Error correction coding: mathematical methods and algorithms
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Hoboken, N.J.
Wiley-Interscience
2005
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| Schlagworte: | |
| Online-Zugang: | Table of contents only Contributor biographical information Publisher description Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | xlii, 755 p. ill. 24 cm |
| ISBN: | 0471648000 |
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| 100 | 1 | |a Moon, Todd K. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Error correction coding |b mathematical methods and algorithms |c Todd K. Moon |
| 264 | 1 | |a Hoboken, N.J. |b Wiley-Interscience |c 2005 | |
| 300 | |a xlii, 755 p. |b ill. |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 7 | |a Coderingstheorie |2 gtt | |
| 650 | 4 | |a Codes correcteurs d'erreurs (Théorie de l'information) | |
| 650 | 7 | |a Datatransmissie |2 gtt | |
| 650 | 7 | |a Foutenleer |2 gtt | |
| 650 | 4 | |a Mathématiques de l'ingénieur | |
| 650 | 4 | |a Engineering mathematics | |
| 650 | 4 | |a Error-correcting codes (Information theory) | |
| 650 | 0 | 7 | |a Fehlerkorrekturcode |0 (DE-588)4124917-3 |2 gnd |9 rswk-swf |
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| 856 | 4 | |u http://www.loc.gov/catdir/enhancements/fy0617/2004031019-b.html |3 Contributor biographical information | |
| 856 | 4 | |u http://www.loc.gov/catdir/enhancements/fy0617/2004031019-d.html |3 Publisher description | |
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Datensatz im Suchindex
| _version_ | 1804136487942356992 |
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| adam_text | ERROR CORRECTION CODING MATHEMATICAL METHODS AND ALGORITHMS TODD K. MOON
UTAH STATE UNIVERSITY IWILEY- INTERSCIENCE A JOHN WILEY & SONS, INC.,
PUBLICATION CONTENTS PREFACE LIST OF PROGRAM FILES LIST OF LABORATORY
EXERCISES LIST OF ALGORITHMS LIST OF FIGURES LIST OF TABLES LIST OF
BOXES VU XXXI XXXII XXXIV XL XLII XLIII PART I INTRODUCTION AND
FOUNDATIONS 1 1 A CONTEXT FOR ERROR CORRECTION CODING 2 1.1 PURPOSE OF
THIS BOOK 2 1.2 INTRODUCTION: WHERE ARE CODES? 2 1.3 THE COMMUNICATIONS
SYSTEM 4 1.4 BASIC DIGITAL COMMUNICATIONS 9 1.4.1 BINARY PHASE-SHIFT
KEYING 10 1.4.2 MORE GENERAL DIGITAL MODULATION 11 1.5 SIGNAL DETECTION
14 1.5.1 THE GAUSSIAN CHANNEL 14 1.5.2 MAP AND ML DETECTION 16 1.5.3
SPECIAL CASE: BINARY DETECTION 18 1.5.4 PROBABILITY OF ERROR FOR BINARY
DETECTION 19 1.5.5 BOUNDS ON PERFORMANCE: THE UNION BOUND 22 1.5.6 THE
BINARY SYMMETRIC CHANNEL 23 1.5.7 THE BSC AND THE GAUSSIAN CHANNEL MODEL
25 1.6 MEMORY LESS CHANNELS 25 1.7 SIMULATION AND ENERGY CONSIDERATIONS
FOR CODED SIGNALS 26 1.8 SOME IMPORTANT DEFINITIONS 27 1.8.1 DETECTION
OF REPETITION CODES OVER A BSC 28 1.8.2 SOFT-DECISION DECODING OF
REPETITION CODES OVER THE AWGN ... 32 1.8.3 SIMULATION OF RESULTS 33
1.8.4 SUMMARY 33 1.9 HAMMING CODES 34 1.9.1 HARD-INPUT DECODING
HAMMING CODES 35 1.9.2 OTHER REPRESENTATIONS OF THE HAMMING CODE 36 AN
ALGEBRAIC REPRESENTATION 37 A POLYNOMIAL REPRESENTATION 37 XW CONTENTS A
TRELLIS REPRESENTATION 38 THE TANNER GRAPH REPRESENTATION 38 1.10 THE
BASIC QUESTIONS 39 1.11 HISTORICAL MILESTONES OF CODING THEORY 40 1.12 A
BIT OF INFORMATION THEORY 40 1.12.1 DEFINITIONS FOR DISCRETE RANDOM
VARIABLES 40 ENTROPY AND CONDITIONAL ENTROPY 40 RELATIVE ENTROPY, MUTUAL
INFORMATION, AND CHANNEL CAPACITY .... 41 1.12.2 DEFINITIONS FOR
CONTINUOUS RANDOM VARIABLES 43 1.12.3 THE CHANNEL CODING THEOREM 45
1.12.4 PROOF OF THE CHANNEL CODING THEOREM 45 1.12.5 CAPACITY FOR THE
CONTINUOUS-TIME AWGN CHANNEL 49 1.12.6 TRANSMISSION AT CAPACITY WITH
ERRORS 51 1.12.7 THE IMPLICATION OF THE CHANNEL CODING THEOREM 52 LAB 1
SIMULATING A COMMUNICATIONS CHANNEL 53 OBJECTIVE 53 BACKGROUND 53 USE OF
CODING IN CONJUNCTION WITH THE BSC 53 ASSIGNMENT 54 PROGRAMMING PART 54
RESOURCES AND IMPLEMENTATION SUGGESTIONS 54 1.13 EXERCISES 56 1.14
REFERENCES 60 PART II BLOCK CODES 61 2 GROUPS AND VECTOR SPACES 62 2.1
INTRODUCTION 62 2.2 GROUPS 62 2.2.1 SUBGROUPS 65 2.2.2 CYCLIC GROUPS AND
THE ORDER OF AN ELEMENT 66 2.2.3 COSETS 67 2.2.4 LAGRANGE S THEOREM 68
2.2.5 INDUCED OPERATIONS; ISOMORPHISM 69 2.2.6 HOMOMORPHISM 72 2.3
FIELDS: A PRELUDE . . . * 73 2.4 REVIEW OF LINEAR ALGEBRA 75 2.5
EXERCISES 80 2.6 REFERENCES 82 3 LINEAR BLOCK CODES 83 3.1 BASIC
DEFINITIONS 83 3.2 THE GENERATOR MATRIX DESCRIPTION OF LINEAR BLOCK
CODES 84 3.2.1 RUDIMENTARY IMPLEMENTATION 86 3.3 THE PARITY CHECK MATRIX
AND DUAL CODES 86 3.3.1 SOME SIMPLE BOUNDS ON BLOCK CODES 88 3.4 ERROR
DETECTION AND CORRECTION OVER HARD-INPUT CHANNELS 90 CONTENTS XV 3.4.1
ERROR DETECTION 90 3.4.2 ERROR CORRECTION: THE STANDARD ARRAY 90 3.5
WEIGHT DISTRIBUTIONS OF CODES AND THEIR DUALS 95 3.6 HAMMING CODES AND
THEIR DUALS 97 3.7 PERFORMANCE OF LINEAR CODES 98 3.7.1 ERROR DETECTION
PERFORMANCE 99 3.7.2 ERROR CORRECTION PERFORMANCE 100 3.7.3 PERFORMANCE
FOR SOFT-DECISION DECODING 103 3.8 ERASURE DECODING 104 3.8.1 BINARY
ERASURE DECODING 105 3.9 MODIFICATIONS TO LINEAR CODES 105 3.10 BEST
KNOWN LINEAR BLOCK CODES 107 3.11 EXERCISES 107 3.12 REFERENCES 112
CYCLIC CODES, RINGS, AND POLYNOMIALS 113 4.1 INTRODUCTION 113 4.2 BASIC
DEFINITIONS 113 4.3 RINGS 114 4.3.1 RINGS OF POLYNOMIALS 115 4.4
QUOTIENT RINGS 116 4.5 IDEALS IN RINGS 118 4.6 ALGEBRAIC DESCRIPTION OF
CYCLIC CODES 120 4.7 NONSYSTEMATIC ENCODING AND PARITY CHECK 122 4.8
SYSTEMATIC ENCODING 124 4.9 SOME HARDWARE BACKGROUND 126 4.9.1
COMPUTATIONAL BUILDING BLOCKS 126 4.9.2 SEQUENCES AND POWER SERIES 127
4.9.3 POLYNOMIAL MULTIPLICATION 128 LAST-ELEMENT-FIRST PROCESSING 128
FIRST-ELEMENT-FIRST PROCESSING 128 4.9.4 POLYNOMIAL DIVISION 129
