Algebraic geometric codes: basic notions
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Math. Soc.
2007
|
| Series: | Mathematical surveys and monographs
139 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIX, 338 S. graph. Darst. |
| ISBN: | 9780821843062 |
Staff View
MARC
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| 100 | 1 | |a Čfasman, Michail A. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Algebraic geometric codes |b basic notions |c Michael Tsfasman ; Serge Vlǎduţ ; Dmitry Nogin |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 2007 | |
| 300 | |a XIX, 338 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematical surveys and monographs |v 139 | |
| 650 | 4 | |a Coding theory | |
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| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Vlǎduţ, Serge G. |e Verfasser |4 aut | |
| 700 | 1 | |a Nogin, Dmitrij J. |d 1966- |e Verfasser |0 (DE-588)135965586 |4 aut | |
| 830 | 0 | |a Mathematical surveys and monographs |v 139 |w (DE-604)BV000018014 |9 139 | |
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Record in the Search Index
| _version_ | 1804137528753651712 |
|---|---|
| adam_text | Contents
Preface
ix
Advice
to the Reader
xvii
Chapter
1.
Codes
1
1.1.
Codes and Their Parameters
1
1.1.1.
Definition of a Code
1
1.1.2.
[n,k,d]q Systems
3
1.1.3.
Spectra and Duality
7
1.1.4.
Bounds
15
1.1.5.
Bounds for Higher Weights
21
1.1.6.
Duality for Generalized Spectra
29
1.2.
Examples and Constructions
33
1.2.1.
Codes of Genus Zero
33
1.2.2.
Code Families
36
1.2.3.
Constructions
44
1.3.
Asymptotic Problems
49
1.3.1.
Main Asymptotic Problem
49
1.3.2.
Asymptotic Bounds
51
1.3.3.
Asymptotic Bounds for Higher Weights
57
1.3.4.
Polynomiality
60
1.3.5.
Other Asymptotics
64
Historical and Bibliographic Notes
67
Chapter
2.
Curves
69
2.1.
Algebraic Curves
69
2.1.1.
Quasiprojective Varieties
70
2.1.2.
Quasiprojective Curves
77
2.1.3.
Divisors
79
2.1.4.
Jacobians
85
2.1.5.
Riemann Surfaces
88
2.2.
Riemann-Roch Theorem
91
2.2.1.
Differential Forms
91
2.2.2.
Riemann-Roch Theorem
95
2.2.3.
Hurwitz Formula
100
2.2.4.
Special Divisors
102
2.2.5.
Cartier
Operator
104
2.3.
Singular Curves
106
2.3.1.
Normalization
106
vi
CONTENTS
2.3.2.
Double-Point Divisor
107
2.3.3.
Plain Curves
108
2.4.
Elliptic Curves
Ш
2.4.1.
Group Law 111
2.4.2.
Isomorphisms and the j-invariant
114
2.4.3. Isogenies 115
2.4.4.
Complex Elliptic Curves
118
2.5.
Curves over Nonclosed Fields
120
2.5.1.
Function Fields
120
2.5.2.
Places of a Function Field
122
2.5.3.
Divisors
125
2.5.4.
Function Fields and Algebraic Curves
126
Historical and Bibliographic Notes
131
Chapter
3.
Curves over Finite Fields
133
3.1.
Zeta
Function
133
3.1.1.
Definition and Rationality
134
3.1.2.
Functional Equation
138
3.1.3.
Weil Theorem and Its Corollaries
141
3.1.4.
Explicit Formula
143
3.1.5.
Pellikaan s Two-Variable
Zeta
Function
144
3.2.
Asymptotics
146
3.2.1.
Drinfeld-Vlăduţ
Theorem
146
3.2.2.
Lower Asymptotic Bounds
147
3.2.3.
Points of Higher Degrees
147
3.2.4.
Asymptotics for the Jacobian
149
3.2.5.
Asymptotically Exact Families
151
3.3.
Elliptic Curves over Finite Fields
158
3.3.1.
Isomorphism Classes
158
3.3.2.
