidefinite Programs . 13.4.2 Matrix *-Algebras. 13.4.3 Example: The Delsarte Linear Programming Bound. 13.4.4 Example: The Schrijver Semidefinite Programming Bound . Extensions and Ramifications. Families of Codes 272 273 275 276 276 277 278 279 283 14 Coding Theory and Galois Geometries Leo Storme 14.1 Galois Geometries . 14.1.1 Basic Properties of Galois Geometries. 14.1.2 Spreads and Partial Spreads. 14.2 Two Links Between Coding Theory and Galois Geometries . 14.2.1 Via the Generator Matrix . 14.2.2 Via the Parity Check Matrix. 14.2.3 Linear MDS Codes and Arcs in Galois Geometries. 14.2.4 Griesmer Bound and Minihypers. 14.3 Projective Reed-Muller Codes
. 14.4 Linear Codes Defined by Incidence Matrices Arising from Galois Geometries 14.5 Subspace Codes and Galois Geometries . 14.5.1 Definitions. 14.5.2 Designs over Fg. 14.5.3 Rank-Metric Codes. 14.5.4 Maximum Scattered Subspaces and MRD Codes. 14.5.5 Semifields and MRD Codes. 14.5.6 Nonlinear MRD Codes. 14.6 A Geometric Result Arising from a Coding Theoretic Result . 15 Algebraic Geometry Codes and Some Applications Alain Couvreur and Hugues Randriambololona 15.1 Notation . 15.2 Curves and Function Fields. 15.2.1 Curves, Points, Function Fields, and Places . 15.2.2 Divisors . 15.2.3 Morphisms of Curves and Pullbacks. 15.2.4 Differential
Forms. 15.2.4.1 Canonical Divisors. 15.2.4.2 Residues. 15.2.5 Genus and the Riemann-Roch Theorem . 15.3 Basics on Algebraic Geometry Codes . 15.3.1 Algebraic Geometry Codes, Definitions, andElementary Results . 15.3.2 Genus 0, Generalized Reed-Solomon and Classical Goppa Codes . 15.3.2.1 The Cl Description. 320 15.3.2.2 The Cn Description. 321 15.4 Asymptotic Parameters of Algebraic Geometry Codes . 15.4.1 Preamble. 323 15.4.2 The Tsfasman-Vladu^-Zink Bound. 324 285 285 285 287 288 288 288 289 292 293 295 297 297 299 299 301 302 303 303 307 311 312 312 313 314 314 315 315 315 316 316 319 323
Contents xii 15.5 15.6 15.7 15.8 15.9 15.4.3 Subfield Subcodes and the Katsman-Tsfasman-Wirtz Bound . . . 15.4.4 Nonlinear Codes. Improved Lower Bounds for the Minimum Distance . 15.5.1 Floor Bounds. 15.5.2 Order Bounds. 15.5.3 Further Bounds. 15.5.4 Geometric Bounds for Codes fromEmbedded Curves. Decoding Algorithms . 15.6.1 Decoding Below Half the Designed Distance. 15.6.1.1 The Basic Algorithm. 15.6.1.2 Getting Rid of Algebraic Geometry: Error-Correcting Pairs. 336 15.6.1.3 Reducing the Gap to Half the Designed Distance . 15.6.1.4 Decoding Up to Half the Designed Distance: The FengRao Algorithm and Error-Correcting Arrays. 337 15.6.2 List Decoding and the Guruswami-Sudan Algorithm. Application to Public-Key Cryptography: A McEliece-Type Cryptosystem 15.7.1 History. 15.7.2 McEliece’s Original Proposal Using Binary Classical Goppa
Codes 15.7.3 Advantages and Drawbacks of the McEliece Scheme. 15.7.4 Janwa and Moreno’s Proposals Using AG Codes. 15.7.5 Security . 15.7.5.1 Concatenated Codes Are Not Secure. 15.7.5.2 Algebraic Geometry Codes Are Not Secure . 15.7.5.3 Conclusion: Only Subfield Subcodes of AG Codes Remain Unbroken. 343 Applications Related to the ^-Product: Frameproof Codes, Multiplication Algorithms, and Secret Sharing. 344 15.8.1 The *-Product from the Perspective of AG Codes. 15.8.1.1 Basic Properties . 15.8.1.2 Dimension of *-Products. 15.8.1.3 Joint Bounds on Dimension and Distance. 15.8.1.4 Automorphisms. 15.8.2 Frameproof Codes and SeparatingSystems. 15.8.3 Multiplication Algorithms. 15.8.4 Arithmetic Secret Sharing. Application to Distributed Storage Systems: Locally Recoverable Codes . 15.9.1
Motivation. 15.9.2 A Bound on the Parameters InvolvingLocality . 15.9.3 Tamo-Barg Codes. 15.9.4 Locally Recoverable Codes from Coverings of Algebraic Curves: Barg-Tamo-Vladu^ Codes. 15.9.5 Improvement: Locally Recoverable Codes with Higher Local Dis tance . 359 15.9.6 Fibre Products of Curves and theAvailability Problem. 16 Codes in Group Algebras Wolfgang Willems 16.1 Introduction . . 16.2 Finite Dimensional Algebras . 16.3 Group Algebras . 326 326 326 328 330 332 332 333 333 333 337 339 340 340 342 342 342 343 343 343 344 344 345 347 347 348 349 352 354 354 356 356 357 359 363 363 364 367
Contents 16.4 16.5 16.6 16.7 16.8 16.9 16.10 16.11 16.12 Group Codes. Self-Dual Group Codes . Idempotent Group Codes . LCP and LCD Group Codes . :. Divisible Group Codes. Checkable Group Codes. Decoding Group Codes . Asymptotic Results . Group Codes over Rings. 17 Constacyclic Codes over Finite Commutative Chain Rings Hai Q. Dinh and Sergio R. Lopez-Permouth 17.1 Introduction . 17.2 Chain Rings, Galois Rings, and Alternative Distances . 17.3 Constacyclic Codes over Arbitrary Commutative Finite Rings . 17.4 Simple-Root Cyclic and Negacyclic Codes over Finite Chain Rings . 17.5 Repeated-Root Constacyclic Codes over Galois Rings . 17.6 Repeated-Root Constacyclic Codes over
R = Fpm + upm, u2 = 0 . 17.6.1 All Constacyclic Codes of Length p3 over 77. 17.6.2 AU Constacyclic Codes of Length 2ps over H. 17.6.3 All Constacyclic Codes of Length 4p3 over Ή. 17.6.4 λ-Constacyclic Codes of Length np3 over R, λ G F*m. 17.7 Extensions . 18 Weight Distribution of Trace Codes over Finite Rings Minjia Shi 18.1 Introduction . 18.2 Preliminaries. 18.3 A Class of Special Finite Rings Rk (TypeI) . 18.3.1 Case (i) к = 1. 18.3.2 Case (ii) к = 2. 18.3.3 Case (in) к 2. 18.3.4 Case (iv) Rk(ph P an Odd Prime. 18.4 A Class of Special Finite Rings R(k.,p, uk= a) (Type II) . 18.5 Three Special Rings . 18.5.1 R(2,p,u2 =
u). 18.5.2 R(3,2,u3 = l). 18.5.3 R(3,3, u3 = l). 18.6 Conclusion . 19 Two-Weight Codes Andries E. Brouwer 19.1 Generalities. 19.2 Codes as Projective Multisets. 19.2.1 Weights . 19.3 Graphs . 19.3.1 Difference Sets. 19.3.2 Using a Projective Setas a Difference Set . 19.3.3 Strongly RegularGraphs. 19.3.4 Parameters. 19.3.5 Complement. xiii 369 373 374 375 377 379 381 382 383 385 385 387 391 392 399 407 407 412 414 418 423 429 429 430 430 431 432 433 435 441 444 444 446 447 448
449 449 450 450 451 451 451 451 452 453
Contents xiv 19.4 19.5 19.6 19.7 19.8 19.9 19.3.6 Duality . 19.3.7 Field Change . Irreducible Cyclic Two-WeightCodes . Cyclotomy . 19.5.1 The Van Lint-SchrijverConstruction. 19.5.2 The De Lange Graphs . 19.5.3 Generalizations. Rank 3 Groups. 19.6.1 One-Dimensional AffineRank 3 Groups. Two-Character Sets in ProjectiveSpace. 19.7.1 Subspaces. 19.7.2 Quadrics. 19.7.3 Maximal Arcs and Hyperovals. 19.7.4 Baer Subspaces. 19.7.5 Hermitian Quadrics. 19.7.6 Sporadic