LAST-ELEMENT-FIRST PROCESSING 129 4.9.5 SIMULTANEOUS POLYNOMIAL DIVISION
AND MULTIPLICATION 132 FIRST-ELEMENT-FIRST PROCESSING 132 4.10 CYCLIC
ENCODING 133 4.11 SYNDROME DECODING 137 4.12 SHORTENED CYCLIC CODES 143
METHOD 1: SIMULATING THE EXTRA CLOCK SHIFTS 144 METHOD 2: CHANGING THE
ERROR PATTERN DETECTION CIRCUIT 147 4.13 BINARY CRC CODES 147 4.13.1
BYTE-ORIENTED ENCODING AND DECODING ALGORITHMS 150 4.13.2 CRC PROTECTING
DATA FILES OR DATA PACKETS 153 APPENDIX 4. A LINEAR FEEDBACK SHIFT
REGISTERS 154 APPENDIX 4.A. 1 BASIC CONCEPTS 154 APPENDIX 4.A.2
CONNECTION WITH POLYNOMIAL DIVISION 157 APPENDIX 4. A.3 SOME ALGEBRAIC
PROPERTIES OF SHIFT SEQUENCES 160 XVI CONTENTS LAB 2 POLYNOMIAL DIVISION
AND LINEAR FEEDBACK SHIFT REGISTERS ... 161 OBJECTIVE 161 PRELIMINARY
EXERCISES 161 PROGRAMMING PART: BINLFSR 161 RESOURCES AND IMPLEMENTATION
SUGGESTIONS 161 PROGRAMMING PART: BINPOLYDIV 162 FOLLOW-ON IDEAS AND
PROBLEMS 162 LAB 3 CRC ENCODING AND DECODING 162 OBJECTIVE 163
PRELIMINARY 163 PROGRAMMING PART 163 RESOURCES AND IMPLEMENTATION
SUGGESTIONS 163 4.14 EXERCISES 165 4.15 REFERENCES 170 5 RUDIMENTS OF
NUMBER THEORY AND ALGEBRA 171 5.1 MOTIVATION 171 5.2 NUMBER THEORETIC
PRELIMINARIES 175 5.2.1 DIVISIBILITY 175 5.2.2 THE EUCLIDEAN ALGORITHM
AND EUCLIDEAN DOMAINS 177 5.2.3 THE SUGIYAMA ALGORITHM 182 5.2.4
CONGRUENCE 184 5.2.5 THE FUNCTION 185 5.2.6 SOME CRYPTOGRAPHIC PAYOFF
186 FERMAT S LITTLE THEOREM 186 RSA ENCRYPTION 187 5.3 THE CHINESE
REMAINDER THEOREM 188 5.3.1 THE CRT AND INTERPOLATION 190 THE EVALUATION
HOMOMORPHISM 190 THE INTERPOLATION PROBLEM 191 5.4 FIELDS 193 5.4.1 AN
EXAMINATION OF R AND C 194 5.4.2 GALOIS FIELD CONSTRUCTION: AN EXAMPLE
196 5.4.3 CONNECTION WITH LINEAR FEEDBACK SHIFT REGISTERS 199 5.5 GALOIS
FIELDS: MATHEMATICAL FACTS 200 5.6 IMPLEMENTING GALOIS FIELD ARITHMETIC
204 5.6.1 ZECH LOGARITHMS 204 5.6.2 HARDWARE IMPLEMENTATIONS 205 5.7
SUBFIELDS OF GALOIS FIELDS 206 5.8 IRREDUCIBLE AND PRIMITIVE POLYNOMIALS
207 5.9 CONJUGATE ELEMENTS AND MINIMAL POLYNOMIALS 209 5.9.1 MINIMAL
POLYNOMIALS 212 5.10 FACTORING* - 1 215 5.11 CYCLOTOMIC COSETS 217
APPENDIX 5.A HOW MANY IRREDUCIBLE POLYNOMIALS ARE THERE? 218 APPENDIX
5.A.1 SOLVING FOR I M EXPLICITLY: THE MOEBIUS FUNCTION 222 LAB 4
PROGRAMMING THE EUCLIDEAN ALGORITHM 223 CONTENTS XVII OBJECTIVE 223
PRELIMINARY EXERCISES 223 BACKGROUND 223 PROGRAMMING PART 223 LAB 5
PROGRAMMING GALOIS FIELD ARITHMETIC 224 OBJECTIVE 224 PRELIMINARY
EXERCISES 224 PROGRAMMING PART 224 5.12 EXERCISES 225 5.13 REFERENCES
234 BCH AND REED-SOLOMON CODES: DESIGNER CYCLIC CODES 235 6.1 BCH CODES
235 6.1.1 DESIGNING BCH CODES 235 6.1.2 THE BCH BOUND 237 6.1.3 WEIGHT
DISTRIBUTIONS FOR SOME BINARY BCH CODES 239 6.1.4 ASYMPTOTIC RESULTS FOR
BCH CODES 240 6.2 REED-SOLOMON CODES 242 6.2.1 REED-SOLOMON CONSTRUCTION
1 242 6.2.2 REED-SOLOMON CONSTRUCTION 2 243 6.2.3 ENCODING REED-SOLOMON
CODES 244 6.2.4 MDS CODES AND WEIGHT DISTRIBUTIONS FOR RS CODES 245 6.3
DECODING BCH AND RS CODES: THE GENERAL OUTLINE 247 6.3.1 COMPUTATION OF
THE SYNDROME 247 6.3.2 THE ERROR LOCATOR POLYNOMIAL 248 6.3.3 CHIEN
SEARCH 248 6.4 FINDING THE ERROR LOCATOR POLYNOMIAL 250 6.4.1
SIMPLIFICATIONS FOR BINARY CODES AND PETERSON S ALGORITHM 251 6.4.2
BERLEKAMP-MASSEY ALGORITHM 253 6.4.3 CHARACTERIZATION OF LFSR LENGTH IN
MASSEY S ALGORITHM 255 6.4.4 SIMPLIFICATIONS FOR BINARY CODES 259 6.5
NON-BINARY BCH AND RS DECODING 261 6.5.1 FORNEY S ALGORITHM 262 6.6
EUCLIDEAN ALGORITHM FOR THE ERROR LOCATOR POLYNOMIAL 266 6.7 ERASURE
DECODING FOR NONBINARY BCH OR RS CODES 267 6.8 GALOIS FIELD FOURIER
TRANSFORM METHODS 269 6.8.1 EQUIVALENCE OF THE TWO REED-SOLOMON CODE
CONSTRUCTIONS .... 274 6.8.2 FREQUENCY-DOMAIN DECODING 275 6.9
VARIATIONS AND EXTENSIONS OF REED-SOLOMON CODES 276 6.9.1 SIMPLE
MODIFICATIONS 276 6.9.2 GENERALIZED REED-SOLOMON CODES AND ALTERNANT
CODES 277 6.9.3 GOPPA CODES 278 6.9.4 DECODING ALTERNANT CODES 280 6.9.5
THE MCELIECE PUBLIC KEY CRYPTOSYSTEM 280 LAB 6 PROGRAMMING THE
BERLEKAMP-MASSEY ALGORITHM 281 BACKGROUND 281 ASSIGNMENT 281 XVIII
CONTENTS PRELIMINARY EXERCISES 281 PROGRAMMING PART 281 RESOURCES AND
IMPLEMENTATION SUGGESTIONS 282 LAB 7 PROGRAMMING THE BCH DECODER 283
OBJECTIVE 283 PRELIMINARY EXERCISES 283 PROGRAMMING PART 283 RESOURCES
AND IMPLEMENTATION SUGGESTIONS 283 FOLLOW-ON IDEAS AND PROBLEMS 284 LAB
8 REED-SOLOMON ENCODING AND DECODING 284 OBJECTIVE 284 BACKGROUND 284
PROGRAMMING PART 284 APPENDIX 6. A PROOF OF NEWTON S IDENTITIES 285 6.10
EXERCISES 287 6.11 REFERENCES 291 7 ALTERNATE DECODING ALGORITHMS FOR
REED-SOLOMON CODES 293 7.1 INTRODUCTION: WORKLOAD FOR REED-SOLOMON
DECODING 293 7.2 DERIVATIONS OF WELCH-BERLEKAMP KEY EQUATION 293 7.2.1
THE WELCH-BERLEKAMP DERIVATION OF THE WB KEY EQUATION 294 7.2.2
DERIVATION FROM THE CONVENTIONAL KEY EQUATION 298 7.3 FINDING THE ERROR
VALUES 300 7.4 METHODS OF SOLVING THE WB KEY EQUATION 302 7.4.1
BACKGROUND: MODULES 302 7.4.2 THE WELCH-BERLEKAMP ALGORITHM 303 7.4.3
MODULAR SOLUTION OF THE WB KEY EQUATION 310 7.5 ERASURE DECODING WITH
THE WELCH-BERLEKAMP KEY EQUATION 321 7.6 THE GURUSWAMI-SUDAN DECODING
ALGORITHM AND SOFT RS DECODING 322 7.6.1 BOUNDED DISTANCE, ML, AND LIST
DECODING 322 7.6.2 ERROR CORRECTION BY INTERPOLATION 323 7.6.3
POLYNOMIALS IN TWO VARIABLES 324 DEGREE AND MONOMIAL ORDER 325 ZEROS AND
MULTIPLE ZEROS 328 7.6.4 THE GS DECODER: THE MAIN THEOREMS 330 THE
INTERPOLATION THEOREM 331 THE FACTORIZATION THEOREM 331 THE CORRECTION
DISTANCE 333 THE NUMBER OF POLYNOMIALS IN THE DECODING LIST 335 7.6.5
ALGORITHMS FOR COMPUTING THE INTERPOLATION STEP 337 FINDING LINEARLY
DEPENDENT COLUMNS: THE FENG-TZENG ALGORITHM 338 FINDING THE INTERSECTION
