Algebraic coding theory
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| Format: | Buch |
| Sprache: | English |
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World Scientific
2015
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| Ausgabe: | Revised edition |
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| 245 | 1 | 0 | |a Algebraic coding theory |c Elwyn Berlekamp |
| 250 | |a Revised edition | ||
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Datensatz im Suchindex
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| adam_text | rfc/n
Algebraic
Coding Theory
In 1948, Claude Shannon showed the existence of codes which could correct
errors. Over the next two decades, he and others explored the theoretical limits
of the performance of long block codes, resulting in curves such as the one shown
above. In the 1960s, mathematicians introduced the notion that the sophisticated
algebra of finite fields might be used to design and implement such codes.
When the first edition of this book appeared in 1968, it immediately became a
landmark in the field. It introduced a novel algorithm for factoring polynomials,
the crux of which is the top formula on the left of the cover. It also introduced a
new algorithm for determining the polynomial that needs to be factored in order
to decode Reed-Solomon codes. The crux of this algorithm, which appears on the
bottom left of the cover, became widely known as the Berlekamp-Massey algorithm.
These advances made algebraic error-correcting codes feasible for a wide range of
applications in both computer memories and in digital communications. Building
on the foundations of this book, a subsequent breakthrough in the 1970s reduced
the complexity of nonbinary Reed-Solomon encoders to that of much simpler
binary codes of comparable redundancies. A circuit diagram for such an encoder
appears on the right of the cover. It became a NASA standard for deep space
communications.
ISBN 978-981 -4635-89-9
9II; ^898 14 635 ► 899
World Scientific
www.worldscientific.com
9407 he
CONTENTS
R
Preface to the Revised Edition vii
Preface to the Second Edition xiii
Preface xv
Acknowledgments xix
1. Basic Binary Codes 1
1.1 Repetition Codes and Singie-Parity-Check Codes 1
1.2 Linear Codes 4
1.3 Hamming Codes 8
1.4 Manipulative Introduction to
Double-Error-Correcting BCH Codes 12
Problems 20
2. Arithmetic Operations Modulo
an Irreducible Binary Polynomial 21
2.1 A Closer Look at Euclid’s Algorithm 21
*2.2 Logical Circuitry 30
*2.3 Multiplicative Inversion 36
*2.4 Multiplication 44
*2.5 The Solution of Simultaneous Linear Equations 51
*2.6 Special Methods for Solving Simultaneous Linear
Equations When the Coefficient Matrix is Mostly Zeros 63
Problems 69
3. The Number of Irreducible g-ary
Polynomials of Given Degree 70
3.1 A Brute-Force Attack 70
3.2 Generating Functions 73
3.3 The Number of Irreducible Monic g-ary
Polynomials of Given Degree—A Refined Approach 76
*3.4 The Moebius Inversion Formulas 81
Problems 85
* STARRED SECTIONS MAY BE SKIPPED ON FIRST READING.
XXI
XXII
CONTENTS
4. The Structure of Finite Fields 87
4.1 Definitions 87
4.2 Multiplicative Structure of Finite Fields 88
4.3 Cyclotomie Polynomials 90
4.4 Algebraic Structure of Finite Fields 96
4.5 Examples 105
★4.6 Algebraic Closure 111
★4.7 Determining Minimal Polynomials 112
Problems 117
5. Cyclic Binary Codes 119
5.1 Reordering the Columns of the Parity-Check
Matrix of Hamming Codes 119
5.2 Reordering the Columns of the Parity-Check Matrix
of Double-Error-Correcting Binary BCH Codes 125
5.3 General Properties of Cyclic Codes 129
5.4 The Chien Search 132
5.5 Outline of General Decoder for Any Cyclic
Binary Code 136
5.6 Example 138
5.7 Example 139
5.8 Equivalence of Cyclic Codes Defined in Terms
of Different Primitive nth Roots of Unity 141
Problems 144
6. The Factorization of
Polynomials Over Finite Fields 146
6.1 A General Algorithm 146
★6.2 Determining the Period of a Polynomial 150
★6.3 Trinomials Over GF(2) 153
6.4 Factoring xn — 1 Explicitly 154
★6.5 Determining the Degrees of the irreducible
Factors of the Cyclotomie Polynomials 156
★6.6 Is the Number of Irreducible
Factors of f(x) Over GF(q) Odd or Even? 159
★6.7 Quadratic Reciprocity 171
Problems 173
7. Binary BCH Codes for
Correcting Multiple Errors 176
7.1 Examples 176
7.2 The Key Equation for Decoding Binary BCH Codes 178
7.3 Heuristic Solution of the Key Equation 180
7.4 An Algorithm for Solving the Key Equation
Over Any Field 184
★7.5 Relation to Matrix Decoding Methods 188
★7.6 Simplifications in Algorithm 7.4
for Binary BCH Codes 192
7.7 Implementation of Binary BCH Decoders 195
Problems 199
CONTENTS xxiii
8. Nonbinary Coding 200
8.1 Modulation Schemes 200
8.2 Weight Functions 204
Problem 206
9. Negacyclic Codes for the Lee Metric 207