Examples. Nonprojective Codes. Brouwer-Van Eupen Duality . 19.9.1 From Projective Code to Two-Weight Code . 19.9.2 From Two-Weight Code to Projective Code . 20 Linear Codes from Functions 453 453 454 454 455 456 456 456 456 457 458 458 459 459 459 459 461 461 461 462 463 Sihem Mesnager 20.1 Introduction . 465 20.2 Preliminaries. 466 20.2.1 The Trace Function. 466 20.2.2 Vectorial Functions. 466 20.2.2.1 Representations of p-Ary Functions. 466 20.2.2.2 The Walsh Transform ofa Vectorial Function. 468 20.2.3 Nonlinearity of Vectorial Boolean Functions and Bent Boolean Functions. 469 20.2.4 Plateaued Functions and More about Bent Functions. 471 20.2.5 Differential Uniformity of Vectorial Boolean Functions, PN, and APN
Functions. 473 20.2.6 APN and Planar Functions over Fg™ . 474 20.2.7 Dickson Polynomials. 475 20.3 Generic Constructions of Linear Codes . 476 20.3.1 The First Generic Construction. 476 20.3.2 The Second Generic Construction. 478 20.3.2.1 The Defining Set Construction of Linear Codes. 478 20.3.2.2 Generalizations of the Defining Set Construction of Lin ear Codes. 479 20.3.2.3 A Modified Defining Set Construction of Linear Codes . 479 20.4 Binary Codes with Few Weights from Boolean Functions and Vectorial Boolean Functions. 479 20.4.1 A First Example of Codes from Boolean Functions: Reed-Muller Codes. . 479 20.4.2 A General Construction of Binary Linear Codes from Boolean Functions. 480 20.4.3 Binary Codes from thePreimage/-1(Ь)of BooleanFunctions f . 480 20.4.4 Codes withFew Weights fromBentBoolean Functions. 481
Contents XV Codes with Few Weights from Semi-Bent Boolean Functions . . . 482 Linear Codes from Quadratic Boolean Functions. 483 Binary Codes Cot with Three Weights. 484 Binary Codes Cot with Four Weights. 484 Binary Codes Cd{ with at Most Five Weights. 485 A Class of Two-Weight Binary Codes from the Preimage of a Type of Boolean Function. 486 20.4.11 Binary Codes from Boolean Functions Whose Supports are Rela tive Difference Sets. 487 20.4.12 Binary Codes with Few Weights from Plateaued Boolean Functions 487 20.4.13 Binary Codes with Few Weights from Almost Bent Functions . . 488 20.4.14 Binary Codes C^g) from Functions on Fa™ of the Form G(x) — F(x) + F(x + 1) + 1 . 489 20.4.15 Binary Codes from the Images of Certain Functions on Fa™ . . . 489 Constructions of Cyclic Codes from Functions: The Sequence Approach . 490 20.5.1 A Generic Construction of Cyclic Codes with Polynomials . 490 20.5.2 Binary Cyclic Codes from APN Functions. 492 20.5.3 Non-Binary Cyclic Codes from Monomials and Trinomials . 495 20.5.4 Cyclic Codes from Dickson Polynomials. 498 Codes with Few Weights from p-Ary Functions with p Odd . 503 20.6.1 Codes with Few Weights from
p-Ary Weakly Regular Bent Func tions Based on the First Generic Construction. 503 20.6.2 Linear Codes with Few Weights from Cyclotomie Classes and Weakly Regular Bent Functions. 504 20.6.3 Codes with Few Weights from p-Ary Weakly Regular Bent Func tions Based on the Second Generic Construction. 507 20.6.4 Codes with Few Weights from p-Ary Weakly Regular Plateaued Functions Based on the First Generic Construction. 508 20.6.5 Codes with Few Weights from p-Ary Weakly Regular Plateaued Functions Based on the Second Generic Construction. 510 Optimal Linear Locally Recoverable Codes from p-Ary Functions. 521 20.7.1 Constructions of r-Good Polynomials for Optimal LRC Codes . . 523 20.7.1.1 Good Polynomials from Power Functions. 523 20.7.1.2 Good Polynomials from Linearized Functions. 523 20.7.1.3 Good Polynomials from Function Composition. 523 20.7.1.4 Good Polynomials from Dickson Polynomials of the First Kind. 524 20.7.1.5 Good Polynomials from the Composition of Functions In volving Dickson Polynomials. 525 20.4.5 20.4.6 20.4.7 20.4.8 20.4.9 20.4.10 20.5 20.6 20.7 21 Codes over Graphs Christine A. Kelley 21.1 Introduction . 21.2 Low-Density Parity Check Codes.
21.3 Decoding . 21.3.1 Decoder Analysis. 21.4 Codes from Finite Geometries . 21.5 Codes from Expander Graphs . 21.6 Protograph Codes . 21.7 Density Evolution . 21.8 Other Families of Codes over Graphs . 527 527 528 532 537 540 542 545 548 550
Contents xvi 21.8.1 21.8.2 21.8.3 III Turbo Codes. Repeat Accumulate Codes. Spatially-Coupled LDPC Codes. Applications 22 Alternative Metrics Marcelo Firer 22.1 Introduction . 22.2 Metrics Generated by Subspaces . 22.2.1 Projective Metrics. 22.2.2 Combinatorial Metrics. 22.2.2.1 Block Metrics. 22.2.2.2 -Burst Metrics. 22.2.2.3 i X b2-Burst Metrics. 22.3 Poset Metrics. 22.3.1 Poset-Block Metrics. 22.3.2 Graph Metrics. 22.4 Additive Generalizations of the Lee Metric . 22.4.1 Metrics over Rings of
Integers. 22.4.2 Z-Dimensional Lee Weights. 22.4.3 Kaushik-Sharma Metrics. 22.5 Non-Additive Metrics Digging into the Alphabet. 22.5.1 Pomset Metrics. 22.5.2 m-Spotty Metrics. 22.6 Metrics for Asymmetric Channels . 22.6.1 The Asymmetric Metric . 22.6.2 The Generalized Asymmetric Metric . 22.7 Editing Metrics. 22.7.1 Bounds for Editing Codes . 22.8 Permutation Metrics. 23 Algorithmic Methods Alfred Wassermann 23.1 Introduction . 23.2 Linear Codes with Prescribed Minimum Distance . 23.3 Linear Codes as Sets of Points in Projective Geometry. 23.3.1 Automorphisms ofProjective Point Sets. 23.4 Projective Point
Sets with Prescribed Automorphism Groups. 23.4.1 Strategiesfor Choosing Groups. 23.4.2 Observations for Permutation Groups. 23.4.3 Observations for Cyclic Groups. 23.5 Solving Strategies . 23.6 Construction of Codes with Additional Restrictions . 23.6.1 Projective Codes . 23.6.2 Codes with Few Weights. 23.6.3 Divisible Codes. 23.6.4 Codes with PrescribedGram Matrix. 23.6.5 Self-Orthogonal Codes. 23.6.6 LCD Codes . 23.7 Extensions of Codes. 550 551 551 553 555 555 558 558 559 561 561 562 562 565 565 566 566 567 567 568 568 569 570 570 571 571 572 573 575 575 576 577 580 582 585 585 586 587 587 587 587 588 589 591 592 594
