Algebraic-Geometric Codes

✍ Scribed by M. A. Tsfasman, S. G. Vlăduţ (auth.)

Publisher: Springer Netherlands
Year: 1991
Tongue: English
Leaves: 671
Series: Mathematics and Its Applications (Soviet Series) 58
Edition: 1
Category: Library

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✦ Synopsis

Codes.- 1.1. Codes and their parameters.- 1.2. Examples and constructions.- 1.3. Asymptotic problems.- 2. Curves.- 2.1. Algebraic curves.- 2.2. Riemann-Roch theorem.- 2.3. Rational points.- 2.4. Elliptic curves.- 2.5. Singular curves.- 2.6. Reductions and schemes.- 3. AG-Codes.- 3.1. Constructions and properties.- 3.2. Examples.- 3.3. Decoding.- 3.4. Asymptotic results.- 4. Modular Codes.- 4.1. Codes on classical modular curves.- 4.2. Codes on Drinfeld curves.- 4.3. Polynomiality.- 5. Sphere Packings.- 5.1. Definitions and examples.- 5.2. Asymptotically dense packings.- 5.3. Number fields.- 5.4. Analogues of AG-codes.- Appendix. Summary of results and tables.- A.1. Codes of finite length.- A.1.1. Bounds.- A.1.2. Parameters of certain codes.- A.1.3. Parameters of certain constructions.- A.1.4. Binary codes from AG-codes.- A.2. Asymptotic bounds.- A.2.1. List of bounds.- A.2.2. Diagrams of comparison.- A.2.3. Behaviour at the ends.- A.2.4. Numerical values.- A.3. Additional bounds.- A.3.1. Constant weight codes.- A.3.2. Self-dual codes.- A.4. Sphere packings.- A.4.1. Small dimensions.- A.4.2. Certain families.- A.4.3. Asymptotic results.- Author index.- List of symbols.

✦ Table of Contents

Front Matter....Pages i-xxiv
Front Matter....Pages 1-4
Codes and Their Parameters....Pages 5-35
Examples and Constructions....Pages 37-65
Asymptotic Problems....Pages 67-91
Back Matter....Pages 92-94
Front Matter....Pages 95-100
Algebraic Curves....Pages 101-139
Riemann-Roch Theorem....Pages 141-167
Rational Points....Pages 169-190
Elliptic Curves....Pages 191-213
Singular Curves....Pages 215-232
Reductions and Schemes....Pages 233-255
Back Matter....Pages 256-259
Front Matter....Pages 261-264
Constructions and Properties....Pages 265-296
Examples....Pages 297-329
Decoding....Pages 331-347
Asymptotic Results....Pages 349-385
Back Matter....Pages 386-388
Front Matter....Pages 389-393
Codes on Classical Modular Curves....Pages 395-434
Codes on Drinfeld Curves....Pages 435-469
Polynomiality....Pages 471-510
Back Matter....Pages 511-513
Front Matter....Pages 515-518
Definitions and Examples....Pages 519-537
Asymptotically Dense Packings....Pages 539-549
Number Fields....Pages 551-566
Analogues of AG-Codes....Pages 567-593
Back Matter....Pages 594-596
Back Matter....Pages 597-667

✦ Subjects

Algebraic Geometry; Number Theory; Electrical Engineering; Theory of Computation

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