OF KERNELS: THE KOTTER ALGORITHM 342 7.6.6 A SPECIAL CASE: M - 1 AND L =
1 348 7.6.7 THE ROTH-RUCKENSTEIN ALGORITHM 350 WHAT TO DO WITH LISTS OF
FACTORS? 354 7.6.8 SOFT-DECISION DECODING OF REED-SOLOMON CODES 358
NOTATION 358 CONTENTS XIX A FACTORIZATION THEOREM 360 MAPPING FROM
RELIABILITY TO MULTIPLICITY 361 THE GEOMETRY OF THE DECODING REGIONS 363
COMPUTING THE RELIABILITY MATRIX 364 7.7 EXERCISES 365 7.8 REFERENCES
368 8 OTHER IMPORTANT BLOCK CODES 369 8.1 INTRODUCTION 369 8.2 HADAMARD
MATRICES, CODES, AND TRANSFORMS 369 8.2.1 INTRODUCTION TO HADAMARD
MATRICES 369 8.2.2 THE PALEY CONSTRUCTION OF HADAMARD MATRICES 371 8.2.3
HADAMARD CODES 374 8.3 REED-MULLER CODES 375 8.3.1 BOOLEAN FUNCTIONS 375
8.3.2 DEFINITION OF THE REED-MULLER CODES 376 8.3.3 ENCODING AND
DECODING ALGORITHMS FOR FIRST-ORDER RM CODES . . . 379 ENCODING RM(L,M)
CODES 379 DECODING RM( ,M) CODES 379 EXPEDITING DECODING USING THE FAST
HADAMARD TRANSFORM 382 8.3.4 THE REED DECODING ALGORITHM FOR RM{R, M)
CODES, R 1 384 DETAILS FOR AN RM(2, 4) CODE 384 A GEOMETRIC VIEWPOINT
387 8.3.5 OTHER CONSTRUCTIONS OF REED-MULLER CODES 391 8.4 BUILDING LONG
CODES FROM SHORT CODES: THE SQUARING CONSTRUCTION .... 392 8.5 QUADRATIC
RESIDUE CODES 396 8.6 GOLAY CODES 398 8.6.1 DECODING THE GOLAY CODE 400
ALGEBRAIC DECODING OF THE 823 GOLAY CODE 400 ARITHMETIC DECODING OF THE
S24 CODE 401 8.7 EXERCISES 403 8.8 REFERENCES 404 9 BOUNDS ON CODES 406
9.1 THE GILBERT-VARSHAMOV BOUND 409 9.2 THE PLOTKIN BOUND 410 9.3 THE
GRIESMER BOUND 411 9.4 THE LINEAR PROGRAMMING AND RELATED BOUNDS 413
9.4.1 KRAWTCHOUK POLYNOMIALS 415 9.4.2 CHARACTER 415 9.4.3 KRAWTCHOUK
POLYNOMIALS AND CHARACTERS 416 9.5 THE MCELIECE-RODEMICH-RUMSEY-WELCH
BOUND 418 9.6 EXERCISES 420 9.7 REFERENCES 424 XX CONTENTS 10 BURSTY
CHANNELS, INTERLEAVES, AND CONCATENATION 425 10.1 INTRODUCTION TO BURSTY
CHANNELS 425 10.2 INTERLEAVES 425 10.3 AN APPLICATION OF INTERLEAVED RS
CODES: COMPACT DISCS 427 10.4 PRODUCT CODES 430 10.5 REED-SOLOMON CODES
431 10.6 CONCATENATED CODES 432 10.7 FIRE CODES 433 10.7.1 FIRE CODE
DEFINITION 433 10.7.2 DECODING FIRE CODES: ERROR TRAPPING DECODING 435
10.8 EXERCISES 437 10.9 REFERENCES 438 11 SOFT-DECISION DECODING
ALGORITHMS 439 11.1 INTRODUCTION AND GENERAL NOTATION 439 11.2
GENERALIZED MINIMUM DISTANCE DECODING 441 11.2.1 DISTANCE MEASURES AND
PROPERTIES 442 11.3 THE CHASE DECODING ALGORITHMS 445 11.4 HALTING THE
SEARCH: AN OPTIMALITY CONDITION 445 11.5 ORDERED STATISTIC DECODING 447
11.6 EXERCISES 449 11.7 REFERENCES 450 PART III CODES ON GRAPHS 451 12
CONVOLUTIONAL CODES 452 12.1 INTRODUCTION AND BASIC NOTATION 452 12.1.1
THE STATE 456 12.2 DEFINITION OF CODES AND EQUIVALENT CODES 458 12.2.1
CATASTROPHIC ENCODERS 461 12.2.2 POLYNOMIAL AND RATIONAL ENCODERS 464
12.2.3 CONSTRAINT LENGTH AND MINIMAL ENCODERS 465 12.2.4 SYSTEMATIC
ENCODERS 468 12.3 DECODING CONVOLUTIONAL CODES 469 12.3.1 INTRODUCTION
AND NOTATION 469 12.3.2 THE VITERBI ALGORITHM 471 12.3.3 SOME
IMPLEMENTATION ISSUES 481 THE BASIC OPERATION: ADD-COMPARE-SELECT 481
DECODING STREAMS OF DATA: WINDOWS ON THE TRELLIS 481 OUTPUT DECISIONS
482 HARD AND SOFT DECODING; QUANTIZATION 484 SYNCHRONIZATION ISSUES 486
12.4 SOME PERFORMANCE RESULTS 487 12.5 ERROR ANALYSIS FOR CONVOLUTIONAL
CODES 491 12.5.1 ENUMERATING PATHS THROUGH THE TRELLIS 493 ENUMERATING
ON MORE COMPLICATED GRAPHS: MASON S RULE 496 CONTENTS XXI 12.5.2
CHARACTERIZING THE NODE ERROR PROBABILITY P E AND THE BIT ERROR RATE PB
498 12.5.3 A BOUND ON PD FOR DISCRETE CHANNELS 501 PERFORMANCE BOUND ON
THE BSC 503 12.5.4 A BOUND ON P D FOR BPSK SIGNALING OVER THE AWGN
CHANNEL ... 503 12.5.5 ASYMPTOTIC CODING GAIN 504 12.6 TABLES OF GOOD
CODES 505 12.7 PUNCTURING 507 12.7.1 PUNCTURING TO ACHIEVE VARIABLE RATE
509 12.8 SUBOPTIMAL DECODING ALGORITHMS FOR CONVOLUTIONAL CODES 510
12.8.1 TREE REPRESENTATIONS 511 12.8.2 THE FANO METRIC 511 12.8.3 THE
STACK ALGORITHM 515 12.8.4 THE FANO ALGORITHM 517 12.8.5 OTHER ISSUES
FOR SEQUENTIAL DECODING 520 12.8.6 A VARIATION ON THE VITERBI ALGORITHM:
THE M ALGORITHM 521 12.9 CONVOLUTIONAL CODES AS BLOCK CODES 522 12.10
TRELLIS REPRESENTATIONS OF BLOCK AND CYCLIC CODES 523 12.10.1 BLOCK
CODES 523 12.10.2 CYCLIC CODES 524 12.10.3 TRELLIS DECODING OF BLOCK
CODES 525 LAB 9 PROGRAMMING CONVOLUTIONAL ENCODERS 526 OBJECTIVE 526
BACKGROUND 526 PROGRAMMING PART 526 LAB 10 CONVOLUTIONAL DECODERS: THE
VITERBI ALGORITHM 528 OBJECTIVE 528 BACKGROUND 528 PROGRAMMING PART 528
12.11 EXERCISES 529 12.12 REFERENCES 533 13 TRELLIS CODED MODULATION 535
13.1 ADDING REDUNDANCY BY ADDING SIGNALS 535 13.2 BACKGROUND ON SIGNAL
CONSTELLATIONS 535 13.3 TCM EXAMPLE 537 13.3.1 THE GENERAL UNGERBOECK
CODING FRAMEWORK 544 13.3.2 THE SET PARTITIONING IDEA 545 13.4 SOME
ERROR ANALYSIS FOR TCM CODES 546 13.4.1 GENERAL CONSIDERATIONS 546
13.4.2 A DESCRIPTION OF THE ERROR EVENTS 548 13.4.3 KNOWN GOOD TCM CODES
552 13.5 DECODING TCM CODES 554 13.6 ROTATIONAL INVARIANCE 556
DIFFERENTIAL ENCODING 558 CONSTELLATION LABELS AND PARTITIONS 559 13.7
MULTIDIMENSIONAL TCM 561 XXII CONTENTS 13.7.1 SOME ADVANTAGES OF
MULTIDIMENSIONAL TCM 562 13.7.2 LATTICES AND SUBLATTICES 563 BASIC
DEFINITIONS 563 COMMON LATTICES 565 SUBLATTICES AND COSETS 566 THE
LATTICE CODE IDEA 567 SOURCES OF CODING GAIN IN LATTICE CODES 567 SOME
GOOD LATTICE CODES 571 13.8 THE V.34 MODEM STANDARD 571 LAB 11
TRELLIS-CODED MODULATION ENCODING AND DECODING 578 OBJECTIVE 578
BACKGROUND 578 PROGRAMMING PART 578 13.9 EXERCISES 578 13.10 REFERENCES
580 PART IV ITERATIVELY DECODED CODES 581 14 TURBO CODES 582 14.1
INTRODUCTION 582 14.2 ENCODING PARALLEL CONCATENATED CODES 584 14.3
TURBO DECODING ALGORITHMS 586 14.3.1 THE MAP DECODING ALGORITHM 588