Contents 23.8 Determining the Minimum Distance and Weight Distribution . xvii 595 24 Interpolation Decoding 599 Swastik Kopparty 24.1 Introduction . 599 24.2 The Berlekamp-Welch Algorithm . 600 24.2.1 Correctness of the Algorithm RSDecode . 602 24.3 List-decoding of Reed-Solomon Codes. 603 24.3.1 The Sudan Algorithm. 603 24.3.2 Correctness of the Algorithm RSListDecodeVl . 604 24.4 List-decoding of Reed-Solomon Codes Using Multiplicities . 605 24.4.1 Preparations. 606 24.4.2 The Guruswami-Sudan Algorithm. . 607 24.4.3 Correctness of the Algorithm RSListDecodeV2. 608 24.4.4 Why Do Multiplicities Help?. 608 24.5 Decoding of Interleaved Reed-Solomon Codes under Random Error . . . 609 24.6 Further Reading . 611 25 Pseudo-Noise Sequences 613 Tor Helleseth and Chunlei Li 25.1 Introduction
. 613 25.2 Sequences with Low Correlation . 614 25.2.1 Correlation Measuresof Sequences . 614 25.2.2 Sequences with Low Periodic Autocorrelation. 619 25.2.3 Sequence Families with Low Periodic Correlation. 624 25.3 Shift Register Sequences. 626 25.3.1 Feedback Shift Registers. 627 25.3.2 Cycle Structure. 628 25.3.3 Cycle Joining and Splitting. 630 25.3.4 Cycle Structure of LFSRs . 631 25.4 Generation of De Bruijn Sequences. 634 25.4.1 Graphical Approach. 634 25.4.2 Combinatorial Approach. 636 25.4.3 Algebraic Approach. 640 26 Lattice Coding 645 Frédérique Oggier 26.1 Introduction . 645 26.2 Lattice Coding for the
Gaussian Channel . . 646 26.3 Modulation Schemes for Fading Channels. 648 26.3.1 Channel Model and Design Criteria. 648 26.3.2 Lattices from Quadratic Fields. 649 26.4 Lattices from Linear Codes . 651 26.4.1 Construction A. 651 26.4.2 Constructions D and D. 653 26.5 Variations of Lattice Coding Problems. 654 26.5.1 Index Codes. 655 26.5.2 Wiretap Codes. 655
xviii Contents 27 Quantum Error-Control Codes Martianus Frederic Ezerman 27.1 Introduction . 27.2 Preliminaries. 27.3 The Stabilizer Formalism . 27.4 Constructions via Classical Codes . 27.5 Going Asymmetric. 27.6 Other Approaches and a Conclusion. 28 Space-Time Coding Frédérique Oggier 28.1 Introduction . 28.2 Channel Models and Design Criteria. 28.2.1 Coherent Space-Time Coding . 28.2.2 Differential Space-Time Coding. 28.3 Some Examples of Space-Time Codes . 28.3.1 The Alamouti Code. 28.3.2 Linear Dispersion Codes. 28.3.3 The Golden
Code. 28.3.4 Cayley Codes. 28.4 Variations of Space-Time Coding Problems . 28.4.1 Distributed Space-Time Coding. 28.4.2 Space-Time Coded Modulation . 28.4.3 Fast Decodable Space-Time Codes. 28.4.4 Secure Space-Time Coding. 29 Network Codes Frank R. Kschischang 29.1 Packet Networks. 29.2 Multicasting from a Single Source . 29.2.1 Combinational Packet Networks. 29.2.2 Network Information Flow Problems. 29.2.3 The Unicast Problem. 29.2.4 Linear Network Coding Achieves Multicast Capacity. 29.2.5 Multicasting from Multiple Sources. 29.3 Random Linear Network Coding. 29.4 Operator
Channels. 29.4.1 Vector Space, Matrix, and Combinatorial Preliminaries. 29.4.2 The Operator Channel. 29.5 Codes and Metrics in Pq(n). 29.5.1 Subspace Codes. 29.5.2 Coding Metrics on Pq(n). 29.6 Bounds on Constant-Dimension Codes. 29.6.1 The Sphere Packing Bound. 29.6.2 The Singleton Bound. 29.6.3 The Anticode Bound. 29.6.4 Johnson-Type Bounds . 29.6.5 The Ahiswede and Aydinian Bound. 29.6.6 A Gilbert-Varshamov-Type Bound. 29.7 Constructions . 29.7.1 Lifted Rank-Metric Codes. 657 657 658 660 665 669 671 673 673 674 675 676 677 677 678 678 679 680 680 681 681 683 685 685 687 687 689 689 690 691 692 694 694 697
698 698 699 701 701 702 702 703 704 704 705 705
Contents 29.8 29.9 29.7.2 Padded Codes. 29.7.3 Lifted Ferrers Diagram Codes. 29.7.4 Codes Obtained by Integer Linear Programming. 29.7.5 Further Constructions . Encoding and Decoding. 29.8.1 Encoding a Union of Lifted FD Codes . 29.8.2 Decoding Lifted Delsarte-Gabidulin Codes. 29.8.3 Decoding a Union of Lifted FD Codes. Conclusions. 30 Coding for Erasures and Fountain Codes Ian F. Blake 30.1 Introduction . 30.2 Tornado Codes. 30.3 LT Codes. 30.4 Raptor Codes. 31 Codes for Distributed Storage Vinayak Ramkumar, Myna Vajha, S. B. Balaji, Μ. Nikhil Krishnan, Birenjith Sasidharan, and P. Vijay Kumar 31.1 Reed-Solomon Codes
. 31.2 Regenerating Codes . 31.2.1 An Example of a Regenerating Code and Sub-Packetization . . . 31.2.2 General Definition of a Regenerating Code. 31.2.3 Bound on File Size. 31.2.4 MSR and MBR Codes. 31.2.5 Storage Bandwidth Tradeoff for Exact-Repair . . . . 31.2.6 Polygon MBR Codes. . . . 31.2.7 The Product-Matrix MSR and MBR Constructions. 31.2.7.1 PM-MSR Codes. 31.2.7.2 PM-MBR Codes . 31.2.8 The Clay Code . 31.2.9 Variants of Regenerating Codes. 31.3 Locally Recoverable Codes . 31.3.1 Information Symbol Locality. 31.3.1.1 Pyramid Codes. 31.3.1.2 Windows Azure LRC. 31.3.2 All Symbol
Locality. 31.3.3 LRCs over Small Field Size. 31.3.4 Recovery from Multiple Erasures . 31.3.4.1 Codes with Sequential Recovery. 31.3.4.2 Codes with Parallel Recovery. 31.3.4.3 Codes with Availability . 31.3.4.4 Codes with Cooperative Recovery. 31.3.4.5 Codes with (r, 5) Locality. 31.3.4.6 Hierarchical Codes. 31.3.5 Maximally Recoverable Codes. 31.4 Locally Regenerating Codes. 31.5 Efficient Repair of Reed-Solomon Codes. 31.6 Codes for Distributed Storage in Practice. xix 707 708 710 710 711 711 711 712 713 715 715 717 722 727 735 737 738 739 739 740 741 742 742 743 743 744 745 748 749 750 750 751 751 753 754 754 755 755 756 756 757 757 758 759 760
Contents XX 32 Polar Codes Noam Presman and Simon Litsyn 32.1 Introduction . 32.2 Kernel Based ECCs . 32.2.1 Kernel Based ECCs are Recursive GCCs. 32.3 Channel Splitting and Combining and the SC Algorithm . 32.4 Polarization Conditions . 32.4.1 Polarization Rate. 32.5 Polar Codes . 32.5.1 Polar Code Design . 32.6 Polar Codes Encoding Algorithms . 32.7 Polar Codes Decoding Algorithms . 32.7.1 The SC Decoding Algorithm. 32.7.1.1 SC for (u + v,v) . 32.7.1.2 SC for General Kernels. 32.7.1.3 SC Complexity. 32.7.2 The SCL Decoding Algorithm. 32.8 Summary and
