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The Resource Rational points on curves over finite fields : theory and applications, H. Niederreiter and Chaoping Xing

Rational points on curves over finite fields : theory and applications, H. Niederreiter and Chaoping Xing

Label
Rational points on curves over finite fields : theory and applications
Title
Rational points on curves over finite fields
Title remainder
theory and applications
Statement of responsibility
H. Niederreiter and Chaoping Xing
Creator
Contributor
Subject
Language
eng
Cataloging source
BDS
http://library.link/vocab/creatorDate
1944-
http://library.link/vocab/creatorName
Niederreiter, Harald
Index
no index present
Literary form
non fiction
http://library.link/vocab/relatedWorkOrContributorDate
1963-
http://library.link/vocab/relatedWorkOrContributorName
Xing, Chaoping
Series statement
London Mathematical Society lecture note series
Series volume
285
http://library.link/vocab/subjectName
  • Curves, Algebraic
  • Finite fields (Algebra)
Label
Rational points on curves over finite fields : theory and applications, H. Niederreiter and Chaoping Xing
Instantiates
Publication
Bibliography note
Includes bibliographical references and index
Carrier category
volume
Carrier category code
  • nc
Carrier MARC source
rdacarrier
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
  • Preface
  • 1.
  • Background on Function Fields.
  • p. 1
  • 1.1.
  • Riemann-Roch Theorem.
  • p. 1
  • 1.2.
  • Divisor Class Groups and Ideal Class Groups.
  • p. 6
  • 1.3.
  • Algebraic Extensions and the Hurwitz Formula.
  • p. 10
  • 1.4.
  • Ramification Theory of Galois Extensions.
  • p. 14
  • 1.5.
  • Constant Field Extensions.
  • p. 20
  • 1.6.
  • Zeta Functions and Rational Places.
  • p. 26
  • 2.
  • Class Field Theory.
  • p. 36
  • 2.1.
  • Local Fields.
  • p. 36
  • 2.2.
  • Newton Polygons.
  • p. 38
  • 2.3.
  • Ramification Groups and Conductors.
  • p. 39
  • 2.4.
  • Global Fields.
  • p. 44
  • 2.5.
  • Ray Class Field and Hilbert Class Fields.
  • p. 47
  • 2.6.
  • Narrow Ray Class Fields.
  • p. 50
  • 2.7.
  • Class Field Towers.
  • p. 55
  • 3.
  • Explicit Function Fields.
  • p. 62
  • 3.1.
  • Kummer and Artin-Schreier Extensions.
  • p. 62
  • 3.2.
  • Cyclotomic Function Fields.
  • p. 65
  • 3.3.
  • Drinfeld Modules of Rank 1.
  • p. 72
  • 4.
  • Function Fields with Many Rational Places.
  • p. 76
  • 4.1.
  • Function Fields from Hilbert Class Fields.
  • p. 76
  • 4.2.
  • Function Fields from Narrow Ray Class Fields.
  • p. 82
  • 4.3.
  • Function Fields from Cyclotomic Fields.
  • p. 108
  • 4.4.
  • Explicit Function Fields.
  • p. 113
  • 4.5.
  • Tables.
  • p. 118
  • 5.
  • Asymptotic Results.
  • p. 122
  • 5.1.
  • Asymptotic Behavior of Towers.
  • p. 122
  • 5.2.
  • Lower Bound of Serre.
  • p. 126
  • 5.3.
  • Further Lower Bounds for A(q[superscript m]).
  • p. 133
  • 5.4.
  • Explicit Towers.
  • p. 136
  • 5.5.
  • Lower Bounds on A(2), A(3), and A(5).
  • p. 138
  • 6.
  • Applications to Algebraic Coding Theory.
  • p. 141
  • 6.1.
  • Goppa's Algebraic-Geometry Codes.
  • p. 141
  • 6.2.
  • Beating the Asymptotic Gilbert-Varshamov Bound.
  • p. 150
  • 6.3.
  • NXL Codes.
  • p. 156
  • 6.4.
  • XNL Codes.
  • p. 160
  • 6.5.
  • Propagation Rule for Linear Codes.
  • p. 164
  • 7.
  • Applications to Cryptography.
  • p. 170
  • 7.1.
  • Background on Stream Ciphers and Linear Complexity.
  • p. 170
  • 7.2.
  • Constructions of Almost Perfect Sequences.
  • p. 177
  • 7.3.
  • Construction of Perfect Hash Families.
  • p. 184
  • 7.4.
  • Hash Families and Authentication Schemes.
  • p. 186
  • 8.
  • Applications to Low-Discrepancy Sequences.
  • p. 191
  • 8.1.
