Algebraic-Geometric Codes

Inhaltsverzeichnis

• 1. 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.