Concluding Remarks . 33 Secret Sharing with Linear Codes Cunsheng Ding 33.1 Introduction to Secret Sharing Schemes . 33.2 The First Construction of Secret Sharing Schemes . 33.3 The Second Construction of Secret Sharing Schemes. 33.3.1 Minimal Linear Codes and the Covering Problem. 33.3.2 The Second Construction of Secret Sharing Schemes. 33.3.3 Secret Sharing Schemes from the Duals of Minimal Codes . 33.3.4 Other Works on the Second Construction. 33.4 Multisecret Sharing with Linear Codes . 33.4.1 The Relation Between Multisecret Sharing and Codes. 33.4.2 Linear Threshold Schemes and MDS Codes. 34 Code-Based Cryptography 763 763 764 766 769 771 774 775 775 776 777 778 778 780 781 781 783 785 785 787 790 790 791 792 793 794 795 796 799 Philippe Gaborit and Jean-Christophe Deneuville 34.1 Preliminaries. 800 34.1.1 Notation. 800 34.1.2 Background on Coding Theory. 801 34.2 Difficult Problems for Code-Based Cryptography: The Syndrome Decoding Problem
and Its Variations . 803 34.3 Best-Known Attacks for the Syndrome Decoding Problem. 804 34.4 Public-Key Encryption from Coding Theory with Hidden Structure . . . 807 34.4.1 The McEliece and Niederreiter Frameworks . 807 34.4.2 Group-Structured McEliece Framework. 809 34.4.3 Moderate-Density Parity Check (MDPC) Codes. 810 34.5 PKE Schemes with Reduction to Decoding Random Codes without Hidden Structure . 812 34.5.1 Alekhnovich’s Approach. 812 34.5.2 HQC: Efficient Encryption from Random Quasi-Cyclic Codes . . 813 34.5.3 Ouroboros Key-Exchange Protocol . 814 34.6 Examples of Parameters for Code-Based Encryption and Key Exchange . 815 34.7 Authentication: The Stern Zero-Knowledge Protocol. 816
Contents xxi 34.8 Digital Signatures from Coding Theory . 34.8.1 Signature from a Zero-Knowledge Authentication Scheme with the Fiat-Shamir Heuristic . 818 34.8.2 The CFS Signature Scheme . 34.8.3 The WAVE Signature. 34.8.4 Few-Times Signature Schemes and Variations. 34.9 OtherPrimitives. 34.10 Rank-Based Cryptography . 817 818 819 820 820 820 Bibliography 823 Index 941 |
| adam_txt |
Contents Preface xxiii Contributors xxix I Coding Fundamentals 1 Basics of Coding Theory W. Cary Huffman, Jon-Lark Kim, and Patrick Sold 1.1 Introduction . 1.2 Finite Fields . 1.3 Codes. 1.4 Generator and Parity Check Matrices . 1.5 Orthogonality . 1.6 Distance and Weight. 1.7 Puncturing, Extending, and Shortening Codes . 1.8 Equivalence and Automorphisms. 1.9 Bounds on Codes . 1.9.1 The Sphere Packing Bound. 1.9.2 The Singleton Bound. 1.9.3 The Plotkin Bound. 1.9.4 The Griesmer Bound. 1.9.5 The Linear Programming Bound
. 1.9.6 The Gilbert Bound. 1.9.7 The Varshamov Bound. 1.9.8 Asymptotic Bounds. 1.10 Hamming Codes . 1.11 Reed-Muller Codes . 1.12 Cyclic Codes . 1.13 Golay Codes . 1.14 BCH and Reed-Solomon Codes . 1.15 Weight Distributions. 1.16 Encoding . 1.17 Decoding . 1.18 Shannon’s Theorem . 2 Cyclic Codes over Finite Fields 2.1 2.2 2.3 2.4 2.5 Cunsheng Ding Notation and Introduction . Subfield Subcodes
. Fundamental Constructionsof Cyclic Codes. The Minimum Distances ofCyclic Codes . Irreducible Cyclic Codes. 1 3 3 5 7 8 9 10 12 13 15 16 17 17 18 18 19 20 20 21 22 23 28 30 33 35 38 42 45 45 46 47 48 49 vii
viü Contents 2.6 2.7 2.8 2.9 2.10 BCH Codes and Their Properties . 2.6.1 The Minimum Distances of BCH Codes. 2.6.2 The Dimensions of BCH Codes . 2.6.3 Other Aspects of BCH Codes. Duadic Codes . Punctured Generalized Reed-Muller Codes . Another Generalization of the Punctured Binary Reed-Muller Codes . . Reversible Cyclic Codes. 3 Construction and Classification of Codes Patric R. J. Ostergdrd Introduction . Equivalence and Isomorphism . 3.2.1 Prescribing Symmetries. 3.2.2 Determining Symmetries. 3.3 Some Central Classes of Codes. 3.3.1 Perfect Codes. 3.3.2 MDS Codes. 3.3.3 Binary Error-Correcting Codes
. 3.1 3.2 4 Self-Dual Codes Stefka Bouyuklieva 4.1 Introduction . 4.2 Weight Enumerators. 4.3 Bounds for the Minimum Distance. 4.4 Construction Methods. 4.4.1 Gluing Theory. 4.4.2 Circulant Constructions . 4.4.3 Subtraction Procedure. 4.4.4 Recursive Constructions. 4.4.5 Constructions of Codes with Prescribed Automorphisms. 4.5 Enumeration and Classification. 5 Codes and Designs 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Vladimir D. Tonchev Introduction . Designs Supported by Codes . Perfect Codes and Designs . The Assmus-Mattson
Theorem. Designs from Codes Meeting the Johnson Bound. Designs and Majority Logic Decoding . Concluding Remarks. 6 Codes over Rings Steven T. Dougherty Introduction . Quaternary Codes . The Gray Map. 6.3.1 Kernels of Quaternary Codes. 6.4 Rings . 6.4.1 Codes over FrobeniusRings. 6.1 6.2 6.3 , 50 51 52 53 54 55 57 58 61 61 62 63 66 67 68 72 73 79 79 81 84 88 88 89 90 90 92 93 97 97 98 99 101 102 105 109 111 Ill 112 113 114 115 115
Contents 6.5 6.6 6.7 6.8 6.4.2 Families of Rings. 6.4.3 The Chinese RemainderTheorem. The MacWilliams Identities. Generating Matrices. The Singleton Bound and MDRCodes . Conclusion . 7 Quasi-Cyclic Codes Cem Güneri, San Ling, and Buket Özkaya Introduction . Algebraic Structure . Decomposition of Quasi-Cyclic Codes . 7.3.1 The Chinese Remainder Theorem and Concatenated Decomposi tions of QC Codes. 132 7.3.2 Applications. 7.3.2.1 Trace Representation. 7.3.2.2 Self-Dual and Complementary Dual QC Codes. 7.4 Minimum Distance Bounds . 7.4.1 The Jensen Bound
. 7.4.2 The Lally Bound. 7.4.3 Spectral Bounds. 7.4.3.1 Cyclic Codes and Distance Bounds From Their Zeros . . 7.4.3.2 Spectral Theory of QC Codes. 7.4.3.3 Spectral Bounds for QC Codes . 7.5 Asymptotics . 7.5.1 Good Self-Dual QC Codes Exist. 7.5.2 Complementary Dual QC Codes Are Good. 7.6 Connection to Convolutional Codes . 7.1 7.2 7.3 8 Introduction to Skew-Polynomial Rings and Skew-Cyclic Codes 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 Heide Gluesing-Luerssen Introduction . Basic Properties of Skew-Polynomial Rings . Skew Polynomials and Linearized Polynomials . Evaluation of Skew Polynomials and Roots . Algebraic Sets and Wedderburn Polynomials . A Circulant Approach Toward Cyclic Block Codes. Algebraic
Theory of Skew-Cyclic Codes with General Modulus . Skew-Constacyclic Codes and Their Duals . The Minimum Distance of Skew-Cyclic Codes . 9 Additive Cyclic Codes Jiirgen Bierbrauer, Stefano Marcugini, and Fernanda Pambianco 9.1 Introduction . 9.2 Basic Notions. 9.3 Code Equivalenceand Cyclicity . 9.4 Additive Codes Which AreCyclic in the Permutation Sense. 9.4.1 The Linear Case m = l. 9.4.2 The General Case m l. 9.4.3 Equivalence. 9.4.4 Duality and Quantum Codes. ix 117 120 121 123 126 127 129 129 130 132 134 134 135 137 137 137 138 138 140 141 143 143 147 148 151 151 152 157 158 162 166 169 174 176 181 181 182 182 184 185 187 190 191
