Codes on algebraic curves:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
New York [u.a.]
Kluwer Acad. / Plenum Publ.
1999
|
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009426352&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XII, 350 S. graph. Darst. |
| ISBN: | 0306461447 |
Internformat
MARC
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| 100 | 1 | |a Stepanov, Sergej A. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Codes on algebraic curves |c Serguei A. Stepanov |
| 264 | 1 | |a New York [u.a.] |b Kluwer Acad. / Plenum Publ. |c 1999 | |
| 300 | |a XII, 350 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 7 | |a Courbes algébriques |2 ram | |
| 650 | 7 | |a Goppa, Codes de |2 ram | |
| 650 | 4 | |a Curves, Algebraic | |
| 650 | 4 | |a Goppa codes | |
| 650 | 0 | 7 | |a Algebraische Kurve |0 (DE-588)4001165-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Goppa-Code |0 (DE-588)4218589-0 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
| _version_ | 1819383318143041536 |
|---|---|
| adam_text | Contents
I. Error Correcting Codes
Chapter 1
Codes and Their Parameters 3
1.1. Introduction 3
1.2. Finite Fields 5
1.3. Linear Codes 13
1.4. Spectrum and Duality 15
Exercises 20
Chapter 2
Bounds on Codes 25
2.1. Upper Bounds 26
2.2. The Linear Programming Bound 32
2.3. Lower Bounds 35
Exercises 37
Chapter 3
Examples and Constructions 41
3.1. Codes of Genus Zero 41
3.2. Some Families of Codes 46
3.3. Constructing Codes from other Codes 60
Exercises 64
ix
x Contents
II. Algebraic Curves and Varieties
Chapter 4
Algebraic Curves 71
4.1. Algebraic Varieties 71
4.2. Non Singular Curves 77
4.3. Divisors on Algebraic Curves 80
4.4. The Riemann Roch Theorem 85
4.5. Hurwitz and Pliicker Genus Formulas 93
4.6. Special Divisors 96
Exercises 98
Chapter 5
Curves over a Finite Field 103
5.1. Rational Points and Divisors 105
5.2. The Zeta Function of a Curve Ill
5.3. L Functions of Artin 120
5.4. Algebraic Function Fields 130
Exercises 139
Chapter 6
Counting Points on Curves over Finite Fields 143
6.1. The Number of Rational Points on a Curve 143
6.2. Character Sums 147
6.3. Asymptotics 157
Exercises 170
III. Elliptic and Modular Curves
Chapter 7
Elliptic Curves 175
7.1. The Group Law 175
7.2. The ./ Invariant 178
7.3. Isogenies 180
7.4. Elliptic Curves over Finite Fields 184
7.5. Elliptic Functions 186
Exercises 190
Chapter 8
Classical Modular Curves 193
8.1. Congruence Subgroups 193
8.2. The CurvesX{N),X0(N), andXi(N) 195
Contents xi
8.3. Hecke Operators 199
8.4. The Petersson Inner Product 212
Exercises 215
Chapter 9
Reductions of Modular Curves 219
9.1. Reductions and Moduli Spaces 219
9.2. The Igusa Theorem 224
9.3. The Eichler Shimura Congruence Relation 231
9.4. The Eichler Selberg Trace Formula 236
Exercises 239
IV. Geometric Goppa Codes
Chapter 10
Constructions and Properties 243
10.1. /. Construction 243
10.2. ^ Construction 245
10.3. Parameters 248
10.4. Duality and Spectra 251
Exercises 254
Chapter 11
Examples 257
11.1. Codes of Small Genera 257
11.2. Elliptic and Hermitian Codes 261
11.3. Codes on Fiber Products 267
11.4. Codes on Classical Modular Curves 274
11.5. Codes on Artin—Schreier Coverings 276
11.6. Codes on Trace Norm Curves 284
Exercises 287
Chapter 12
Decoding Geometric Goppa Codes 289
12.1. The Decoding Problem 289
12.2. The Basic and Modified Algorithms 292
12.3. An Improvement of the Modified Algorithm 301
12.4. Majority Voting for Unknown Syndromes 306
12.5. Faster Decoding 309
Exercises 310
xii Contents
Chapter 13
Bounds 315
13.1. Asymptotic Bounds 315
13.2. Constructive Bounds 316
13.3. Other Bounds 319
Exercises 321
Bibliography 323
List of Notations 335
Index 343
|
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| id | DE-604.BV013788988 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T10:53:32Z |
| institution | BVB |
| isbn | 0306461447 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009426352 |
| oclc_num | 40135329 |
| open_access_boolean | |
| owner | DE-20 DE-706 DE-11 DE-188 |
| owner_facet | DE-20 DE-706 DE-11 DE-188 |
| physical | XII, 350 S. graph. Darst. |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | Kluwer Acad. / Plenum Publ. |
| record_format | marc |
| spellingShingle | Stepanov, Sergej A. Codes on algebraic curves Courbes algébriques ram Goppa, Codes de ram Curves, Algebraic Goppa codes Algebraische Kurve (DE-588)4001165-3 gnd Goppa-Code (DE-588)4218589-0 gnd |
| subject_GND | (DE-588)4001165-3 (DE-588)4218589-0 |
| title | Codes on algebraic curves |
| title_auth | Codes on algebraic curves |
| title_exact_search | Codes on algebraic curves |
| title_full | Codes on algebraic curves Serguei A. Stepanov |
| title_fullStr | Codes on algebraic curves Serguei A. Stepanov |
| title_full_unstemmed | Codes on algebraic curves Serguei A. Stepanov |
| title_short | Codes on algebraic curves |
| title_sort | codes on algebraic curves |
| topic | Courbes algébriques ram Goppa, Codes de ram Curves, Algebraic Goppa codes Algebraische Kurve (DE-588)4001165-3 gnd Goppa-Code (DE-588)4218589-0 gnd |
| topic_facet | Courbes algébriques Goppa, Codes de Curves, Algebraic Goppa codes Algebraische Kurve Goppa-Code |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009426352&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT stepanovsergeja codesonalgebraiccurves |