Algebraic Geometry

Download e-book for kindle: Algebraic-Geometric Codes by M. Tsfasman, S.G. Vladut

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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. rate reductions 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 yes codes.- A.1.3. Parameters of definite 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. convinced families.- A.4.3. Asymptotic results.- writer index.- checklist of symbols.

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Extra resources for Algebraic-Geometric Codes

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1 11 C c IF~ degenerate iff the subspace of vectors We call a linear code C s;; IF qn - 1 c IF q , where IF qn - 1 is having 0 in some fixed position. k ~ 1 and d ~ 1. 6. one-to-one correspondence between the set of equi val ence classes of non-degenerate linear [n,k,d]q-codes and the set of equivalence classes of projective [n,k,d] -systems. 7. • Prove the theorem. projective systems have an advantage of dispensing with a choice of some particular code in its equivalence class; they also look more natural than codes since there is no choice of basis involved.

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. 36 (the Singleton bound). linear [n,k,d]q-code For any Part 1 CODES 28 Proof: Let us argue in terms of [n,k,d]q-systems.

Particular that the parameters do coincide. Projective systems. object. e. the space ~(V) linear space over V) IFq . [n,k,d]q-system is a finite unordered family of ~ which (note that we write does I 'P I ~ n = I'PI, dim 'P c multiplicities). follows: not ~ ~ lie in = dim ~ + 1, 'P (projective) + 1 = dim V ). 1 11 C c IF~ degenerate iff the subspace of vectors We call a linear code C s;; IF qn - 1 c IF q , where IF qn - 1 is having 0 in some fixed position. k ~ 1 and d ~ 1. 6. one-to-one correspondence between the set of equi val ence classes of non-degenerate linear [n,k,d]q-codes and the set of equivalence classes of projective [n,k,d] -systems.

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