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. savings and schemes.- three. AG-Codes.- 3.1. structures 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 definite codes.- A.1.3. Parameters of sure constructions.- A.1.4. Binary codes from AG-codes.- A.2. Asymptotic bounds.- A.2.1. record 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. sure families.- A.4.3. Asymptotic results.- writer index.- record of symbols.

**Read or Download Algebraic-Geometric Codes PDF**

**Best algebraic geometry books**

**Computer Graphics and Geometric Modelling: Mathematics**

Probably the main accomplished review of special effects as visible within the context of geometric modelling, this quantity paintings covers implementation and idea in a radical and systematic type. special effects and Geometric Modelling: arithmetic, comprises the mathematical historical past wanted for the geometric modeling themes in special effects lined within the first quantity.

This ebook constitutes the court cases of the 2000 Howard convention on "Infinite Dimensional Lie teams in Geometry and illustration Theory". It provides a few very important contemporary advancements during this region. It opens with a topological characterization of standard teams, treats between different issues the integrability challenge of varied endless dimensional Lie algebras, provides sizeable contributions to big matters in sleek geometry, and concludes with fascinating functions to illustration thought.

**Foundations of free noncommutative function theory**

During this booklet the authors strengthen a idea of unfastened noncommutative services, in either algebraic and analytic settings. Such capabilities are outlined as mappings from sq. matrices of all sizes over a module (in specific, a vector area) to sq. matrices over one other module, which appreciate the scale, direct sums, and similarities of matrices.

- Theory of Stein spaces
- Hilbert schemes
- Arrangements of Hyperplanes
- The Zeta Functions of Picard Modular Surfaces
- Outer Billiards on Kites
- Geometry of the plane Cremona maps

**Extra info for Algebraic-Geometric Codes**

**Example text**

36 it cannot be better), a + 1 L(a) c L(a = k + d (Hint: = IF * i = '; - Prove that a - 2 and Use the fact that < q - 1). q Then arbitrary let "P S; us IF q find the dual code for the case of an . Set n P . e"P (x - Pi) -1 , ~ Cons ider the vector space n (a) spanned by the functions CODES 40 ge (x) F (x) Part 1 e ~ n - a - 2. Consider a function = f (x) . go (x) , f (x) being a polynomial. Recall the definition of the residue of F at Pi for 0 ~ Res p F = f ( P . ) . n (P. 3. e'P ~ Res'P Prove that for (the residue formula).

E V . By H. e'R Hi) . Then ~ B. 26. Self-dual codes. iff C = CL exists a . ~ '" 0 C i Part 1 CODES 24 1, ... L . Here is called formally self-dual self-dual code if W C = W quasi-self-dual, is quasi-self-dual code is formally self-dual. L . Of and any 2 then any quasi-self-dual code is self-dual. 28. Let q = 2 or q = 3 . Show that the weights of all code vectors of a self-dual divisible by q-ary code are q. Note that for formally self-dual codes this statement (1,0) E 1F2 is wrong: a [2,1,l]2-code spanned by the vector 2 is formally self-dual.

3. Spectra and duality An important invariant of a code is its weight enumerator or spectrum. 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.