Contents X 9.5 Additive Codes Which Are Cyclic in the Monomial Sense . 9.5.1 The Linear Case m = 1. 9.5.2 The General Case m 1. 10 Convolutional Codes Julia Lieb, Raquel Pinto, and Joachim Rosenthal 10.1 Introduction . 10.2 Foundational Aspects of Convolutional Codes. 10.2.1 Definition of Convolutional Codes via Generator and Parity Check Matrices. 198 10.2.2 Distances of Convolutional Codes. 10.3 Constructions of Codes with Optimal Distance. 10.3.1 Constructions of MDS Convolutional Codes. 10.3.2 Constructions of MDP Convolutional Codes. 10.4 Connections to Systems Theory . 10.5 Decoding of Convolutional Codes. 10.5.1 Decoding over the Erasure Channel. 10.5.1.1 The Case 5 = 0. 10.5.1.2 The General Case. 10.5.2 The Viterbi Decoding Algorithm. 10.6 Two-
Dimensional Convolutional Codes . 10.6.1 Definition of 2D Convolutional Codes via Generator and Parity Check Matrices. 218 10.6.2 ISO Representations . 10.7 Connections to Symbolic Dynamics . 11 Rank-Metric Codes Elisa Gorla 11.1 Definitions, Isometries, and Equivalenceof Codes. 11.2 The Notion of Support in the Rank Metric . 11.3 MRD Codes and Optimal Anticodes. 11.4 Duality and the MacWilliams Identities. 11.5 Generalized Weights. 11.6 q-Polymatroids and Code Invariants . 12 Linear Programming Bounds Peter Boyvalenkov and Danyo Danev 12.1 Preliminaries - Krawtchouk Polynomials, Codes, andDesigns. 12.2 General Linear Programming Theorems. 12.3 Universal Bounds . 12.4 Linear Programming on Sn-1. 12.5 Linear Programming in Other
Coding Theory Problems. 13 Semidefinite Programming Bounds for Error-Correcting Codes Frank Vallentin 13.1 Introduction . 13.2 Conic Programming . 13.2.1 Conic Programming andits Duality Theory. 13.2.2 Linear Programming. 13.2.3 Semidefinite Programming. 13.3 Independent Sets in Graphs. 193 194 195 197 197 198 203 207 207 208 211 214 214 214 215 217 218 221 223 227 227 230 232 236 239 245 251 251 253 255 261 265 267 267 268 268 270 270 272
xi Contents 13.4 13.5 II 13.3.1 Independence Number and Codes. 13.3.2 Semidefinite Programming Bounds for the Independence Number Symmetry Reduction and Matrix *-Algebras . 13.4.1 Symmetry Reduction of Semidefinite Programs . 13.4.2 Matrix *-Algebras. 13.4.3 Example: The Delsarte Linear Programming Bound. 13.4.4 Example: The Schrijver Semidefinite Programming Bound . Extensions and Ramifications. Families of Codes 272 273 275 276 276 277 278 279 283 14 Coding Theory and Galois Geometries Leo Storme 14.1 Galois Geometries . 14.1.1 Basic Properties of Galois Geometries. 14.1.2 Spreads and Partial Spreads. 14.2 Two Links Between Coding Theory and Galois Geometries . 14.2.1 Via the Generator Matrix . 14.2.2 Via the Parity Check Matrix. 14.2.3 Linear MDS Codes and Arcs in Galois Geometries. 14.2.4 Griesmer Bound and Minihypers. 14.3 Projective Reed-Muller Codes
. 14.4 Linear Codes Defined by Incidence Matrices Arising from Galois Geometries 14.5 Subspace Codes and Galois Geometries . 14.5.1 Definitions. 14.5.2 Designs over Fg. 14.5.3 Rank-Metric Codes. 14.5.4 Maximum Scattered Subspaces and MRD Codes. 14.5.5 Semifields and MRD Codes. 14.5.6 Nonlinear MRD Codes. 14.6 A Geometric Result Arising from a Coding Theoretic Result . 15 Algebraic Geometry Codes and Some Applications Alain Couvreur and Hugues Randriambololona 15.1 Notation . 15.2 Curves and Function Fields. 15.2.1 Curves, Points, Function Fields, and Places . 15.2.2 Divisors . 15.2.3 Morphisms of Curves and Pullbacks. 15.2.4 Differential
Forms. 15.2.4.1 Canonical Divisors. 15.2.4.2 Residues. 15.2.5 Genus and the Riemann-Roch Theorem . 15.3 Basics on Algebraic Geometry Codes . 15.3.1 Algebraic Geometry Codes, Definitions, andElementary Results . 15.3.2 Genus 0, Generalized Reed-Solomon and Classical Goppa Codes . 15.3.2.1 The Cl Description. 320 15.3.2.2 The Cn Description. 321 15.4 Asymptotic Parameters of Algebraic Geometry Codes . 15.4.1 Preamble. 323 15.4.2 The Tsfasman-Vladu^-Zink Bound. 324 285 285 285 287 288 288 288 289 292 293 295 297 297 299 299 301 302 303 303 307 311 312 312 313 314 314 315 315 315 316 316 319 323
Contents xii 15.5 15.6 15.7 15.8 15.9 15.4.3 Subfield Subcodes and the Katsman-Tsfasman-Wirtz Bound . . . 15.4.4 Nonlinear Codes. Improved Lower Bounds for the Minimum Distance . 15.5.1 Floor Bounds. 15.5.2 Order Bounds. 15.5.3 Further Bounds. 15.5.4 Geometric Bounds for Codes fromEmbedded Curves. Decoding Algorithms . 15.6.1 Decoding Below Half the Designed Distance. 15.6.1.1 The Basic Algorithm. 15.6.1.2 Getting Rid of Algebraic Geometry: Error-Correcting Pairs. 336 15.6.1.3 Reducing the Gap to Half the Designed Distance . 15.6.1.4 Decoding Up to Half the Designed Distance: The FengRao Algorithm and Error-Correcting Arrays. 337 15.6.2 List Decoding and the Guruswami-Sudan Algorithm. Application to Public-Key Cryptography: A McEliece-Type Cryptosystem 15.7.1 History. 15.7.2 McEliece’s Original Proposal Using Binary Classical Goppa
Codes 15.7.3 Advantages and Drawbacks of the McEliece Scheme. 15.7.4 Janwa and Moreno’s Proposals Using AG Codes. 15.7.5 Security . 15.7.5.1 Concatenated Codes Are Not Secure. 15.7.5.2 Algebraic Geometry Codes Are Not Secure . 15.7.5.3 Conclusion: Only Subfield Subcodes of AG Codes Remain Unbroken. 343 Applications Related to the ^-Product: Frameproof Codes, Multiplication Algorithms, and Secret Sharing. 344 15.8.1 The *-Product from the Perspective of AG Codes. 15.8.1.1 Basic Properties . 15.8.1.2 Dimension of *-Products. 15.8.1.3 Joint Bounds on Dimension and Distance. 15.8.1.4 Automorphisms. 15.8.2 Frameproof Codes and SeparatingSystems. 15.8.3 Multiplication Algorithms. 15.8.4 Arithmetic Secret Sharing. Application to Distributed Storage Systems: Locally Recoverable Codes . 15.9.1
Motivation. 15.9.2 A Bound on the Parameters InvolvingLocality . 15.9.3 Tamo-Barg Codes. 15.9.4 Locally Recoverable Codes from Coverings of Algebraic Curves: Barg-Tamo-Vladu^ Codes. 15.9.5 Improvement: Locally Recoverable Codes with Higher Local Dis tance . 359 15.9.6 Fibre Products of Curves and theAvailability Problem. 16 Codes in Group Algebras Wolfgang Willems 16.1 Introduction . . 16.2 Finite Dimensional Algebras . 16.3 Group Algebras . 326 326 326 328 330 332 332 333 333 333 337 339 340 340 342 342 342 343 343 343 344 344 345 347 347 348 349 352 354 354 356 356 357 359 363 363 364 367