14.3.2 NOTATION 588 14.3.3 POSTERIOR PROBABILITY 590 14.3.4 COMPUTING A
T AND F} T 592 14.3.5 COMPUTING)/, 593 14.3.6 NORMALIZATION 594 14.3.7
SUMMARY OF THE BCJR ALGORITHM 596 14.3.8 A MATRIX/VECTOR FORMULATION 597
14.3.9 COMPARISON OF THE VITERBI AND BCJR ALGORITHMS 598 14.3.10 THE
BCJR ALGORITHM FOR SYSTEMATIC CODES 598 14.3.11 TURBO DECODING USING THE
BCJR ALGORITHM 600 THE TERMINAL STATE OF THE ENCODERS 602 14.3.12
LIKELIHOOD RATIO DECODING 602 LOG PRIOR RATIO X PJ 603 LOG POSTERIOR X]
605 14.3.13 STATEMENT OF THE TURBO DECODING ALGORITHM 605 14.3.14 TURBO
DECODING STOPPING CRITERIA 605 THE CROSS ENTROPY STOPPING CRITERION 606
THE SIGN CHANGE RATIO (SCR) CRITERION 607 THE HARD DECISION AIDED (HDA)
CRITERION 608 14.3.15 MODIFICATIONS OF THE MAP ALGORITHM 608 THE
MAX-LOG-MAP ALGORITHM 608 14.3.16 CORRECTIONS TO THE MAX-LOG-MAP
ALGORITHM 609 14.3.17 THE SOFT OUTPUT VITERBI ALGORITHM 610 14.4 ON THE
ERROR FLOOR AND WEIGHT DISTRIBUTIONS 612 CONTENTS XX1U 14.4.1 THE ERROR
FLOOR 612 14.4.2 SPECTRAL THINNING AND RANDOM INTERLEAVES 614 14.4.3 ON
INTERLEAVES 618 14.5 EXIT CHART ANALYSIS 619 14.5.1 THE EXIT CHART 622
14.6 BLOCK TURBO CODING 623 14.7 TURBO EQUALIZATION 626 14.7.1
INTRODUCTION TO TURBO EQUALIZATION 626 14.7.2 THE FRAMEWORK FOR TURBO
EQUALIZATION 627 LAB 12 TURBO CODE DECODING 629 OBJECTIVE 629 BACKGROUND
629 PROGRAMMING PART 629 14.8 EXERCISES 629 14.9 REFERENCES 632 15
LOW-DENSITY PARITY-CHECK CODES 634 15.1 INTRODUCTION 634 15.2 LDPC
CODES: CONSTRUCTION AND NOTATION 635 15.3 TANNER GRAPHS 638 15.4
TRANSMISSION THROUGH A GAUSSIAN CHANNEL 638 15.5 DECODING LDPC CODES 640
15.5.1 THE VERTICAL STEP: UPDATING Q MN (X) 641 15.5.2 HORIZONTAL STEP:
UPDATING R MN (X) 644 15.5.3 TERMINATING AND INITIALIZING THE DECODING
ALGORITHM 647 15.5.4 SUMMARY OF THE ALGORITHM 648 15.5.5 MESSAGE PASSING
VIEWPOINT 649 15.5.6 LIKELIHOOD RATIO DECODER FORMULATION 649 15.6 WHY
LOW-DENSITY PARITY-CHECK CODES? 653 15.7 THE ITERATIVE DECODER ON
GENERAL BLOCK CODES 654 15.8 DENSITY EVOLUTION 655 15.9 EXIT CHARTS FOR
LDPC CODES 659 15.10 IRREGULAR LDPC CODES 660 15.10.1 DEGREE
DISTRIBUTION PAIRS 662 15.10.2 SOME GOOD CODES 664 15.10.3 DENSITY
EVOLUTION FOR IRREGULAR CODES 664 15.10.4 COMPUTATION AND OPTIMIZATION
OF DENSITY EVOLUTION 667 15.10.5 USING IRREGULAR CODES 668 15.11 MORE ON
LDPC CODE CONSTRUCTION 668 15.11.1 A CONSTRUCTION BASED ON FINITE
GEOMETRIES 668 15.11.2 CONSTRUCTIONS BASED ON OTHER COMBINATORIC OBJECTS
669 15.12 ENCODING LDPC CODES 669 15.13 A VARIATION: LOW-DENSITY
GENERATOR MATRIX CODES 671 15.14 SERIAL CONCATENATED CODES;
REPEAT-ACCUMULATE CODES 671 15.14.1 IRREGULAR RA CODES 673 LAB 13
PROGRAMMING AN LDPC DECODER 674 OBJECTIVE 674 XXIV CONTENTS BACKGROUND
674 ASSIGNMENT 675 NUMERICAL CONSIDERATIONS 675 15.15 EXERCISES 676
15.16 REFERENCES 679 16 DECODING ALGORITHMS ON GRAPHS 680 16.1
INTRODUCTION 680 16.2 OPERATIONS IN SEMIRINGS 681 16.3 FUNCTIONS ON
LOCAL DOMAINS 681 16.4 FACTOR GRAPHS AND MARGINALIZATION 686 16.4.1
MARGINALIZING ON A SINGLE VARIABLE 687 16.4.2 MARGINALIZING ON ALL
INDIVIDUAL VARIABLES 691 16.5 APPLICATIONS TO CODING 694 16.5.1
BLOCKCODES 694 16.5.2 MODIFICATIONS TO MESSAGE PASSING FOR BINARY
VARIABLES 695 16.5.3 TRELLIS PROCESSING AND THE FORWARD/BACKWARD
ALGORITHM 696 16.5.4 TURBO CODES 699 16.6 SUMMARY OF DECODING ALGORITHMS
ON GRAPHS 699 16.7 TRANSFORMATIONS OF FACTOR GRAPHS 700 16.7.1
CLUSTERING 700 16.7.2 STRETCHING VARIABLE NODES 701 16.7.3 EXACT
COMPUTATION OF GRAPHS WITH CYCLES 702 16.8 EXERCISES 706 16.9 REFERENCES
708 PARTV SPACE-TIME CODING 709 17 FADING CHANNELS AND SPACE-TIME CODES
710 17.1 INTRODUCTION 710 17.2 FADING CHANNELS 710 17.2.1 RAYLEIGH
FADING 712 17.3 DIVERSITY TRANSMISSION AND RECEPTION: THE MJMO CHANNEL
714 17.3.1 THE NARROWBAND MIMO CHANNEL 716 17.3.2 DIVERSITY PERFORMANCE
WITH MAXIMAL-RATIO COMBINING 717 17.4 SPACE-TIME BLOCK CODES 719 17.4.1
THE ALAMOUTI CODE 719 17.4.2 A MORE GENERAL FORMULATION 721 17.4.3
PERFORMANCE CALCULATION 721 REAL ORTHOGONAL DESIGNS 723 ENCODING AND
DECODING BASED ON ORTHOGONAL DESIGNS 724 GENERALIZED REAL ORTHOGONAL
DESIGNS 726 17.4.4 COMPLEX ORTHOGONAL DESIGNS 727 FUTURE WORK 728 17.5
SPACE-TIME TRELLIS CODES 728 17.5.1 CONCATENATION 729 17.6 HOW MANY
ANTENNAS? 732 CONTENTS XXV 17.7 ESTIMATING CHANNEL INFORMATION 733 17.8
EXERCISES 733 17.9 REFERENCES 734 A LOG LIKELIHOOD ALGEBRA 735 A.I
EXERCISES 737 REFERENCES 739 INDEX 750
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ERROR CORRECTION CODING MATHEMATICAL METHODS AND ALGORITHMS TODD K. MOON
UTAH STATE UNIVERSITY IWILEY- INTERSCIENCE A JOHN WILEY & SONS, INC.,
PUBLICATION CONTENTS PREFACE LIST OF PROGRAM FILES LIST OF LABORATORY
EXERCISES LIST OF ALGORITHMS LIST OF FIGURES LIST OF TABLES LIST OF
BOXES VU XXXI XXXII XXXIV XL XLII XLIII PART I INTRODUCTION AND
FOUNDATIONS 1 1 A CONTEXT FOR ERROR CORRECTION CODING 2 1.1 PURPOSE OF
THIS BOOK 2 1.2 INTRODUCTION: WHERE ARE CODES? 2 1.3 THE COMMUNICATIONS
SYSTEM 4 1.4 BASIC DIGITAL COMMUNICATIONS 9 1.4.1 BINARY PHASE-SHIFT
KEYING 10 1.4.2 MORE GENERAL DIGITAL MODULATION 11 1.5 SIGNAL DETECTION
14 1.5.1 THE GAUSSIAN CHANNEL 14 1.5.2 MAP AND ML DETECTION 16 1.5.3
SPECIAL CASE: BINARY DETECTION 18 1.5.4 PROBABILITY OF ERROR FOR BINARY
DETECTION 19 1.5.5 BOUNDS ON PERFORMANCE: THE UNION BOUND 22 1.5.6 THE
BINARY SYMMETRIC CHANNEL 23 1.5.7 THE BSC AND THE GAUSSIAN CHANNEL MODEL
25 1.6 MEMORY LESS CHANNELS 25 1.7 SIMULATION AND ENERGY CONSIDERATIONS
FOR CODED SIGNALS 26 1.8 SOME IMPORTANT DEFINITIONS 27 1.8.1 DETECTION
OF REPETITION CODES OVER A BSC 28 1.8.2 SOFT-DECISION DECODING OF
REPETITION CODES OVER THE AWGN . 32 1.8.3 SIMULATION OF RESULTS 33
