Rational points on curves over finite fields: theory and applications
Ever since the seminal work of Goppa on algebraic-geometry codes, rational points on algebraic curves over finite fields have been an important research topic for algebraic geometers and coding theorists. The focus in this application of algebraic geometry to coding theory is on algebraic curves ove...
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| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
2001
|
| Series: | London Mathematical Society lecture note series
285 |
| Subjects: | |
| Online Access: | BSB01 FHN01 Volltext |
| Summary: | Ever since the seminal work of Goppa on algebraic-geometry codes, rational points on algebraic curves over finite fields have been an important research topic for algebraic geometers and coding theorists. The focus in this application of algebraic geometry to coding theory is on algebraic curves over finite fields with many rational points (relative to the genus). Recently, the authors discovered another important application of such curves, namely to the construction of low-discrepancy sequences. These sequences are needed for numerical methods in areas as diverse as computational physics and mathematical finance. This has given additional impetus to the theory of, and the search for, algebraic curves over finite fields with many rational points. This book aims to sum up the theoretical work on algebraic curves over finite fields with many rational points and to discuss the applications of such curves to algebraic coding theory and the construction of low-discrepancy sequences |
| Item Description: | Title from publisher's bibliographic system (viewed on 05 Oct 2015) |
| Physical Description: | 1 online resource (x, 245 pages) |
| ISBN: | 9781107325951 |
| DOI: | 10.1017/CBO9781107325951 |
Staff View
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| 505 | 8 | 0 | |t Background on Function Fields |t Riemann-Roch Theorem |t Divisor Class Groups and Ideal Class Groups |t Algebraic Extensions and the Hurwitz Formula |t Ramification Theory of Galois Extensions |t Constant Field Extensions |t Zeta Functions and Rational Places |t Class Field Theory |t Local Fields |t Newton Polygons |t Ramification Groups and Conductors |t Global Fields |t Ray Class Fields and Hilbert Class Fields |t Narrow Ray Class Fields |t Class Field Towers |t Explicit Function Fields |t Kummer and Artin-Schreier Extensions |t Cyclotomic Function Fields |t Drinfeld Modules of Rank 1 |t Function Fields with Many Rational Places |t Function Fields from Hilbert Class Fields |t Function Fields from Narrow Ray Class Fields |t The First Construction |t The Second Construction |t The Third Construction |t Function Fields from Cyclotomic Fields |t Explicit Function Fields |t Asymptotic Results |t Asymptotic Behavior of Towers |t The Lower Bound of Serre |t Further Lower Bounds for A(q[superscript m]) |t Explicit Towers |t Lower Bounds on A(2), A(3), and A(5) |t Applications to Algebraic Coding Theory |t Goppa's Algebraic-Geometry Codes |t Beating the Asymptotic Gilbert-Varshamov Bound |t NXL Codes |t XNL Codes |t A Propagation Rule for Linear Codes |t Applications to Cryptography |t Background on Stream Ciphers and Linear Complexity |t Constructions of Almost Perfect Sequences |t A Construction of Perfect Hash Families |t Hash Families and Authentication Schemes |t Applications to Low-Discrepancy Sequences |
| 520 | |a Ever since the seminal work of Goppa on algebraic-geometry codes, rational points on algebraic curves over finite fields have been an important research topic for algebraic geometers and coding theorists. The focus in this application of algebraic geometry to coding theory is on algebraic curves over finite fields with many rational points (relative to the genus). Recently, the authors discovered another important application of such curves, namely to the construction of low-discrepancy sequences. These sequences are needed for numerical methods in areas as diverse as computational physics and mathematical finance. This has given additional impetus to the theory of, and the search for, algebraic curves over finite fields with many rational points. This book aims to sum up the theoretical work on algebraic curves over finite fields with many rational points and to discuss the applications of such curves to algebraic coding theory and the construction of low-discrepancy sequences | ||
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Record in the Search Index
| _version_ | 1804176884263550976 |
|---|---|
| any_adam_object | |
| author | Niederreiter, Harald 1944- |
| author_facet | Niederreiter, Harald 1944- |
| author_role | aut |
| author_sort | Niederreiter, Harald 1944- |
| author_variant | h n hn |
| building | Verbundindex |
| bvnumber | BV043941945 |
| classification_rvk | SI 320 SK 180 |
| collection | ZDB-20-CBO |