Isogeny Classes
161
3.3.3.
Endoniorphism Ring and the
Zeta
Function
162
3.3.4.
Structure of E(Fq)
163
3.4.
Remarkable Examples
165
3.4.1.
Hermitian, Sub-Hermitian, and Maximal Curves
165
3.4.2. Kummer
and Artin-Schreier Covers
167
3.4.3.
Garcia-Stichtenoth Towers
177
3.4.4.
Curves of Small Genera
178
3.5.
Connection with Exponential Sums
180
3.5.1.
Number of Points on
Fermat
Curves
180
3.5.2.
¿-functions of Characters
181
3.5.3.
Estimates for Exponential Sums
183
Historical and Bibliographic Notes
187
Chapter
4.
Algebraic Geometry Codes
191
4.1.
Constructions and Properties
191
4.1.1.
Basic Algebraic Geometry Constructions and
Their Parameters
192
4.1.2.
Duality and Spectra
198
4.1.3.
Decoding Problem
202
CONTENTS
vii
4.2.
Additional Bounds and Constructions
206
4.2.1.
Extra Bounds
206
4.2.2.
Variants of the Basic Construction
216
4.2.3.
Partial Algebraic Geometry Codes
219
4.3.
Characterization of Algebraic Geometry Codes
225
4.3.1.
Three
AG
Levels
225
4.3.2.
All Linear Codes Are Weakly
AG
226
4.3.3.
Criteria
228
4.4.
Examples
232
4.4.1.
Codes of Small Genera
232
4.4.2.
Elliptic Codes
234
4.4.3.
Hermitian Codes
241
4.4.4.
Other Examples
245
4.4.5.
Generalized Algebraic Geometry Codes
247
4.5.
Asymptotic Results
250
4.5.1.
The Basic Algebraic Geometry Bound and Its Variants
250
4.5.2.
Expurgation Bound and Codes witli Many Light Vectors
253
4.5.3.
Constructive Bounds
263
4.5.4.
Other Bounds
266
4.6.
Nonlinear Algebraic Geometry Constructions
271
4.6.1.
Elkies Codes
271
4.6.2.
Xing Codes
280
Historical and Bibliographic Notes
284
Appendix. Summary of Results and Tables
287
A.I. Codes of Finite Length
287
A.I.I. Bounds
287
A.
1.2.
Parameters of Some Codes
288
A.
1.3.
Parameters of Some Constructions
289
A.
2.
Asymptotic Bounds
292
À.2.I.
List of Bounds
292
A.
2.2.
Comparison Diagrams
295
A.
2.3.
Behaviour at the
Endpoints 299
A.
2.4.
Numerical Values
299
A.3. Additional Bounds
306
A.3.1. Constant-Weight Codes
306
A.3.2. Self-dual Codes
306
A.3.3. Bounds for Higher Weights
307
Bibliography
309
List of Names
329
Index
331
|
| adam_txt |
Contents
Preface
ix
Advice
to the Reader
xvii
Chapter
1.
Codes
1
1.1.
Codes and Their Parameters
1
1.1.1.
Definition of a Code
1
1.1.2.
[n,k,d]q Systems
3
1.1.3.
Spectra and Duality
7
1.1.4.
Bounds
15
1.1.5.
Bounds for Higher Weights
21
1.1.6.
Duality for Generalized Spectra
29
1.2.
Examples and Constructions
33
1.2.1.
Codes of Genus Zero
33
1.2.2.
Code Families
36
1.2.3.
Constructions
44
1.3.
Asymptotic Problems
49
1.3.1.
Main Asymptotic Problem
49
1.3.2.
Asymptotic Bounds
51
1.3.3.
Asymptotic Bounds for Higher Weights
57
1.3.4.
Polynomiality
60
1.3.5.
Other Asymptotics
64
Historical and Bibliographic Notes
67
Chapter
2.
Curves
69
2.1.
Algebraic Curves
69
2.1.1.
Quasiprojective Varieties
70
2.1.2.
Quasiprojective Curves
77
2.1.3.
Divisors
79
2.1.4.