1.8.4 SUMMARY ' 33 1.9 HAMMING CODES 34 1.9.1 HARD-INPUT DECODING
HAMMING CODES 35 1.9.2 OTHER REPRESENTATIONS OF THE HAMMING CODE 36 AN
ALGEBRAIC REPRESENTATION 37 A POLYNOMIAL REPRESENTATION 37 XW CONTENTS A
TRELLIS REPRESENTATION 38 THE TANNER GRAPH REPRESENTATION 38 1.10 THE
BASIC QUESTIONS 39 1.11 HISTORICAL MILESTONES OF CODING THEORY 40 1.12 A
BIT OF INFORMATION THEORY 40 1.12.1 DEFINITIONS FOR DISCRETE RANDOM
VARIABLES 40 ENTROPY AND CONDITIONAL ENTROPY 40 RELATIVE ENTROPY, MUTUAL
INFORMATION, AND CHANNEL CAPACITY . 41 1.12.2 DEFINITIONS FOR
CONTINUOUS RANDOM VARIABLES 43 1.12.3 THE CHANNEL CODING THEOREM 45
1.12.4 "PROOF OF THE CHANNEL CODING THEOREM 45 1.12.5 CAPACITY FOR THE
CONTINUOUS-TIME AWGN CHANNEL 49 1.12.6 TRANSMISSION AT CAPACITY WITH
ERRORS 51 1.12.7 THE IMPLICATION OF THE CHANNEL CODING THEOREM 52 LAB 1
SIMULATING A COMMUNICATIONS CHANNEL 53 OBJECTIVE 53 BACKGROUND 53 USE OF
CODING IN CONJUNCTION WITH THE BSC 53 ASSIGNMENT 54 PROGRAMMING PART 54
RESOURCES AND IMPLEMENTATION SUGGESTIONS 54 1.13 EXERCISES 56 1.14
REFERENCES 60 PART II BLOCK CODES 61 2 GROUPS AND VECTOR SPACES 62 2.1
INTRODUCTION 62 2.2 GROUPS 62 2.2.1 SUBGROUPS 65 2.2.2 CYCLIC GROUPS AND
THE ORDER OF AN ELEMENT 66 2.2.3 COSETS 67 2.2.4 LAGRANGE'S THEOREM 68
2.2.5 INDUCED OPERATIONS; ISOMORPHISM 69 2.2.6 HOMOMORPHISM 72 2.3
FIELDS: A PRELUDE . . . * 73 2.4 REVIEW OF LINEAR ALGEBRA 75 2.5
EXERCISES 80 2.6 REFERENCES 82 3 LINEAR BLOCK CODES 83 3.1 BASIC
DEFINITIONS 83 3.2 THE GENERATOR MATRIX DESCRIPTION OF LINEAR BLOCK
CODES 84 3.2.1 RUDIMENTARY IMPLEMENTATION 86 3.3 THE PARITY CHECK MATRIX
AND DUAL CODES 86 3.3.1 SOME SIMPLE BOUNDS ON BLOCK CODES 88 3.4 ERROR
DETECTION AND CORRECTION OVER HARD-INPUT CHANNELS 90 CONTENTS XV 3.4.1
ERROR DETECTION 90 3.4.2 ERROR CORRECTION: THE STANDARD ARRAY 90 3.5
WEIGHT DISTRIBUTIONS OF CODES AND THEIR DUALS 95 3.6 HAMMING CODES AND
THEIR DUALS 97 3.7 PERFORMANCE OF LINEAR CODES 98 3.7.1 ERROR DETECTION
PERFORMANCE 99 3.7.2 ERROR CORRECTION PERFORMANCE 100 3.7.3 PERFORMANCE
FOR SOFT-DECISION DECODING 103 3.8 ERASURE DECODING 104 3.8.1 BINARY
ERASURE DECODING 105 3.9 MODIFICATIONS TO LINEAR CODES 105 3.10 BEST
KNOWN LINEAR BLOCK CODES 107 3.11 EXERCISES 107 3.12 REFERENCES 112
CYCLIC CODES, RINGS, AND POLYNOMIALS 113 4.1 INTRODUCTION 113 4.2 BASIC
DEFINITIONS 113 4.3 RINGS 114 4.3.1 RINGS OF POLYNOMIALS 115 4.4
QUOTIENT RINGS 116 4.5 IDEALS IN RINGS 118 4.6 ALGEBRAIC DESCRIPTION OF
CYCLIC CODES 120 4.7 NONSYSTEMATIC ENCODING AND PARITY CHECK 122 4.8
SYSTEMATIC ENCODING 124 4.9 SOME HARDWARE BACKGROUND 126 4.9.1
COMPUTATIONAL BUILDING BLOCKS 126 4.9.2 SEQUENCES AND POWER SERIES 127
4.9.3 POLYNOMIAL MULTIPLICATION 128 LAST-ELEMENT-FIRST PROCESSING 128
FIRST-ELEMENT-FIRST PROCESSING 128 4.9.4 POLYNOMIAL DIVISION 129
LAST-ELEMENT-FIRST PROCESSING 129 4.9.5 SIMULTANEOUS POLYNOMIAL DIVISION
AND MULTIPLICATION 132 FIRST-ELEMENT-FIRST PROCESSING 132 4.10 CYCLIC
ENCODING 133 4.11 SYNDROME DECODING 137 4.12 SHORTENED CYCLIC CODES 143
METHOD 1: SIMULATING THE EXTRA CLOCK SHIFTS 144 METHOD 2: CHANGING THE
ERROR PATTERN DETECTION CIRCUIT 147 4.13 BINARY CRC CODES 147 4.13.1
BYTE-ORIENTED ENCODING AND DECODING ALGORITHMS 150 4.13.2 CRC PROTECTING
DATA FILES OR DATA PACKETS 153 APPENDIX 4. A LINEAR FEEDBACK SHIFT
REGISTERS 154 APPENDIX 4.A. 1 BASIC CONCEPTS 154 APPENDIX 4.A.2
CONNECTION WITH POLYNOMIAL DIVISION 157 APPENDIX 4. A.3 SOME ALGEBRAIC
PROPERTIES OF SHIFT SEQUENCES 160 XVI CONTENTS LAB 2 POLYNOMIAL DIVISION
AND LINEAR FEEDBACK SHIFT REGISTERS . 161 OBJECTIVE 161 PRELIMINARY
EXERCISES 161 PROGRAMMING PART: BINLFSR 161 RESOURCES AND IMPLEMENTATION
SUGGESTIONS 161 PROGRAMMING PART: BINPOLYDIV 162 FOLLOW-ON IDEAS AND
PROBLEMS 162 LAB 3 CRC ENCODING AND DECODING 162 OBJECTIVE 163
PRELIMINARY 163 PROGRAMMING PART 163 RESOURCES AND IMPLEMENTATION
SUGGESTIONS 163 4.14 EXERCISES 165 4.15 REFERENCES 170 5 RUDIMENTS OF
NUMBER THEORY AND ALGEBRA 171 5.1 MOTIVATION 171 5.2 NUMBER THEORETIC
PRELIMINARIES 175 5.2.1 DIVISIBILITY 175 5.2.2 THE EUCLIDEAN ALGORITHM
AND EUCLIDEAN DOMAINS 177 5.2.3 THE SUGIYAMA ALGORITHM 182 5.2.4
CONGRUENCE 184 5.2.5 THE FUNCTION 185 5.2.6 SOME CRYPTOGRAPHIC PAYOFF
186 FERMAT'S LITTLE THEOREM 186 RSA ENCRYPTION 187 5.3 THE CHINESE
REMAINDER THEOREM 188 5.3.1 THE CRT AND INTERPOLATION 190 THE EVALUATION
HOMOMORPHISM 190 THE INTERPOLATION PROBLEM 191 5.4 FIELDS 193 5.4.1 AN
EXAMINATION OF R AND C 194 5.4.2 GALOIS FIELD CONSTRUCTION: AN EXAMPLE
196 5.4.3 CONNECTION WITH LINEAR FEEDBACK SHIFT REGISTERS 199 5.5 GALOIS
FIELDS: MATHEMATICAL FACTS 200 5.6 IMPLEMENTING GALOIS FIELD ARITHMETIC
204 5.6.1 ZECH LOGARITHMS 204 5.6.2 HARDWARE IMPLEMENTATIONS 205 5.7
SUBFIELDS OF GALOIS FIELDS 206 5.8 IRREDUCIBLE AND PRIMITIVE POLYNOMIALS
207 5.9 CONJUGATE ELEMENTS AND MINIMAL POLYNOMIALS 209 5.9.1 MINIMAL
POLYNOMIALS 212 5.10 FACTORING*" - 1 215 5.11 CYCLOTOMIC COSETS 217
APPENDIX 5.A HOW MANY IRREDUCIBLE POLYNOMIALS ARE THERE? 218 APPENDIX
5.A.1 SOLVING FOR I M EXPLICITLY: THE MOEBIUS FUNCTION 222 LAB 4
PROGRAMMING THE EUCLIDEAN ALGORITHM 223 CONTENTS XVII OBJECTIVE 223
PRELIMINARY EXERCISES 223 BACKGROUND 223 PROGRAMMING PART 223 LAB 5
PROGRAMMING GALOIS FIELD ARITHMETIC 224 OBJECTIVE 224 PRELIMINARY
EXERCISES 224 PROGRAMMING PART 224 5.12 EXERCISES 225 5.13 REFERENCES
234 BCH AND REED-SOLOMON CODES: DESIGNER CYCLIC CODES 235 6.1 BCH CODES
235 6.1.1 DESIGNING BCH CODES 235 6.1.2 THE BCH BOUND 237 6.1.3 WEIGHT
DISTRIBUTIONS FOR SOME BINARY BCH CODES 239 6.1.4 ASYMPTOTIC RESULTS FOR
BCH CODES 240 6.2 REED-SOLOMON CODES 242 6.2.1 REED-SOLOMON CONSTRUCTION
1 242 6.2.2 REED-SOLOMON CONSTRUCTION 2 243 6.2.3 ENCODING REED-SOLOMON
