Download Algebraic-Geometric Codes by M. Tsfasman, S.G. Vladut PDF

Algebraic Geometry

By M. Tsfasman, S.G. Vladut

ISBN-10: 1402003358

ISBN-13: 9781402003356

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. discounts 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 convinced codes.- A.1.3. Parameters of yes 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.- record of symbols.

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Example text

X } generator = i O,l, ... ,a; The dual code. line with is also a matrix in c of = IFq "P Reed-Solomon code with a Let 'P [n, n - a - 1, a + 2]q . i is Let us start with the simplest case. or for L(a) 1,2, ... ,n . 1. parameters L xi = 0 on 1 , n s q + 1 Choose the basis xelF which is In section 3. 1. 1. 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 .

1. 2) . 1 Choose CODES AND THEIR PARAMETERS a hyperplane H o 27 c IPk - 1 such one more point from max IHo n:P1 • Add to H (it does not matter whether it already belongs to not) . Let vectors. Exclude from to Qo (it is possible by 1, and d and (n b) that H o or be a linearly dependent set of d Q any vector which does not belong since d < n ). Then n is lessened k) do not change. c) Chose H as in a) and exclude from o d 2: 2 . Since which does not belong to Ho system again (the remaining points cannot all hyperplane); k and (n - d) do not change.

16 to codes of genus zero, we get a complete answer. 4. If zero then xn + k-1 L i=O C is an [n,k,d]q-Code of genus . 5. • Establish this formula for Solomon codes by a direct computation. Check that for Reedi * 0 Decoding. Consider an [n, n - a - 1, a + 2]q-code C dual to a Reed-Solomon code of degree a. Recall that fd ; decoding up to t = 11 means an algorithm it possible, starting with some v E F~ which is distance t from some code vector u e C, to u. Let is called v - u = e, lIeU:s t , e vector .

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