By M. Tsfasman, S.G. Vladut
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. discount rates and schemes.- three. AG-Codes.- 3.1. buildings and properties.- 3.2. Examples.- 3.3. Decoding.- 3.4. Asymptotic results.- four. Modular Codes.- 4.1. Codes on classical modular curves.- 4.2. Codes on Drinfeld curves.- 4.3. Polynomiality.- five. Sphere Packings.- 5.1. Definitions and examples.- 5.2. Asymptotically dense packings.- 5.3. quantity fields.- 5.4. Analogues of AG-codes.- Appendix. precis of effects and tables.- A.1. Codes of finite length.- A.1.1. Bounds.- A.1.2. Parameters of convinced codes.- A.1.3. Parameters of convinced constructions.- A.1.4. Binary codes from AG-codes.- A.2. Asymptotic bounds.- A.2.1. checklist of bounds.- A.2.2. Diagrams of comparison.- A.2.3. Behaviour on the ends.- A.2.4. Numerical values.- A.3. extra bounds.- A.3.1. consistent weight codes.- A.3.2. Self-dual codes.- A.4. Sphere packings.- A.4.1. Small dimensions.- A.4.2. definite families.- A.4.3. Asymptotic results.- writer index.- checklist of symbols.
Read or Download Algebraic-Geometric Codes PDF
Similar algebraic geometry books
This e-book makes a speciality of the algebraic-topological facets of chance concept, resulting in a much broader and deeper knowing of simple theorems, resembling these at the constitution of constant convolution semigroups and the corresponding approaches with autonomous increments. the tactic utilized in the environment of Banach areas and of in the community compact Abelian teams is that of the Fourier rework.
This can be a monograph approximately non-commutative algebraic geometry, and its program to physics. the most mathematical inputs are the non-commutative deformation conception, moduli thought of representations of associative algebras, a brand new non-commutative thought of section areas, and its canonical Dirac derivation.
This article presents an creation to ergodic thought appropriate for readers understanding uncomplicated degree concept. The mathematical must haves are summarized in bankruptcy zero. it really is was hoping the reader should be able to take on learn papers after examining the publication. the 1st a part of the textual content is anxious with measure-preserving modifications of chance areas; recurrence homes, blending homes, the Birkhoff ergodic theorem, isomorphism and spectral isomorphism, and entropy concept are mentioned.
- Arithmetic on Elliptic Curves with Complex Multiplication (Lecture Notes in Mathematics)
- Geometric Algebra (Interscience Tracts in Pure and Applied Mathematics)
- Several Complex Variables with Connections to Algebraic Geometry and Lie Groups (Graduate Studies in Mathematics)
- Hodge Theory and Complex Algebraic Geometry I: Volume 1 (Cambridge Studies in Advanced Mathematics)
- Singularity Theory I, 1st Edition
Extra info for Algebraic-Geometric Codes
S v < n - d is a code vector of would Ho (that is so already The parameters of the corresponding Ho). code of which is not a linear combination of v We get a matrix Ho. 8, + 1 dl. Hence k+d=n + n - n which row k < then we we have is not of kl. < n ). Consider a [6,4,2] 2 -code C generated by the matrix [ ~ i ~ ~ ~ ~l 000 110 000 011 Calculate the spectra of (Ansver: B 4 = 6 , B B' 0 = 0 3 15, B' 1 C B 1 = B'2 = and cl. 42 6 in terms of B B' 3 2 2 ) . 15, recall that an additive character of homomorphism X ~q from the additive group of 3 B.
We are going to study spectra of linear codes. Let e be a linear [n,k,d]q-code, define as the number of code vectors of weight r for course, Ar 2! y r=O r it is easy to see that '\' L. y IIvll Sometimes non-homogeneous coordinates convenient, then we consider polynomials n n r '\' L. y are more r r=O r A code e has only one vector of weight 0 and has no other vectors of weight less that d. 1 15 Since in many cases we know not the precise value of d but only some lower estimate for it, the following form is rather convenient.
State and prove the spoiling lemma for non-linear codes. (Hint: In this case (k - 1) is changed by logq(lqk-l Exercise r) ). In many cases it is difficult to calculate precise values of parameters but it is possible to bound them. The spoiling lemma makes it possible to pass from an [:s n, d]q-code C to an [n,k,d]q-code. In such a situation we say that up to a spoiling the code C is an [n,k,d]q-code. 2: k, 2: So we can always spoil parameters, but of course we cannot always make them better. Here are some restrictions.