CODES 244 6.2.4 MDS CODES AND WEIGHT DISTRIBUTIONS FOR RS CODES 245 6.3
DECODING BCH AND RS CODES: THE GENERAL OUTLINE 247 6.3.1 COMPUTATION OF
THE SYNDROME 247 6.3.2 THE ERROR LOCATOR POLYNOMIAL 248 6.3.3 CHIEN
SEARCH 248 6.4 FINDING THE ERROR LOCATOR POLYNOMIAL 250 6.4.1
SIMPLIFICATIONS FOR BINARY CODES AND PETERSON'S ALGORITHM 251 6.4.2
BERLEKAMP-MASSEY ALGORITHM 253 6.4.3 CHARACTERIZATION OF LFSR LENGTH IN
MASSEY'S ALGORITHM 255 6.4.4 SIMPLIFICATIONS FOR BINARY CODES 259 6.5
NON-BINARY BCH AND RS DECODING 261 6.5.1 FORNEY'S ALGORITHM 262 6.6
EUCLIDEAN ALGORITHM FOR THE ERROR LOCATOR POLYNOMIAL 266 6.7 ERASURE
DECODING FOR NONBINARY BCH OR RS CODES 267 6.8 GALOIS FIELD FOURIER
TRANSFORM METHODS 269 6.8.1 EQUIVALENCE OF THE TWO REED-SOLOMON CODE
CONSTRUCTIONS . 274 6.8.2 FREQUENCY-DOMAIN DECODING 275 6.9
VARIATIONS AND EXTENSIONS OF REED-SOLOMON CODES 276 6.9.1 SIMPLE
MODIFICATIONS 276 6.9.2 GENERALIZED REED-SOLOMON CODES AND ALTERNANT
CODES 277 6.9.3 GOPPA CODES 278 6.9.4 DECODING ALTERNANT CODES 280 6.9.5
THE MCELIECE PUBLIC KEY CRYPTOSYSTEM 280 LAB 6 PROGRAMMING THE
BERLEKAMP-MASSEY ALGORITHM 281 BACKGROUND 281 ASSIGNMENT 281 XVIII
CONTENTS PRELIMINARY EXERCISES 281 PROGRAMMING PART 281 RESOURCES AND
IMPLEMENTATION SUGGESTIONS 282 LAB 7 PROGRAMMING THE BCH DECODER 283
OBJECTIVE 283 PRELIMINARY EXERCISES 283 PROGRAMMING PART 283 RESOURCES
AND IMPLEMENTATION SUGGESTIONS 283 FOLLOW-ON IDEAS AND PROBLEMS 284 LAB
8 REED-SOLOMON ENCODING AND DECODING 284 OBJECTIVE 284 BACKGROUND 284
PROGRAMMING PART 284 APPENDIX 6. A PROOF OF NEWTON'S IDENTITIES 285 6.10
EXERCISES 287 6.11 REFERENCES 291 7 ALTERNATE DECODING ALGORITHMS FOR
REED-SOLOMON CODES 293 7.1 INTRODUCTION: WORKLOAD FOR REED-SOLOMON
DECODING 293 7.2 DERIVATIONS OF WELCH-BERLEKAMP KEY EQUATION 293 7.2.1
THE WELCH-BERLEKAMP DERIVATION OF THE WB KEY EQUATION 294 7.2.2
DERIVATION FROM THE CONVENTIONAL KEY EQUATION 298 7.3 FINDING THE ERROR
VALUES 300 7.4 METHODS OF SOLVING THE WB KEY EQUATION 302 7.4.1
BACKGROUND: MODULES 302 7.4.2 THE WELCH-BERLEKAMP ALGORITHM 303 7.4.3
MODULAR SOLUTION OF THE WB KEY EQUATION 310 7.5 ERASURE DECODING WITH
THE WELCH-BERLEKAMP KEY EQUATION 321 7.6 THE GURUSWAMI-SUDAN DECODING
ALGORITHM AND SOFT RS DECODING 322 7.6.1 BOUNDED DISTANCE, ML, AND LIST
DECODING 322 7.6.2 ERROR CORRECTION BY INTERPOLATION 323 7.6.3
POLYNOMIALS IN TWO VARIABLES 324 DEGREE AND MONOMIAL ORDER 325 ZEROS AND
MULTIPLE ZEROS 328 7.6.4 THE GS DECODER: THE MAIN THEOREMS 330 THE
INTERPOLATION THEOREM 331 THE FACTORIZATION THEOREM 331 THE CORRECTION
DISTANCE 333 THE NUMBER OF POLYNOMIALS IN THE DECODING LIST 335 7.6.5
ALGORITHMS FOR COMPUTING THE INTERPOLATION STEP 337 FINDING LINEARLY
DEPENDENT COLUMNS: THE FENG-TZENG ALGORITHM 338 FINDING THE INTERSECTION
OF KERNELS: THE KOTTER ALGORITHM 342 7.6.6 A SPECIAL CASE: M - 1 AND L =
1 348 7.6.7 THE ROTH-RUCKENSTEIN ALGORITHM 350 WHAT TO DO WITH LISTS OF
FACTORS? 354 7.6.8 SOFT-DECISION DECODING OF REED-SOLOMON CODES 358
NOTATION 358 CONTENTS XIX A FACTORIZATION THEOREM 360 MAPPING FROM
RELIABILITY TO MULTIPLICITY 361 THE GEOMETRY OF THE DECODING REGIONS 363
COMPUTING THE RELIABILITY MATRIX 364 7.7 EXERCISES 365 7.8 REFERENCES
368 8 OTHER IMPORTANT BLOCK CODES 369 8.1 INTRODUCTION 369 8.2 HADAMARD
MATRICES, CODES, AND TRANSFORMS 369 8.2.1 INTRODUCTION TO HADAMARD
MATRICES 369 8.2.2 THE PALEY CONSTRUCTION OF HADAMARD MATRICES 371 8.2.3
HADAMARD CODES 374 8.3 REED-MULLER CODES 375 8.3.1 BOOLEAN FUNCTIONS 375
8.3.2 DEFINITION OF THE REED-MULLER CODES 376 8.3.3 ENCODING AND
DECODING ALGORITHMS FOR FIRST-ORDER RM CODES . . . 379 ENCODING RM(L,M)
CODES 379 DECODING RM(\,M) CODES 379 EXPEDITING DECODING USING THE FAST
HADAMARD TRANSFORM 382 8.3.4 THE REED DECODING ALGORITHM FOR RM{R, M)
CODES, R 1 384 DETAILS FOR AN RM(2, 4) CODE 384 A GEOMETRIC VIEWPOINT
387 8.3.5 OTHER CONSTRUCTIONS OF REED-MULLER CODES 391 8.4 BUILDING LONG
CODES FROM SHORT CODES: THE SQUARING CONSTRUCTION . 392 8.5 QUADRATIC
RESIDUE CODES 396 8.6 GOLAY CODES 398 8.6.1 DECODING THE GOLAY CODE 400
ALGEBRAIC DECODING OF THE 823 GOLAY CODE 400 ARITHMETIC DECODING OF THE
S24 CODE 401 8.7 EXERCISES 403 8.8 REFERENCES 404 9 BOUNDS ON CODES 406
9.1 THE GILBERT-VARSHAMOV BOUND 409 9.2 THE PLOTKIN BOUND 410 9.3 THE
GRIESMER BOUND 411 9.4 THE LINEAR PROGRAMMING AND RELATED BOUNDS 413
9.4.1 KRAWTCHOUK POLYNOMIALS 415 9.4.2 CHARACTER 415 9.4.3 KRAWTCHOUK
POLYNOMIALS AND CHARACTERS 416 9.5 THE MCELIECE-RODEMICH-RUMSEY-WELCH
BOUND 418 9.6 EXERCISES 420 9.7 REFERENCES 424 XX CONTENTS 10 BURSTY
CHANNELS, INTERLEAVES, AND CONCATENATION 425 10.1 INTRODUCTION TO BURSTY
CHANNELS 425 10.2 INTERLEAVES 425 10.3 AN APPLICATION OF INTERLEAVED RS
CODES: COMPACT DISCS 427 10.4 PRODUCT CODES 430 10.5 REED-SOLOMON CODES
431 10.6 CONCATENATED CODES 432 10.7 FIRE CODES 433 10.7.1 FIRE CODE
DEFINITION 433 10.7.2 DECODING FIRE CODES: ERROR TRAPPING DECODING 435
10.8 EXERCISES 437 10.9 REFERENCES 438 11 SOFT-DECISION DECODING
ALGORITHMS 439 11.1 INTRODUCTION AND GENERAL NOTATION 439 11.2
GENERALIZED MINIMUM DISTANCE DECODING 441 11.2.1 DISTANCE MEASURES AND
PROPERTIES 442 11.3 THE CHASE DECODING ALGORITHMS 445 11.4 HALTING THE
SEARCH: AN OPTIMALITY CONDITION 445 11.5 ORDERED STATISTIC DECODING 447
11.6 EXERCISES 449 11.7 REFERENCES 450 PART III CODES ON GRAPHS 451 12
CONVOLUTIONAL CODES 452 12.1 INTRODUCTION AND BASIC NOTATION 452 12.1.1
THE STATE 456 12.2 DEFINITION OF CODES AND EQUIVALENT CODES 458 12.2.1
CATASTROPHIC ENCODERS 461 12.2.2 POLYNOMIAL AND RATIONAL ENCODERS 464
12.2.3 CONSTRAINT LENGTH AND MINIMAL ENCODERS 465 12.2.4 SYSTEMATIC
ENCODERS 468 12.3 DECODING CONVOLUTIONAL CODES 469 12.3.1 INTRODUCTION