Contents 16.4 16.5 16.6 16.7 16.8 16.9 16.10 16.11 16.12 Group Codes. Self-Dual Group Codes . Idempotent Group Codes . LCP and LCD Group Codes . :. Divisible Group Codes. Checkable Group Codes. Decoding Group Codes . Asymptotic Results . Group Codes over Rings. 17 Constacyclic Codes over Finite Commutative Chain Rings Hai Q. Dinh and Sergio R. Lopez-Permouth 17.1 Introduction . 17.2 Chain Rings, Galois Rings, and Alternative Distances . 17.3 Constacyclic Codes over Arbitrary Commutative Finite Rings . 17.4 Simple-Root Cyclic and Negacyclic Codes over Finite Chain Rings . 17.5 Repeated-Root Constacyclic Codes over Galois Rings . 17.6 Repeated-Root Constacyclic Codes over
R = Fpm + upm, u2 = 0 . 17.6.1 All Constacyclic Codes of Length p3 over 77. 17.6.2 AU Constacyclic Codes of Length 2ps over H. 17.6.3 All Constacyclic Codes of Length 4p3 over Ή. 17.6.4 λ-Constacyclic Codes of Length np3 over R, λ G F*m. 17.7 Extensions . 18 Weight Distribution of Trace Codes over Finite Rings Minjia Shi 18.1 Introduction . 18.2 Preliminaries. 18.3 A Class of Special Finite Rings Rk (TypeI) . 18.3.1 Case (i) к = 1. 18.3.2 Case (ii) к = 2. 18.3.3 Case (in) к 2. 18.3.4 Case (iv) Rk(ph P an Odd Prime. 18.4 A Class of Special Finite Rings R(k.,p, uk= a) (Type II) . 18.5 Three Special Rings . 18.5.1 R(2,p,u2 =
u). 18.5.2 R(3,2,u3 = l). 18.5.3 R(3,3, u3 = l). 18.6 Conclusion . 19 Two-Weight Codes Andries E. Brouwer 19.1 Generalities. 19.2 Codes as Projective Multisets. 19.2.1 Weights . 19.3 Graphs . 19.3.1 Difference Sets. 19.3.2 Using a Projective Setas a Difference Set . 19.3.3 Strongly RegularGraphs. 19.3.4 Parameters. 19.3.5 Complement. xiii 369 373 374 375 377 379 381 382 383 385 385 387 391 392 399 407 407 412 414 418 423 429 429 430 430 431 432 433 435 441 444 444 446 447 448
449 449 450 450 451 451 451 451 452 453
Contents xiv 19.4 19.5 19.6 19.7 19.8 19.9 19.3.6 Duality . 19.3.7 Field Change . Irreducible Cyclic Two-WeightCodes . Cyclotomy . 19.5.1 The Van Lint-SchrijverConstruction. 19.5.2 The De Lange Graphs . 19.5.3 Generalizations. Rank 3 Groups. 19.6.1 One-Dimensional AffineRank 3 Groups. Two-Character Sets in ProjectiveSpace. 19.7.1 Subspaces. 19.7.2 Quadrics. 19.7.3 Maximal Arcs and Hyperovals. 19.7.4 Baer Subspaces. 19.7.5 Hermitian Quadrics. 19.7.6 Sporadic
Examples. Nonprojective Codes. Brouwer-Van Eupen Duality . 19.9.1 From Projective Code to Two-Weight Code . 19.9.2 From Two-Weight Code to Projective Code . 20 Linear Codes from Functions 453 453 454 454 455 456 456 456 456 457 458 458 459 459 459 459 461 461 461 462 463 Sihem Mesnager 20.1 Introduction . 465 20.2 Preliminaries. 466 20.2.1 The Trace Function. 466 20.2.2 Vectorial Functions. 466 20.2.2.1 Representations of p-Ary Functions. 466 20.2.2.2 The Walsh Transform ofa Vectorial Function. 468 20.2.3 Nonlinearity of Vectorial Boolean Functions and Bent Boolean Functions. 469 20.2.4 Plateaued Functions and More about Bent Functions. 471 20.2.5 Differential Uniformity of Vectorial Boolean Functions, PN, and APN
Functions. 473 20.2.6 APN and Planar Functions over Fg™ . 474 20.2.7 Dickson Polynomials. 475 20.3 Generic Constructions of Linear Codes . 476 20.3.1 The First Generic Construction. 476 20.3.2 The Second Generic Construction. 478 20.3.2.1 The Defining Set Construction of Linear Codes. 478 20.3.2.2 Generalizations of the Defining Set Construction of Lin ear Codes. 479 20.3.2.3 A Modified Defining Set Construction of Linear Codes . 479 20.4 Binary Codes with Few Weights from Boolean Functions and Vectorial Boolean Functions. 479 20.4.1 A First Example of Codes from Boolean Functions: Reed-Muller Codes. . 479 20.4.2 A General Construction of Binary Linear Codes from Boolean Functions. 480 20.4.3 Binary Codes from thePreimage/-1(Ь)of BooleanFunctions f . 480 20.4.4 Codes withFew Weights fromBentBoolean Functions. 481
Contents XV Codes with Few Weights from Semi-Bent Boolean Functions . . . 482 Linear Codes from Quadratic Boolean Functions. 483 Binary Codes Cot with Three Weights. 484 Binary Codes Cot with Four Weights. 484 Binary Codes Cd{ with at Most Five Weights. 485 A Class of Two-Weight Binary Codes from the Preimage of a Type of Boolean Function. 486 20.4.11 Binary Codes from Boolean Functions Whose Supports are Rela tive Difference Sets. 487 20.4.12 Binary Codes with Few Weights from Plateaued Boolean Functions 487 20.4.13 Binary Codes with Few Weights from Almost Bent Functions . . 488 20.4.14 Binary Codes C^g) from Functions on Fa™ of the Form G(x) — F(x) + F(x + 1) + 1 . 489 20.4.15 Binary Codes from the Images of Certain Functions on Fa™ . . . 489 Constructions of Cyclic Codes from Functions: The Sequence Approach . 490 20.5.1 A Generic Construction of Cyclic Codes with Polynomials . 490 20.5.2 Binary Cyclic Codes from APN Functions. 492 20.5.3 Non-Binary Cyclic Codes from Monomials and Trinomials . 495 20.5.4 Cyclic Codes from Dickson Polynomials. 498 Codes with Few Weights from p-Ary Functions with p Odd . 503 20.6.1 Codes with Few Weights from
p-Ary Weakly Regular Bent Func tions Based on the First Generic Construction. 503 20.6.2 Linear Codes with Few Weights from Cyclotomie Classes and Weakly Regular Bent Functions. 504 20.6.3 Codes with Few Weights from p-Ary Weakly Regular Bent Func tions Based on the Second Generic Construction. 507 20.6.4 Codes with Few Weights from p-Ary Weakly Regular Plateaued Functions Based on the First Generic Construction. 508 20.6.5 Codes with Few Weights from p-Ary Weakly Regular Plateaued Functions Based on the Second Generic Construction. 510 Optimal Linear Locally Recoverable Codes from p-Ary Functions. 521 20.7.1 Constructions of r-Good Polynomials for Optimal LRC Codes . . 523 20.7.1.1 Good Polynomials from Power Functions. 523 20.7.1.2 Good Polynomials from Linearized Functions. 523 20.7.1.3 Good Polynomials from Function Composition. 523 20.7.1.4 Good Polynomials from Dickson Polynomials of the First Kind. 524 20.7.1.5 Good Polynomials from the Composition of Functions In volving Dickson Polynomials. 525 20.4.5 20.4.6 20.4.7 20.4.8 20.4.9 20.4.10 20.5 20.6 20.7 21 Codes over Graphs Christine A. Kelley 21.1 Introduction . 21.2 Low-Density Parity Check Codes.