| contents | Background on Function Fields Riemann-Roch Theorem Divisor Class Groups and Ideal Class Groups Algebraic Extensions and the Hurwitz Formula Ramification Theory of Galois Extensions Constant Field Extensions Zeta Functions and Rational Places Class Field Theory Local Fields Newton Polygons Ramification Groups and Conductors Global Fields Ray Class Fields and Hilbert Class Fields Narrow Ray Class Fields Class Field Towers Explicit Function Fields Kummer and Artin-Schreier Extensions Cyclotomic Function Fields Drinfeld Modules of Rank 1 Function Fields with Many Rational Places Function Fields from Hilbert Class Fields Function Fields from Narrow Ray Class Fields The First Construction The Second Construction The Third Construction Function Fields from Cyclotomic Fields Asymptotic Results Asymptotic Behavior of Towers The Lower Bound of Serre Further Lower Bounds for A(q[superscript m]) Explicit Towers Lower Bounds on A(2), A(3), and A(5) Applications to Algebraic Coding Theory Goppa's Algebraic-Geometry Codes Beating the Asymptotic Gilbert-Varshamov Bound NXL Codes XNL Codes A Propagation Rule for Linear Codes Applications to Cryptography Background on Stream Ciphers and Linear Complexity Constructions of Almost Perfect Sequences A Construction of Perfect Hash Families Hash Families and Authentication Schemes Applications to Low-Discrepancy Sequences |
| ctrlnum | (ZDB-20-CBO)CR9781107325951 (OCoLC)852492790 (DE-599)BVBBV043941945 |
| dewey-full | 516.3/52 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/52 |
| dewey-search | 516.3/52 |
| dewey-sort | 3516.3 252 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1017/CBO9781107325951 |
| format | Electronic eBook |
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| id | DE-604.BV043941945 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:39:16Z |
| institution | BVB |
| isbn | 9781107325951 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-029350915 |
| oclc_num | 852492790 |
| open_access_boolean | |
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| owner_facet | DE-12 DE-92 |
| physical | 1 online resource (x, 245 pages) |
| psigel | ZDB-20-CBO ZDB-20-CBO BSB_PDA_CBO ZDB-20-CBO FHN_PDA_CBO |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | Cambridge University Press |
| record_format | marc |
| series2 | London Mathematical Society lecture note series |
| spelling | Niederreiter, Harald 1944- Verfasser aut Rational points on curves over finite fields theory and applications Harald Niederreiter, Chaoping Xing Cambridge Cambridge University Press 2001 1 online resource (x, 245 pages) txt rdacontent c rdamedia cr rdacarrier London Mathematical Society lecture note series 285 Title from publisher's bibliographic system (viewed on 05 Oct 2015) Background on Function Fields Riemann-Roch Theorem Divisor Class Groups and Ideal Class Groups Algebraic Extensions and the Hurwitz Formula Ramification Theory of Galois Extensions Constant Field Extensions Zeta Functions and Rational Places Class Field Theory Local Fields Newton Polygons Ramification Groups and Conductors Global Fields Ray Class Fields and Hilbert Class Fields Narrow Ray Class Fields Class Field Towers Explicit Function Fields Kummer and Artin-Schreier Extensions Cyclotomic Function Fields Drinfeld Modules of Rank 1 Function Fields with Many Rational Places Function Fields from Hilbert Class Fields Function Fields from Narrow Ray Class Fields The First Construction The Second Construction The Third Construction Function Fields from Cyclotomic Fields Explicit Function Fields Asymptotic Results Asymptotic Behavior of Towers The Lower Bound of Serre Further Lower Bounds for A(q[superscript m]) Explicit Towers Lower Bounds on A(2), A(3), and A(5) Applications to Algebraic Coding Theory Goppa's Algebraic-Geometry Codes Beating the Asymptotic Gilbert-Varshamov Bound NXL Codes XNL Codes A Propagation Rule for Linear Codes Applications to Cryptography Background on Stream Ciphers and Linear Complexity Constructions of Almost Perfect Sequences A Construction of Perfect Hash Families Hash Families and Authentication Schemes Applications to Low-Discrepancy Sequences Ever since the seminal work of Goppa on algebraic-geometry codes, rational points on algebraic curves over finite fields have been an important research topic for algebraic geometers and coding theorists. The focus in this application of algebraic geometry to coding theory is on algebraic curves over finite fields with many rational points (relative to the genus). Recently, the authors discovered another important application of such curves, namely to the construction of low-discrepancy sequences. These sequences are needed for numerical methods in areas as diverse as computational physics and mathematical finance. This has given additional impetus to the theory of, and the search for, algebraic curves over finite fields with many rational points. This book aims to sum up the theoretical work on algebraic curves over finite fields with many rational points and to discuss the applications of such curves to algebraic coding theory and the construction of low-discrepancy sequences Curves, Algebraic Finite fields (Algebra) Rational points (Geometry) Coding theory Algebraische Kurve (DE-588)4001165-3 gnd rswk-swf Galois-Feld (DE-588)4155896-0 gnd rswk-swf Feld Mathematik (DE-588)4280861-3 gnd rswk-swf Algebraische Codierung (DE-588)4141834-7 gnd rswk-swf Rationaler Punkt (DE-588)4177004-3 gnd rswk-swf