AND NOTATION 469 12.3.2 THE VITERBI ALGORITHM 471 12.3.3 SOME
IMPLEMENTATION ISSUES 481 THE BASIC OPERATION: ADD-COMPARE-SELECT 481
DECODING STREAMS OF DATA: WINDOWS ON THE TRELLIS 481 OUTPUT DECISIONS
482 HARD AND SOFT DECODING; QUANTIZATION 484 SYNCHRONIZATION ISSUES 486
12.4 SOME PERFORMANCE RESULTS 487 12.5 ERROR ANALYSIS FOR CONVOLUTIONAL
CODES 491 12.5.1 ENUMERATING PATHS THROUGH THE TRELLIS 493 ENUMERATING
ON MORE COMPLICATED GRAPHS: MASON'S RULE 496 CONTENTS XXI 12.5.2
CHARACTERIZING THE NODE ERROR PROBABILITY P E AND THE BIT ERROR RATE PB
498 12.5.3 A BOUND ON PD FOR DISCRETE CHANNELS 501 PERFORMANCE BOUND ON
THE BSC 503 12.5.4 A BOUND ON P D FOR BPSK SIGNALING OVER THE AWGN
CHANNEL . 503 12.5.5 ASYMPTOTIC CODING GAIN 504 12.6 TABLES OF GOOD
CODES 505 12.7 PUNCTURING 507 12.7.1 PUNCTURING TO ACHIEVE VARIABLE RATE
509 12.8 SUBOPTIMAL DECODING ALGORITHMS FOR CONVOLUTIONAL CODES 510
12.8.1 TREE REPRESENTATIONS 511 12.8.2 THE FANO METRIC 511 12.8.3 THE
STACK ALGORITHM 515 12.8.4 THE FANO ALGORITHM 517 12.8.5 OTHER ISSUES
FOR SEQUENTIAL DECODING 520 12.8.6 A VARIATION ON THE VITERBI ALGORITHM:
THE M ALGORITHM 521 12.9 CONVOLUTIONAL CODES AS BLOCK CODES 522 12.10
TRELLIS REPRESENTATIONS OF BLOCK AND CYCLIC CODES 523 12.10.1 BLOCK
CODES 523 12.10.2 CYCLIC CODES 524 12.10.3 TRELLIS DECODING OF BLOCK
CODES 525 LAB 9 PROGRAMMING CONVOLUTIONAL ENCODERS 526 OBJECTIVE 526
BACKGROUND 526 PROGRAMMING PART 526 LAB 10 CONVOLUTIONAL DECODERS: THE
VITERBI ALGORITHM 528 OBJECTIVE 528 BACKGROUND 528 PROGRAMMING PART 528
12.11 EXERCISES 529 12.12 REFERENCES 533 13 TRELLIS CODED MODULATION 535
13.1 ADDING REDUNDANCY BY ADDING SIGNALS 535 13.2 BACKGROUND ON SIGNAL
CONSTELLATIONS 535 13.3 TCM EXAMPLE 537 13.3.1 THE GENERAL UNGERBOECK
CODING FRAMEWORK 544 13.3.2 THE SET PARTITIONING IDEA 545 13.4 SOME
ERROR ANALYSIS FOR TCM CODES 546 13.4.1 GENERAL CONSIDERATIONS 546
13.4.2 A DESCRIPTION OF THE ERROR EVENTS 548 13.4.3 KNOWN GOOD TCM CODES
552 13.5 DECODING TCM CODES 554 13.6 ROTATIONAL INVARIANCE 556
DIFFERENTIAL ENCODING 558 CONSTELLATION LABELS AND PARTITIONS 559 13.7
MULTIDIMENSIONAL TCM 561 XXII CONTENTS 13.7.1 SOME ADVANTAGES OF
MULTIDIMENSIONAL TCM 562 13.7.2 LATTICES AND SUBLATTICES 563 BASIC
DEFINITIONS 563 COMMON LATTICES 565 SUBLATTICES AND COSETS 566 THE
LATTICE CODE IDEA 567 SOURCES OF CODING GAIN IN LATTICE CODES 567 SOME
GOOD LATTICE CODES 571 13.8 THE V.34 MODEM STANDARD 571 LAB 11
TRELLIS-CODED MODULATION ENCODING AND DECODING 578 OBJECTIVE 578
BACKGROUND 578 PROGRAMMING PART 578 13.9 EXERCISES 578 13.10 REFERENCES
580 PART IV ITERATIVELY DECODED CODES 581 14 TURBO CODES 582 14.1
INTRODUCTION 582 14.2 ENCODING PARALLEL CONCATENATED CODES 584 14.3
TURBO DECODING ALGORITHMS 586 14.3.1 THE MAP DECODING ALGORITHM 588
14.3.2 NOTATION 588 14.3.3 POSTERIOR PROBABILITY 590 14.3.4 COMPUTING A
T AND F} T 592 14.3.5 COMPUTING)/, 593 14.3.6 NORMALIZATION 594 14.3.7
SUMMARY OF THE BCJR ALGORITHM 596 14.3.8 A MATRIX/VECTOR FORMULATION 597
14.3.9 COMPARISON OF THE VITERBI AND BCJR ALGORITHMS 598 14.3.10 THE
BCJR ALGORITHM FOR SYSTEMATIC CODES 598 14.3.11 TURBO DECODING USING THE
BCJR ALGORITHM 600 THE TERMINAL STATE OF THE ENCODERS 602 14.3.12
LIKELIHOOD RATIO DECODING 602 LOG PRIOR RATIO X PJ 603 LOG POSTERIOR X]
605 14.3.13 STATEMENT OF THE TURBO DECODING ALGORITHM 605 14.3.14 TURBO
DECODING STOPPING CRITERIA 605 THE CROSS ENTROPY STOPPING CRITERION 606
THE SIGN CHANGE RATIO (SCR) CRITERION 607 THE HARD DECISION AIDED (HDA)
CRITERION 608 14.3.15 MODIFICATIONS OF THE MAP ALGORITHM 608 THE
MAX-LOG-MAP ALGORITHM 608 14.3.16 CORRECTIONS TO THE MAX-LOG-MAP
ALGORITHM 609 14.3.17 THE SOFT OUTPUT VITERBI ALGORITHM 610 14.4 ON THE
ERROR FLOOR AND WEIGHT DISTRIBUTIONS 612 CONTENTS XX1U 14.4.1 THE ERROR
FLOOR 612 14.4.2 SPECTRAL THINNING AND RANDOM INTERLEAVES 614 14.4.3 ON
INTERLEAVES 618 14.5 EXIT CHART ANALYSIS 619 14.5.1 THE EXIT CHART 622
14.6 BLOCK TURBO CODING 623 14.7 TURBO EQUALIZATION 626 14.7.1
INTRODUCTION TO TURBO EQUALIZATION 626 14.7.2 THE FRAMEWORK FOR TURBO
EQUALIZATION 627 LAB 12 TURBO CODE DECODING 629 OBJECTIVE 629 BACKGROUND
629 PROGRAMMING PART 629 14.8 EXERCISES 629 14.9 REFERENCES 632 15
LOW-DENSITY PARITY-CHECK CODES 634 15.1 INTRODUCTION 634 15.2 LDPC
CODES: CONSTRUCTION AND NOTATION 635 15.3 TANNER GRAPHS 638 15.4
TRANSMISSION THROUGH A GAUSSIAN CHANNEL 638 15.5 DECODING LDPC CODES 640
15.5.1 THE VERTICAL STEP: UPDATING Q MN (X) 641 15.5.2 HORIZONTAL STEP:
UPDATING R MN (X) 644 15.5.3 TERMINATING AND INITIALIZING THE DECODING
ALGORITHM 647 15.5.4 SUMMARY OF THE ALGORITHM 648 15.5.5 MESSAGE PASSING
VIEWPOINT 649 15.5.6 LIKELIHOOD RATIO DECODER FORMULATION 649 15.6 WHY
LOW-DENSITY PARITY-CHECK CODES? 653 15.7 THE ITERATIVE DECODER ON
GENERAL BLOCK CODES 654 15.8 DENSITY EVOLUTION 655 15.9 EXIT CHARTS FOR
LDPC CODES 659 15.10 IRREGULAR LDPC CODES 660 15.10.1 DEGREE
DISTRIBUTION PAIRS 662 15.10.2 SOME GOOD CODES 664 15.10.3 DENSITY
EVOLUTION FOR IRREGULAR CODES 664 15.10.4 COMPUTATION AND OPTIMIZATION
OF DENSITY EVOLUTION 667 15.10.5 USING IRREGULAR CODES 668 15.11 MORE ON
LDPC CODE CONSTRUCTION 668 15.11.1 A CONSTRUCTION BASED ON FINITE
GEOMETRIES 668 15.11.2 CONSTRUCTIONS BASED ON OTHER COMBINATORIC OBJECTS
669 15.12 ENCODING LDPC CODES 669 15.13 A VARIATION: LOW-DENSITY
GENERATOR MATRIX CODES 671 15.14 SERIAL CONCATENATED CODES;
REPEAT-ACCUMULATE CODES 671 15.14.1 IRREGULAR RA CODES 673 LAB 13
PROGRAMMING AN LDPC DECODER 674 OBJECTIVE 674 XXIV CONTENTS BACKGROUND
674 ASSIGNMENT 675 NUMERICAL CONSIDERATIONS 675 15.15 EXERCISES 676
15.16 REFERENCES 679 16 DECODING ALGORITHMS ON GRAPHS 680 16.1