9.1 Error Locations and the Error-Locator Polynomial 207
9.2 Double-Error-Correcting Codes 209
9.3 Negacyclic Codes 211
Problems 217
10. Gorenstein-Zierler Generalized Nonbinary
BCH Codes for the Hamming Metric 218
10.1 The Generalized BCH Codes and An
Algorithm to Decode Them 218
10.2 Examples 221
*10.3 Alternate BCH Codes and Extended BCH Codes 223
10.4 Decoding Erasures as Well as Errors 229
10.5 Decoding More Than tErrors 231
10.6 Examples 237
Problem 240
11. Linearized Polynomials and Affine Polynomials 241
11.1 How to Find Their Roots 241
11.2 The Least Affine Multiple 245
★11.3 Abstract Properties of Linearized
and Affine Polynomials 247
★11.4 Transformations of f(z) 255
★11.5 Root Counting 257
★11.6 Low-Weight Codewords in Certain Codes 262
Problems 271
12. The Enumeration of Information Symbols
in BCH Codes 273
12.1 Converting the Problem to an Enumeration
of Certain Integers mod n 273
★12.2 Converting the Problem for Primitive BCH
Codes to an Enumeration of Certain g-ary Sequences 275
★12.3 Sequence Enumeration Theorems 278
★12.4 Examples 284
★12.5 The Enumeration of Information
Symbols in Nonprimitive BCH Codes 287
12.6 Asymptotic Results 290
★12.7 Actual Distance 294
Problems 296
13. The Information Rate of the Optimum Codes 298
13.1 The Hamming-Rao High-Rate Volume Bound 298
*13.2 Perfect Codes 302
*13.3 The Bound d n + 1 — fc 309
13.4 Plotkin’s Low-Rate Average Distance Bound 311
*13.5 Equidistant Codes 315
13.6 The Elias Bound 318
xxiv
13.7 The Gilbert Bound
13.8 Asymptotic Error Bounds and Finite Special Cases
14. Codes Derived by Modifying or
Combining Other Codes
14.1 Extending a Code by Annexing More Check Digits
14.2 Puncturing a Code by Deleting Check Digits
14.3 Augmenting Additional Codewords to the Code
14.4 Expurgating Codewords from the Code
14.5 Lengthening the Code by Annexing
More Message Digits
14.6 Shortening the Code by Omitting Message Digits
14.7 Subfield Subcodes
14.8 Direct-Product Codes and Their Relatives
14.9 Concatenated Codes
15. Other Important Coding and Decoding Methods
15.1 Srivastava Codes—Noncyclic Codes with
an Algebraic Decoding Algorithm
15.2 Residue Codes—Good Codes Which
are Hard to Decode
15.3 Reed-Muller Codes—Weak Codes
Which are Easy to Decode
15.4 Threshold Decoding—The Best Known
Algorithm for Decoding Certain Codes
15.5 Orthogonalizable Codes Based on Finite Geometries
15.6 Convolutional Codes—A Survey
Problems
16. Weight Enumerators
16.1 The Relationship between Weight Enumerators
and the Probability of Decoding Failure
16.2 The MacWilliams-Pless Equations for the
Weight Enumerators of Dual Codes
★16.3 Weight Restrictions
★16.4 Kasami’s Weight Enumerators for Certain
Subcodes of the Second-Order RM Code
★16.5 The Weight Enumerator for the Reed-Solomon Codes
Appendix A
Appendix B
References
References to the Second Edition
Index
CONTENTS
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| any_adam_object | 1 |
| author | Berlekamp, Elwyn R. 1940-2019 |
| author_GND | (DE-588)123359732 |
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| classification_rvk | SK 170 |
| classification_tum | DAT 580f |
| ctrlnum | (OCoLC)953885374 (DE-599)BVBBV042612547 |
| discipline | Informatik Mathematik |
| edition | Revised edition |
| format | Book |
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| id | DE-604.BV042612547 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-24T04:29:30Z |
| institution | BVB |
| isbn | 9789814635905 9814635901 9789814635899 9814635898 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-028045398 |
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| physical | xxiv, 474 pages |
| publishDate | 2015 |
| publishDateSearch | 2015 |
| publishDateSort | 2015 |
| publisher | World Scientific |
| record_format | marc |
| spellingShingle | Berlekamp, Elwyn R. 1940-2019 Algebraic coding theory Codierung (DE-588)4070059-8 gnd Codierungstheorie (DE-588)4139405-7 gnd Algebraische Codierung (DE-588)4141834-7 gnd Informationstheorie (DE-588)4026927-9 gnd |
| subject_GND | (DE-588)4070059-8 (DE-588)4139405-7 (DE-588)4141834-7 (DE-588)4026927-9 |
| title | Algebraic coding theory |
| title_auth | Algebraic coding theory |
| title_exact_search | Algebraic coding theory |
| title_full | Algebraic coding theory Elwyn Berlekamp |
| title_fullStr | Algebraic coding theory Elwyn Berlekamp |
| title_full_unstemmed | Algebraic coding theory Elwyn Berlekamp |
| title_short | Algebraic coding theory |
| title_sort | algebraic coding theory |
| topic | Codierung (DE-588)4070059-8 gnd Codierungstheorie (DE-588)4139405-7 gnd Algebraische Codierung (DE-588)4141834-7 gnd Informationstheorie (DE-588)4026927-9 gnd |
| topic_facet | Codierung Codierungstheorie Algebraische Codierung Informationstheorie |
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