Algebraische Kurve (DE-588)4001165-3 s Galois-Feld (DE-588)4155896-0 s Rationaler Punkt (DE-588)4177004-3 s Algebraische Codierung (DE-588)4141834-7 s 1\p DE-604 Feld Mathematik (DE-588)4280861-3 s 2\p DE-604 Xing, Chaoping 1963- Sonstige oth Erscheint auch als Druckausgabe 978-0-521-66543-8 https://doi.org/10.1017/CBO9781107325951 Verlag URL des Erstveröffentlichers Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Niederreiter, Harald 1944- Rational points on curves over finite fields theory and applications Background on Function Fields Riemann-Roch Theorem Divisor Class Groups and Ideal Class Groups Algebraic Extensions and the Hurwitz Formula Ramification Theory of Galois Extensions Constant Field Extensions Zeta Functions and Rational Places Class Field Theory Local Fields Newton Polygons Ramification Groups and Conductors Global Fields Ray Class Fields and Hilbert Class Fields Narrow Ray Class Fields Class Field Towers Explicit Function Fields Kummer and Artin-Schreier Extensions Cyclotomic Function Fields Drinfeld Modules of Rank 1 Function Fields with Many Rational Places Function Fields from Hilbert Class Fields Function Fields from Narrow Ray Class Fields The First Construction The Second Construction The Third Construction Function Fields from Cyclotomic Fields Asymptotic Results Asymptotic Behavior of Towers The Lower Bound of Serre Further Lower Bounds for A(q[superscript m]) Explicit Towers Lower Bounds on A(2), A(3), and A(5) Applications to Algebraic Coding Theory Goppa's Algebraic-Geometry Codes Beating the Asymptotic Gilbert-Varshamov Bound NXL Codes XNL Codes A Propagation Rule for Linear Codes Applications to Cryptography Background on Stream Ciphers and Linear Complexity Constructions of Almost Perfect Sequences A Construction of Perfect Hash Families Hash Families and Authentication Schemes Applications to Low-Discrepancy Sequences Curves, Algebraic Finite fields (Algebra) Rational points (Geometry) Coding theory Algebraische Kurve (DE-588)4001165-3 gnd Galois-Feld (DE-588)4155896-0 gnd Feld Mathematik (DE-588)4280861-3 gnd Algebraische Codierung (DE-588)4141834-7 gnd Rationaler Punkt (DE-588)4177004-3 gnd |
| subject_GND | (DE-588)4001165-3 (DE-588)4155896-0 (DE-588)4280861-3 (DE-588)4141834-7 (DE-588)4177004-3 |
| title | Rational points on curves over finite fields theory and applications |
| title_alt | Background on Function Fields Riemann-Roch Theorem Divisor Class Groups and Ideal Class Groups Algebraic Extensions and the Hurwitz Formula Ramification Theory of Galois Extensions Constant Field Extensions Zeta Functions and Rational Places Class Field Theory Local Fields Newton Polygons Ramification Groups and Conductors Global Fields Ray Class Fields and Hilbert Class Fields Narrow Ray Class Fields Class Field Towers Explicit Function Fields Kummer and Artin-Schreier Extensions Cyclotomic Function Fields Drinfeld Modules of Rank 1 Function Fields with Many Rational Places Function Fields from Hilbert Class Fields Function Fields from Narrow Ray Class Fields The First Construction The Second Construction The Third Construction Function Fields from Cyclotomic Fields Asymptotic Results Asymptotic Behavior of Towers The Lower Bound of Serre Further Lower Bounds for A(q[superscript m]) Explicit Towers Lower Bounds on A(2), A(3), and A(5) Applications to Algebraic Coding Theory Goppa's Algebraic-Geometry Codes Beating the Asymptotic Gilbert-Varshamov Bound NXL Codes XNL Codes A Propagation Rule for Linear Codes Applications to Cryptography Background on Stream Ciphers and Linear Complexity Constructions of Almost Perfect Sequences A Construction of Perfect Hash Families Hash Families and Authentication Schemes Applications to Low-Discrepancy Sequences |
| title_auth | Rational points on curves over finite fields theory and applications |
| title_exact_search | Rational points on curves over finite fields theory and applications |
| title_full | Rational points on curves over finite fields theory and applications Harald Niederreiter, Chaoping Xing |
| title_fullStr | Rational points on curves over finite fields theory and applications Harald Niederreiter, Chaoping Xing |
| title_full_unstemmed | Rational points on curves over finite fields theory and applications Harald Niederreiter, Chaoping Xing |
| title_short | Rational points on curves over finite fields |
| title_sort | rational points on curves over finite fields theory and applications |
| title_sub | theory and applications |
| topic | Curves, Algebraic Finite fields (Algebra) Rational points (Geometry) Coding theory Algebraische Kurve (DE-588)4001165-3 gnd Galois-Feld (DE-588)4155896-0 gnd Feld Mathematik (DE-588)4280861-3 gnd Algebraische Codierung (DE-588)4141834-7 gnd Rationaler Punkt (DE-588)4177004-3 gnd |
| topic_facet | Curves, Algebraic Finite fields (Algebra) Rational points (Geometry) Coding theory Algebraische Kurve Galois-Feld Feld Mathematik Algebraische Codierung Rationaler Punkt |
| url | https://doi.org/10.1017/CBO9781107325951 |
| work_keys_str_mv | AT niederreiterharald rationalpointsoncurvesoverfinitefieldstheoryandapplications AT xingchaoping rationalpointsoncurvesoverfinitefieldstheoryandapplications |