Jacobians
85
2.1.5.
Riemann Surfaces
88
2.2.
Riemann-Roch Theorem
91
2.2.1.
Differential Forms
91
2.2.2.
Riemann-Roch Theorem
95
2.2.3.
Hurwitz Formula
100
2.2.4.
Special Divisors
102
2.2.5.
Cartier
Operator
104
2.3.
Singular Curves
106
2.3.1.
Normalization
106
vi
CONTENTS
2.3.2.
Double-Point Divisor
107
2.3.3.
Plain Curves
108
2.4.
Elliptic Curves
Ш
2.4.1.
Group Law 111
2.4.2.
Isomorphisms and the j-invariant
114
2.4.3. Isogenies 115
2.4.4.
Complex Elliptic Curves
118
2.5.
Curves over Nonclosed Fields
120
2.5.1.
Function Fields
120
2.5.2.
Places of a Function Field
122
2.5.3.
Divisors
125
2.5.4.
Function Fields and Algebraic Curves
126
Historical and Bibliographic Notes
131
Chapter
3.
Curves over Finite Fields
133
3.1.
Zeta
Function
133
3.1.1.
Definition and Rationality
134
3.1.2.
Functional Equation
138
3.1.3.
Weil Theorem and Its Corollaries
141
3.1.4.
Explicit Formula
143
3.1.5.
Pellikaan's Two-Variable
Zeta
Function
144
3.2.
Asymptotics
146
3.2.1.
Drinfeld-Vlăduţ
Theorem
146
3.2.2.
Lower Asymptotic Bounds
147
3.2.3.
Points of Higher Degrees
147
3.2.4.
Asymptotics for the Jacobian
149
3.2.5.
Asymptotically Exact Families
151
3.3.
Elliptic Curves over Finite Fields
158
3.3.1.
Isomorphism Classes
158
3.3.2.
Isogeny Classes
161
3.3.3.
Endoniorphism Ring and the
Zeta
Function
162
3.3.4.
Structure of E(Fq)
163
3.4.
Remarkable Examples
165
3.4.1.
Hermitian, Sub-Hermitian, and Maximal Curves
165
3.4.2. Kummer
and Artin-Schreier Covers
167
3.4.3.
Garcia-Stichtenoth Towers
177
3.4.4.
Curves of Small Genera
178
3.5.
Connection with Exponential Sums
180
3.5.1.
Number of Points on
Fermat
Curves
180
3.5.2.
¿-functions of Characters
181
3.5.3.
Estimates for Exponential Sums
183
Historical and Bibliographic Notes
187
Chapter
4.
Algebraic Geometry Codes
191
4.1.
Constructions and Properties
191
4.1.1.
Basic Algebraic Geometry Constructions and
Their Parameters
192
4.1.2.
Duality and Spectra
198
4.1.3.
Decoding Problem
202
CONTENTS
vii
4.2.
Additional Bounds and Constructions
206
4.2.1.
Extra Bounds
206
4.2.2.
Variants of the Basic Construction
216
4.2.3.
Partial Algebraic Geometry Codes
219
4.3.
Characterization of Algebraic Geometry Codes
225
4.3.1.
Three
AG
Levels
225
4.3.2.
All Linear Codes Are Weakly
AG
226
4.3.3.
Criteria
228
4.4.
Examples
232
4.4.1.
Codes of Small Genera
232
4.4.2.
Elliptic Codes
234
4.4.3.
Hermitian Codes
241
4.4.4.
Other Examples
245
4.4.5.
Generalized Algebraic Geometry Codes
247
4.5.
Asymptotic Results
250
4.5.1.
The Basic Algebraic Geometry Bound and Its Variants
250
4.5.2.
Expurgation Bound and Codes witli Many Light Vectors
253
4.5.3.
Constructive Bounds
263
4.5.4.
Other Bounds
266
4.6.
Nonlinear Algebraic Geometry Constructions
271
4.6.1.
Elkies Codes
271
4.6.2.
Xing Codes
280
Historical and Bibliographic Notes
284
Appendix. Summary of Results and Tables
287
A.I. Codes of Finite Length
287
A.I.I. Bounds
287
A.