21.3 Decoding . 21.3.1 Decoder Analysis. 21.4 Codes from Finite Geometries . 21.5 Codes from Expander Graphs . 21.6 Protograph Codes . 21.7 Density Evolution . 21.8 Other Families of Codes over Graphs . 527 527 528 532 537 540 542 545 548 550
Contents xvi 21.8.1 21.8.2 21.8.3 III Turbo Codes. Repeat Accumulate Codes. Spatially-Coupled LDPC Codes. Applications 22 Alternative Metrics Marcelo Firer 22.1 Introduction . 22.2 Metrics Generated by Subspaces . 22.2.1 Projective Metrics. 22.2.2 Combinatorial Metrics. 22.2.2.1 Block Metrics. 22.2.2.2 -Burst Metrics. 22.2.2.3 i X b2-Burst Metrics. 22.3 Poset Metrics. 22.3.1 Poset-Block Metrics. 22.3.2 Graph Metrics. 22.4 Additive Generalizations of the Lee Metric . 22.4.1 Metrics over Rings of
Integers. 22.4.2 Z-Dimensional Lee Weights. 22.4.3 Kaushik-Sharma Metrics. 22.5 Non-Additive Metrics Digging into the Alphabet. 22.5.1 Pomset Metrics. 22.5.2 m-Spotty Metrics. 22.6 Metrics for Asymmetric Channels . 22.6.1 The Asymmetric Metric . 22.6.2 The Generalized Asymmetric Metric . 22.7 Editing Metrics. 22.7.1 Bounds for Editing Codes . 22.8 Permutation Metrics. 23 Algorithmic Methods Alfred Wassermann 23.1 Introduction . 23.2 Linear Codes with Prescribed Minimum Distance . 23.3 Linear Codes as Sets of Points in Projective Geometry. 23.3.1 Automorphisms ofProjective Point Sets. 23.4 Projective Point
Sets with Prescribed Automorphism Groups. 23.4.1 Strategiesfor Choosing Groups. 23.4.2 Observations for Permutation Groups. 23.4.3 Observations for Cyclic Groups. 23.5 Solving Strategies . 23.6 Construction of Codes with Additional Restrictions . 23.6.1 Projective Codes . 23.6.2 Codes with Few Weights. 23.6.3 Divisible Codes. 23.6.4 Codes with PrescribedGram Matrix. 23.6.5 Self-Orthogonal Codes. 23.6.6 LCD Codes . 23.7 Extensions of Codes. 550 551 551 553 555 555 558 558 559 561 561 562 562 565 565 566 566 567 567 568 568 569 570 570 571 571 572 573 575 575 576 577 580 582 585 585 586 587 587 587 587 588 589 591 592 594
Contents 23.8 Determining the Minimum Distance and Weight Distribution . xvii 595 24 Interpolation Decoding 599 Swastik Kopparty 24.1 Introduction . 599 24.2 The Berlekamp-Welch Algorithm . 600 24.2.1 Correctness of the Algorithm RSDecode . 602 24.3 List-decoding of Reed-Solomon Codes. 603 24.3.1 The Sudan Algorithm. 603 24.3.2 Correctness of the Algorithm RSListDecodeVl . 604 24.4 List-decoding of Reed-Solomon Codes Using Multiplicities . 605 24.4.1 Preparations. 606 24.4.2 The Guruswami-Sudan Algorithm. . 607 24.4.3 Correctness of the Algorithm RSListDecodeV2. 608 24.4.4 Why Do Multiplicities Help?. 608 24.5 Decoding of Interleaved Reed-Solomon Codes under Random Error . . . 609 24.6 Further Reading . 611 25 Pseudo-Noise Sequences 613 Tor Helleseth and Chunlei Li 25.1 Introduction
. 613 25.2 Sequences with Low Correlation . 614 25.2.1 Correlation Measuresof Sequences . 614 25.2.2 Sequences with Low Periodic Autocorrelation. 619 25.2.3 Sequence Families with Low Periodic Correlation. 624 25.3 Shift Register Sequences. 626 25.3.1 Feedback Shift Registers. 627 25.3.2 Cycle Structure. 628 25.3.3 Cycle Joining and Splitting. 630 25.3.4 Cycle Structure of LFSRs . 631 25.4 Generation of De Bruijn Sequences. 634 25.4.1 Graphical Approach. 634 25.4.2 Combinatorial Approach. 636 25.4.3 Algebraic Approach. 640 26 Lattice Coding 645 Frédérique Oggier 26.1 Introduction . 645 26.2 Lattice Coding for the
Gaussian Channel . . 646 26.3 Modulation Schemes for Fading Channels. 648 26.3.1 Channel Model and Design Criteria. 648 26.3.2 Lattices from Quadratic Fields. 649 26.4 Lattices from Linear Codes . 651 26.4.1 Construction A. 651 26.4.2 Constructions D and D. 653 26.5 Variations of Lattice Coding Problems. 654 26.5.1 Index Codes. 655 26.5.2 Wiretap Codes. 655
xviii Contents 27 Quantum Error-Control Codes Martianus Frederic Ezerman 27.1 Introduction . 27.2 Preliminaries. 27.3 The Stabilizer Formalism . 27.4 Constructions via Classical Codes . 27.5 Going Asymmetric. 27.6 Other Approaches and a Conclusion. 28 Space-Time Coding Frédérique Oggier 28.1 Introduction . 28.2 Channel Models and Design Criteria. 28.2.1 Coherent Space-Time Coding . 28.2.2 Differential Space-Time Coding. 28.3 Some Examples of Space-Time Codes . 28.3.1 The Alamouti Code. 28.3.2 Linear Dispersion Codes. 28.3.3 The Golden
Code. 28.3.4 Cayley Codes. 28.4 Variations of Space-Time Coding Problems . 28.4.1 Distributed Space-Time Coding. 28.4.2 Space-Time Coded Modulation . 28.4.3 Fast Decodable Space-Time Codes. 28.4.4 Secure Space-Time Coding. 29 Network Codes Frank R. Kschischang 29.1 Packet Networks. 29.2 Multicasting from a Single Source . 29.2.1 Combinational Packet Networks. 29.2.2 Network Information Flow Problems. 29.2.3 The Unicast Problem. 29.2.4 Linear Network Coding Achieves Multicast Capacity. 29.2.5 Multicasting from Multiple Sources. 29.3 Random Linear Network Coding. 29.4 Operator
Channels. 29.4.1 Vector Space, Matrix, and Combinatorial Preliminaries. 29.4.2 The Operator Channel. 29.5 Codes and Metrics in Pq(n). 29.5.1 Subspace Codes. 29.5.2 Coding Metrics on Pq(n). 29.6 Bounds on Constant-Dimension Codes. 29.6.1 The Sphere Packing Bound. 29.6.2 The Singleton Bound. 29.6.3 The Anticode Bound. 29.6.4 Johnson-Type Bounds . 29.6.5 The Ahiswede and Aydinian Bound. 29.6.6 A Gilbert-Varshamov-Type Bound. 29.7 Constructions . 29.7.1 Lifted Rank-Metric Codes. 657 657 658 660 665 669 671 673 673 674 675 676 677 677 678 678 679 680 680 681 681 683 685 685 687 687 689 689 690 691 692 694 694 697
698 698 699 701 701 702 702 703 704 704 705 705
Contents 29.8 29.9 29.7.2 Padded Codes. 29.7.3 Lifted Ferrers Diagram Codes. 29.7.4 Codes Obtained by Integer Linear Programming. 29.7.5 Further Constructions . Encoding and Decoding. 29.8.1 Encoding a Union of Lifted FD Codes . 29.8.2 Decoding Lifted Delsarte-Gabidulin Codes. 29.8.3 Decoding a Union of Lifted FD Codes. Conclusions. 30 Coding for Erasures and Fountain Codes Ian F. Blake 30.1 Introduction . 30.2 Tornado Codes. 30.3 LT Codes. 30.4 Raptor Codes. 31 Codes for Distributed Storage Vinayak Ramkumar, Myna Vajha, S. B. Balaji, Μ. Nikhil Krishnan, Birenjith Sasidharan, and P. Vijay Kumar 31.1 Reed-Solomon Codes