  • Background on (t, m, s)-Nets and (t,s)-Sequences.
  • p. 191
  • 8.2.
  • Digital Method.
  • p. 197
  • 8.3.
  • Construction Using Rational Places.
  • p. 203
  • 8.4.
  • Construction Using Arbitrary Places.
  • p. 212
  • A.
  • Curves and Their Function Fields.
  • p. 219
  • Bibliography.
  • p. 227
  • Index.
  • p. 240
Control code
20000a777545
Dimensions
23 cm.
Extent
x, 245 p.
Isbn
9780521665438
Media category
unmediated
Media MARC source
rdamedia
Media type code
  • n
Other physical details
ill.
Label
Rational points on curves over finite fields : theory and applications, H. Niederreiter and Chaoping Xing
Publication
Bibliography note
Includes bibliographical references and index
Carrier category
volume
Carrier category code
  • nc
Carrier MARC source
rdacarrier
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
  • Preface
  • 1.
  • Background on Function Fields.
  • p. 1
  • 1.1.
  • Riemann-Roch Theorem.
  • p. 1
  • 1.2.
  • Divisor Class Groups and Ideal Class Groups.
  • p. 6
  • 1.3.
  • Algebraic Extensions and the Hurwitz Formula.
  • p. 10
  • 1.4.
  • Ramification Theory of Galois Extensions.
  • p. 14
  • 1.5.
  • Constant Field Extensions.
  • p. 20
  • 1.6.
  • Zeta Functions and Rational Places.
  • p. 26
  • 2.
  • Class Field Theory.
  • p. 36
  • 2.1.
  • Local Fields.
  • p. 36
  • 2.2.
  • Newton Polygons.
  • p. 38
  • 2.3.
  • Ramification Groups and Conductors.
  • p. 39
  • 2.4.
  • Global Fields.
  • p. 44
  • 2.5.
  • Ray Class Field and Hilbert Class Fields.
  • p. 47
  • 2.6.
  • Narrow Ray Class Fields.
  • p. 50
  • 2.7.
  • Class Field Towers.
  • p. 55
  • 3.
  • Explicit Function Fields.
  • p. 62
  • 3.1.
  • Kummer and Artin-Schreier Extensions.
  • p. 62
  • 3.2.
  • Cyclotomic Function Fields.
  • p. 65
  • 3.3.
  • Drinfeld Modules of Rank 1.
  • p. 72
  • 4.
  • Function Fields with Many Rational Places.
  • p. 76
  • 4.1.
  • Function Fields from Hilbert Class Fields.
  • p. 76
  • 4.2.
  • Function Fields from Narrow Ray Class Fields.
  • p. 82
  • 4.3.
  • Function Fields from Cyclotomic Fields.
  • p. 108
  • 4.4.
  • Explicit Function Fields.
  • p. 113
  • 4.5.
  • Tables.
  • p. 118
  • 5.
  • Asymptotic Results.
  • p. 122
  • 5.1.
  • Asymptotic Behavior of Towers.
  • p. 122
  • 5.2.
  • Lower Bound of Serre.
  • p. 126
  • 5.3.
  • Further Lower Bounds for A(q[superscript m]).
  • p. 133
  • 5.4.
  • Explicit Towers.
  • p. 136
  • 5.5.
  • Lower Bounds on A(2), A(3), and A(5).
  • p. 138
  • 6.
  • Applications to Algebraic Coding Theory.
  • p. 141
  • 6.1.
  • Goppa's Algebraic-Geometry Codes.
  • p. 141
  • 6.2.
  • Beating the Asymptotic Gilbert-Varshamov Bound.
  • p. 150
  • 6.3.
  • NXL Codes.
  • p. 156
  • 6.4.
  • XNL Codes.
  • p. 160
  • 6.5.
  • Propagation Rule for Linear Codes.
  • p. 164
  • 7.
  • Applications to Cryptography.
  • p. 170
  • 7.1.
  • Background on Stream Ciphers and Linear Complexity.
  • p. 170
  • 7.2.
  • Constructions of Almost Perfect Sequences.
  • p. 177
  • 7.3.
  • Construction of Perfect Hash Families.
  • p. 184
  • 7.4.
  • Hash Families and Authentication Schemes.
  • p. 186
  • 8.
  • Applications to Low-Discrepancy Sequences.
  • p. 191
  • 8.1.
  • Background on (t, m, s)-Nets and (t,s)-Sequences.
  • p. 191
  • 8.2.
  • Digital Method.
  • p. 197
  • 8.3.
  • Construction Using Rational Places.
  • p. 203
  • 8.4.
  • Construction Using Arbitrary Places.
  • p. 212
  • A.
  • Curves and Their Function Fields.
  • p. 219
  • Bibliography.
  • p. 227
  • Index.
  • p. 240
Control code
20000a777545
Dimensions
23 cm.
Extent
x, 245 p.
Isbn
9780521665438
Media category
unmediated
Media MARC source
rdamedia
Media type code
  • n
Other physical details
ill.

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