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| {{multiple issues|
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| {{Confusing|date=February 2009}}
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| {{No footnotes|date=April 2009}}
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| In [[mathematics]], an '''algebraic geometric code''' ('''AG-code'''), otherwise known as a '''Goppa code''', is a general type of [[linear code]] constructed by using an [[algebraic curve]] <math>X</math> over a [[finite field]] <math>\mathbb{F}_q</math>. Such codes were introduced by [[Valerii Denisovich Goppa]]. In particular cases, they can have interesting [[extremal property|extremal properties]]. They should not be confused with [[Binary Goppa code]]s that are used, for instance, in the [[McEliece cryptosystem]].
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| ==Construction==
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| Traditionally, an AG-code is constructed from a [[non-singular]] [[projective curve]] '''X''' over a finite field <math>\mathbb{F}_q</math> by using a number of fixed distinct <math>\mathbb{F}_q</math> -[[rational points]]
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| :<math>\mathcal{P}</math>:= {''P''<sub>1</sub>, ''P''<sub>2</sub>, ..., ''P''<sub>n</sub>} ⊂ '''X''' ( <math>\mathbb{F}_q</math>) on '''X'''.
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| Let '''G''' be a [[divisor (algebraic geometry)|divisor]] on '''X''', with a [[Support (mathematics)|support]] that consists of only rational points and that is disjoint from the <math>P_i</math>'s.
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| Thus <math>\mathcal{P}</math> ∩ supp('''G''') = Ø
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| By the [[Riemann-Roch]] theorem, there is a unique finite-dimensional vector space, <math>L(G)</math>, with respect to the divisor '''G'''. The vector space is a subspace of the [[function field of an algebraic variety|function field]] of '''X'''.
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| There are two main types of AG-codes that can be constructed using the above information.
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| == Function code ==
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| The function code (or dual code) with respect to a curve '''X''', a divisor '''G''' and the set <math>\mathcal{P}</math> is constructed as follows.<br />
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| Let <math>D = P_1 + P_2 + \cdots + P_n</math>, be a divisor, with the '''<math>P_i</math>''' defined as above. We usually denote a Goppa code by '''C'''('''D''','''G''').
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| We now know all we need to define the Goppa code:<br />
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| :''C''(''D'',''G'') = {(''f''(''P''<sub>1</sub>), ..., ''f''(''P''<sub>n</sub>))|''f'' <math>\in</math> ''L''(''G'')}⊂<math>\mathbb{F}_q^n
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| </math>
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| For a fixed basis
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| :''f''<sub>1</sub>, ''f''<sub>2</sub>, ..., ''f''<sub>k</sub>
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| for ''L''(''G'') over <math>\mathbb{F}_q</math>, the corresponding Goppa code in <math>\mathbb{F}_q^n</math> is spanned over <math>\mathbb{F}_q</math> by the vectors
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| :(''f''<sub>''i''</sub>(''P''<sub>1</sub>), ''f''<sub>''i''</sub>(''P''<sub>2</sub>), ..., ''f''<sub>''i''</sub>(''P''<sub>n</sub>)).
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| Therefore
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| : <math>
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| \begin{bmatrix}
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| f_1(P_1) & ... & f_1(P_n) \\
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| ... & ... & ... \\
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| f_k(P_1) & ... & f_k(P_n) \end{bmatrix}
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| </math>
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| is a generator matrix for '''C'''('''D''','''G''') | |
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| Equivalently, it is defined as the image of
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| :<math>\alpha : L(G) \longrightarrow \mathbb{F}^n</math>, | |
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| where ''f'' is defined by <math>f \longmapsto (f(P_1), \dots ,f(P_n))</math>.
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| The following shows how the parameters of the code relate to classical parameters of [[linear systems of divisors]] ''D'' on ''C'' (cf. [[Riemann–Roch theorem]] for more). The notation ''l''(''D'') means the dimension of ''L''(''D'').
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| '''Proposition A''' The dimension of the Goppa code ''C''(''D'',''G'') is
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| :<math>k = l(G) - l(G-D)</math>,
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| '''Proposition B''' The minimal distance between two code words is
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| :<math>d \geq n - \deg(G)</math>.
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| '''Proof A'''
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| Since
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| :<math>C(D,G) \cong L(G)/\ker(\alpha), </math>
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| we must show that
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| :<math>\ker(\alpha)=L(G-D) </math>.
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| Suppose <math>f \in \ker(\alpha)</math>. Then <math>f(P_i)=0,
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| i=1, \dots ,n</math>, so <math>\mathrm{div}(f) > D </math>. Thus, <math>f \in
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| L(G-D)</math>.<br /> Conversely, suppose <math>f \in L(G-D)</math>.<br /> Then
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| :<math>\mathrm{div}(f)> D</math>
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| since
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| :<math>P_i < G, i=1, \dots ,n</math>.
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| (''G'' doesn't “fix”
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| the problems with the <math>-D</math>, so ''f'' must do that instead.) It follows
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| that
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| :<math>f(P_i)=0, i=1, \dots ,n</math>.
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| '''Proof B'''<br />
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| To show that <math>d \geq n - \deg(G)</math>, suppose the [[Hamming weight]] of
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| <math>\alpha(f)</math> is ''d''. That means that <math>f(P_i)=0</math> for <math>n-d</math> <math>P_i</math>s, say
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| <math>P_{i_1}, \dots ,P_{i_{n-d}}</math>. Then <math>f \in L(G-P_{i_1} - \dots
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| - P_{i_{n-d}})</math>, and
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| :<math>\mathrm{div}(f)+G-P_{i_1} - \dots - P_{i_{n-d}}> 0</math>.
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| Taking degrees on both sides and noting that
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| :<math>\deg(\mathrm{div}(f))=0</math>, | |
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| we get
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| :<math>\deg(G)-(n-d) \geq 0</math>,
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| so
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| :<math>d \geq n - \deg(G)</math>. Q.E.D.
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| == Residue code ==
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| The residue code can be defined as the dual of the function code, or as the residue of some functions at the <math>P_i</math>'s.
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| == References == | |
| * Key One Chung, ''Goppa Codes'', December 2004, Department of Mathematics, Iowa State University.
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| ==External links==
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| * [http://commons.wikimedia.org/wiki/File:Algebraic_Geometric_Coding_Theory.pdf An undergraduate thesis on Algebraic Geometric Coding Theory]
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| * [http://orion.math.iastate.edu/linglong/Math690F04/Goppa%20codes.pdf Goppa Codes by Key One Chung]
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| [[Category:Coding theory]]
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| [[Category:Algebraic curves]]
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| [[Category:Finite fields]]
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| [[Category:Articles containing proofs]]
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