. 31.2 Regenerating Codes . 31.2.1 An Example of a Regenerating Code and Sub-Packetization . . . 31.2.2 General Definition of a Regenerating Code. 31.2.3 Bound on File Size. 31.2.4 MSR and MBR Codes. 31.2.5 Storage Bandwidth Tradeoff for Exact-Repair . . . . 31.2.6 Polygon MBR Codes. . . . 31.2.7 The Product-Matrix MSR and MBR Constructions. 31.2.7.1 PM-MSR Codes. 31.2.7.2 PM-MBR Codes . 31.2.8 The Clay Code . 31.2.9 Variants of Regenerating Codes. 31.3 Locally Recoverable Codes . 31.3.1 Information Symbol Locality. 31.3.1.1 Pyramid Codes. 31.3.1.2 Windows Azure LRC. 31.3.2 All Symbol
Locality. 31.3.3 LRCs over Small Field Size. 31.3.4 Recovery from Multiple Erasures . 31.3.4.1 Codes with Sequential Recovery. 31.3.4.2 Codes with Parallel Recovery. 31.3.4.3 Codes with Availability . 31.3.4.4 Codes with Cooperative Recovery. 31.3.4.5 Codes with (r, 5) Locality. 31.3.4.6 Hierarchical Codes. 31.3.5 Maximally Recoverable Codes. 31.4 Locally Regenerating Codes. 31.5 Efficient Repair of Reed-Solomon Codes. 31.6 Codes for Distributed Storage in Practice. xix 707 708 710 710 711 711 711 712 713 715 715 717 722 727 735 737 738 739 739 740 741 742 742 743 743 744 745 748 749 750 750 751 751 753 754 754 755 755 756 756 757 757 758 759 760
Contents XX 32 Polar Codes Noam Presman and Simon Litsyn 32.1 Introduction . 32.2 Kernel Based ECCs . 32.2.1 Kernel Based ECCs are Recursive GCCs. 32.3 Channel Splitting and Combining and the SC Algorithm . 32.4 Polarization Conditions . 32.4.1 Polarization Rate. 32.5 Polar Codes . 32.5.1 Polar Code Design . 32.6 Polar Codes Encoding Algorithms . 32.7 Polar Codes Decoding Algorithms . 32.7.1 The SC Decoding Algorithm. 32.7.1.1 SC for (u + v,v) . 32.7.1.2 SC for General Kernels. 32.7.1.3 SC Complexity. 32.7.2 The SCL Decoding Algorithm. 32.8 Summary and
Concluding Remarks . 33 Secret Sharing with Linear Codes Cunsheng Ding 33.1 Introduction to Secret Sharing Schemes . 33.2 The First Construction of Secret Sharing Schemes . 33.3 The Second Construction of Secret Sharing Schemes. 33.3.1 Minimal Linear Codes and the Covering Problem. 33.3.2 The Second Construction of Secret Sharing Schemes. 33.3.3 Secret Sharing Schemes from the Duals of Minimal Codes . 33.3.4 Other Works on the Second Construction. 33.4 Multisecret Sharing with Linear Codes . 33.4.1 The Relation Between Multisecret Sharing and Codes. 33.4.2 Linear Threshold Schemes and MDS Codes. 34 Code-Based Cryptography 763 763 764 766 769 771 774 775 775 776 777 778 778 780 781 781 783 785 785 787 790 790 791 792 793 794 795 796 799 Philippe Gaborit and Jean-Christophe Deneuville 34.1 Preliminaries. 800 34.1.1 Notation. 800 34.1.2 Background on Coding Theory. 801 34.2 Difficult Problems for Code-Based Cryptography: The Syndrome Decoding Problem
and Its Variations . 803 34.3 Best-Known Attacks for the Syndrome Decoding Problem. 804 34.4 Public-Key Encryption from Coding Theory with Hidden Structure . . . 807 34.4.1 The McEliece and Niederreiter Frameworks . 807 34.4.2 Group-Structured McEliece Framework. 809 34.4.3 Moderate-Density Parity Check (MDPC) Codes. 810 34.5 PKE Schemes with Reduction to Decoding Random Codes without Hidden Structure . 812 34.5.1 Alekhnovich’s Approach. 812 34.5.2 HQC: Efficient Encryption from Random Quasi-Cyclic Codes . . 813 34.5.3 Ouroboros Key-Exchange Protocol . 814 34.6 Examples of Parameters for Code-Based Encryption and Key Exchange . 815 34.7 Authentication: The Stern Zero-Knowledge Protocol. 816
Contents xxi 34.8 Digital Signatures from Coding Theory . 34.8.1 Signature from a Zero-Knowledge Authentication Scheme with the Fiat-Shamir Heuristic . 818 34.8.2 The CFS Signature Scheme . 34.8.3 The WAVE Signature. 34.8.4 Few-Times Signature Schemes and Variations. 34.9 OtherPrimitives. 34.10 Rank-Based Cryptography . 817 818 819 820 820 820 Bibliography 823 Index 941 |
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| illustrated | Illustrated |
| index_date | 2024-07-03T17:13:06Z |
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| institution | BVB |
| isbn | 9781138551992 9780367709327 |
| language | English |
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| spelling | Concise encyclopedia of coding theory edited by W. Cary Huffman (Loyola University Chicago, USA), Jon-Lark Kim (Sogang University, Republic of Korea), Patrick Solé (University Aix-Marseille, Marseilles, France) First edition Boca Raton ; London ; New York CRC Press [2021] © 2021 xxxii, 965 Seiten Illustrationen txt rdacontent sti rdacontent n rdamedia nc rdacarrier Codierungstheorie (DE-588)4139405-7 gnd rswk-swf Coding theory / Encyclopedias Codierungstheorie (DE-588)4139405-7 s DE-604 Huffman, William C. 1931- (DE-588)133058263 edt Kim, Jon-Lark 1970- (DE-588)1015270689 edt Solé, Patrick (DE-588)1209971208 edt Erscheint auch als Online-Ausgabe, PDF 978-1-351-37510-8 Erscheint auch als Online-Ausgabe, EPUB 978-1-351-37509-2 Erscheint auch als Online-Ausgabe 978-1-315-14790-1 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032671193&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| subject_GND | (DE-588)4139405-7 |
| title | Concise encyclopedia of coding theory |
| title_auth | Concise encyclopedia of coding theory |
| title_exact_search | Concise encyclopedia of coding theory |
| title_exact_search_txtP | Concise encyclopedia of coding theory |
| title_full | Concise encyclopedia of coding theory edited by W. Cary Huffman (Loyola University Chicago, USA), Jon-Lark Kim (Sogang University, Republic of Korea), Patrick Solé (University Aix-Marseille, Marseilles, France) |
| title_fullStr | Concise encyclopedia of coding theory edited by W. Cary Huffman (Loyola University Chicago, USA), Jon-Lark Kim (Sogang University, Republic of Korea), Patrick Solé (University Aix-Marseille, Marseilles, France) |
| title_full_unstemmed | Concise encyclopedia of coding theory edited by W. Cary Huffman (Loyola University Chicago, USA), Jon-Lark Kim (Sogang University, Republic of Korea), Patrick Solé (University Aix-Marseille, Marseilles, France) |
| title_short | Concise encyclopedia of coding theory |
| title_sort | concise encyclopedia of coding theory |
| topic | Codierungstheorie (DE-588)4139405-7 gnd |
| topic_facet | Codierungstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032671193&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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