INTRODUCTION 680 16.2 OPERATIONS IN SEMIRINGS 681 16.3 FUNCTIONS ON
LOCAL DOMAINS 681 16.4 FACTOR GRAPHS AND MARGINALIZATION 686 16.4.1
MARGINALIZING ON A SINGLE VARIABLE 687 16.4.2 MARGINALIZING ON ALL
INDIVIDUAL VARIABLES 691 16.5 APPLICATIONS TO CODING 694 16.5.1
BLOCKCODES 694 16.5.2 MODIFICATIONS TO MESSAGE PASSING FOR BINARY
VARIABLES 695 16.5.3 TRELLIS PROCESSING AND THE FORWARD/BACKWARD
ALGORITHM 696 16.5.4 TURBO CODES 699 16.6 SUMMARY OF DECODING ALGORITHMS
ON GRAPHS 699 16.7 TRANSFORMATIONS OF FACTOR GRAPHS 700 16.7.1
CLUSTERING 700 16.7.2 STRETCHING VARIABLE NODES 701 16.7.3 EXACT
COMPUTATION OF GRAPHS WITH CYCLES 702 16.8 EXERCISES 706 16.9 REFERENCES
708 PARTV SPACE-TIME CODING 709 17 FADING CHANNELS AND SPACE-TIME CODES
710 17.1 INTRODUCTION 710 17.2 FADING CHANNELS 710 17.2.1 RAYLEIGH
FADING 712 17.3 DIVERSITY TRANSMISSION AND RECEPTION: THE MJMO CHANNEL
714 17.3.1 THE NARROWBAND MIMO CHANNEL 716 17.3.2 DIVERSITY PERFORMANCE
WITH MAXIMAL-RATIO COMBINING 717 17.4 SPACE-TIME BLOCK CODES 719 17.4.1
THE ALAMOUTI CODE 719 17.4.2 A MORE GENERAL FORMULATION 721 17.4.3
PERFORMANCE CALCULATION 721 REAL ORTHOGONAL DESIGNS 723 ENCODING AND
DECODING BASED ON ORTHOGONAL DESIGNS 724 GENERALIZED REAL ORTHOGONAL
DESIGNS 726 17.4.4 COMPLEX ORTHOGONAL DESIGNS 727 FUTURE WORK 728 17.5
SPACE-TIME TRELLIS CODES 728 17.5.1 CONCATENATION 729 17.6 HOW MANY
ANTENNAS? 732 CONTENTS XXV 17.7 ESTIMATING CHANNEL INFORMATION 733 17.8
EXERCISES 733 17.9 REFERENCES 734 A LOG LIKELIHOOD ALGEBRA 735 A.I
EXERCISES 737 REFERENCES 739 INDEX 750 |
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| callnumber-first | T - Technology |
| callnumber-label | TA331 |
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| callnumber-sort | TA 3331 |
| callnumber-subject | TA - General and Civil Engineering |
| classification_rvk | SK 170 |
| classification_tum | DAT 584f |
| ctrlnum | (OCoLC)57373781 (DE-599)BVBBV022419497 |
| dewey-full | 621.382/0285/572 |
| dewey-hundreds | 600 - Technology (Applied sciences) |
| dewey-ones | 621 - Applied physics |
| dewey-raw | 621.382/0285/572 |
| dewey-search | 621.382/0285/572 |
| dewey-sort | 3621.382 3285 3572 |
| dewey-tens | 620 - Engineering and allied operations |
| discipline | Informatik Mathematik Elektrotechnik / Elektronik / Nachrichtentechnik |
| discipline_str_mv | Informatik Mathematik Elektrotechnik / Elektronik / Nachrichtentechnik |
| format | Book |
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| id | DE-604.BV022419497 |
| illustrated | Illustrated |
| index_date | 2024-07-02T17:25:09Z |
| indexdate | 2024-07-09T20:57:11Z |
| institution | BVB |
| isbn | 0471648000 |
| language | English |
| lccn | 2004031019 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015627839 |
| oclc_num | 57373781 |
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| owner_facet | DE-Aug4 DE-523 DE-706 DE-29T DE-91 DE-BY-TUM |
| physical | xlii, 755 p. ill. 24 cm |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Wiley-Interscience |
| record_format | marc |
| spelling | Moon, Todd K. Verfasser aut Error correction coding mathematical methods and algorithms Todd K. Moon Hoboken, N.J. Wiley-Interscience 2005 xlii, 755 p. ill. 24 cm txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Coderingstheorie gtt Codes correcteurs d'erreurs (Théorie de l'information) Datatransmissie gtt Foutenleer gtt Mathématiques de l'ingénieur Engineering mathematics Error-correcting codes (Information theory) Fehlerkorrekturcode (DE-588)4124917-3 gnd rswk-swf Fehlerkorrekturcode (DE-588)4124917-3 s DE-604 http://www.loc.gov/catdir/toc/ecip055/2004031019.html Table of contents only http://www.loc.gov/catdir/enhancements/fy0617/2004031019-b.html Contributor biographical information http://www.loc.gov/catdir/enhancements/fy0617/2004031019-d.html Publisher description HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015627839&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Moon, Todd K. Error correction coding mathematical methods and algorithms Coderingstheorie gtt Codes correcteurs d'erreurs (Théorie de l'information) Datatransmissie gtt Foutenleer gtt Mathématiques de l'ingénieur Engineering mathematics Error-correcting codes (Information theory) Fehlerkorrekturcode (DE-588)4124917-3 gnd |
| subject_GND | (DE-588)4124917-3 |
| title | Error correction coding mathematical methods and algorithms |
| title_auth | Error correction coding mathematical methods and algorithms |
| title_exact_search | Error correction coding mathematical methods and algorithms |
| title_exact_search_txtP | Error correction coding mathematical methods and algorithms |
| title_full | Error correction coding mathematical methods and algorithms Todd K. Moon |
| title_fullStr | Error correction coding mathematical methods and algorithms Todd K. Moon |
| title_full_unstemmed | Error correction coding mathematical methods and algorithms Todd K. Moon |
| title_short | Error correction coding |
| title_sort | error correction coding mathematical methods and algorithms |
| title_sub | mathematical methods and algorithms |
| topic | Coderingstheorie gtt Codes correcteurs d'erreurs (Théorie de l'information) Datatransmissie gtt Foutenleer gtt Mathématiques de l'ingénieur Engineering mathematics Error-correcting codes (Information theory) Fehlerkorrekturcode (DE-588)4124917-3 gnd |
| topic_facet | Coderingstheorie Codes correcteurs d'erreurs (Théorie de l'information) Datatransmissie Foutenleer Mathématiques de l'ingénieur Engineering mathematics Error-correcting codes (Information theory) Fehlerkorrekturcode |
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