1.2.
Parameters of Some Codes
288
A.
1.3.
Parameters of Some Constructions
289
A.
2.
Asymptotic Bounds
292
À.2.I.
List of Bounds
292
A.
2.2.
Comparison Diagrams
295
A.
2.3.
Behaviour at the
Endpoints 299
A.
2.4.
Numerical Values
299
A.3. Additional Bounds
306
A.3.1. Constant-Weight Codes
306
A.3.2. Self-dual Codes
306
A.3.3. Bounds for Higher Weights
307
Bibliography
309
List of Names
329
Index
331 |
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| author | Čfasman, Michail A. Vlǎduţ, Serge G. Nogin, Dmitrij J. 1966- |
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| dewey-full | 00300.54 |
| dewey-hundreds | 000 - Computer science, information, general works |
| dewey-ones | 003 - Systems |
| dewey-raw | 003 00 .54 |
| dewey-search | 003 00 .54 |
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| discipline | Informatik Mathematik |
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| id | DE-604.BV023234460 |
| illustrated | Illustrated |
| index_date | 2024-07-02T20:21:25Z |
| indexdate | 2024-07-09T21:13:44Z |
| institution | BVB |
| isbn | 9780821843062 |
| language | English |
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| physical | XIX, 338 S. graph. Darst. |
| publishDate | 2007 |
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| publisher | American Math. Soc. |
| record_format | marc |
| series | Mathematical surveys and monographs |
| series2 | Mathematical surveys and monographs |
| spelling | Čfasman, Michail A. Verfasser aut Algebraic geometric codes basic notions Michael Tsfasman ; Serge Vlǎduţ ; Dmitry Nogin Providence, RI American Math. Soc. 2007 XIX, 338 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematical surveys and monographs 139 Coding theory Geometry, Algebraic Number theory Codierungstheorie (DE-588)4139405-7 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Codierungstheorie (DE-588)4139405-7 s Algebraische Geometrie (DE-588)4001161-6 s DE-604 Vlǎduţ, Serge G. Verfasser aut Nogin, Dmitrij J. 1966- Verfasser (DE-588)135965586 aut Mathematical surveys and monographs 139 (DE-604)BV000018014 139 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016420100&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Čfasman, Michail A. Vlǎduţ, Serge G. Nogin, Dmitrij J. 1966- Algebraic geometric codes basic notions Mathematical surveys and monographs Coding theory Geometry, Algebraic Number theory Codierungstheorie (DE-588)4139405-7 gnd Algebraische Geometrie (DE-588)4001161-6 gnd |
| subject_GND | (DE-588)4139405-7 (DE-588)4001161-6 |
| title | Algebraic geometric codes basic notions |
| title_auth | Algebraic geometric codes basic notions |
| title_exact_search | Algebraic geometric codes basic notions |
| title_exact_search_txtP | Algebraic geometric codes basic notions |
| title_full | Algebraic geometric codes basic notions Michael Tsfasman ; Serge Vlǎduţ ; Dmitry Nogin |
| title_fullStr | Algebraic geometric codes basic notions Michael Tsfasman ; Serge Vlǎduţ ; Dmitry Nogin |
| title_full_unstemmed | Algebraic geometric codes basic notions Michael Tsfasman ; Serge Vlǎduţ ; Dmitry Nogin |
| title_short | Algebraic geometric codes |
| title_sort | algebraic geometric codes basic notions |
| title_sub | basic notions |
| topic | Coding theory Geometry, Algebraic Number theory Codierungstheorie (DE-588)4139405-7 gnd Algebraische Geometrie (DE-588)4001161-6 gnd |
| topic_facet | Coding theory Geometry, Algebraic Number theory Codierungstheorie Algebraische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016420100&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000018014 |
| work_keys_str_mv | AT cfasmanmichaila algebraicgeometriccodesbasicnotions AT vladutsergeg algebraicgeometriccodesbasicnotions AT nogindmitrijj algebraicgeometriccodesbasicnotions |