Factor theorem: Difference between revisions

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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]].
 
==Construction==
 
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]]
 
:<math>\mathcal{P}</math>:= {''P''<sub>1</sub>, ''P''<sub>2</sub>, ..., ''P''<sub>n</sub>} ⊂  '''X''' ( <math>\mathbb{F}_q</math>) on '''X'''.
 
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.
Thus <math>\mathcal{P}</math> ∩ supp('''G''') = Ø
 
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'''.
 
There are two main types of AG-codes that can be constructed using the above information.
 
== Function code ==
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 />
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''').
We now know all we need to define the Goppa code:<br />
:''C''(''D'',''G'') = {(''f''(''P''<sub>1</sub>), ..., ''f''(''P''<sub>n</sub>))|''f'' <math>\in</math> ''L''(''G'')}&sub;<math>\mathbb{F}_q^n
</math>
For a fixed basis
:''f''<sub>1</sub>, ''f''<sub>2</sub>, ..., ''f''<sub>k</sub>
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
 
:(''f''<sub>''i''</sub>(''P''<sub>1</sub>), ''f''<sub>''i''</sub>(''P''<sub>2</sub>), ..., ''f''<sub>''i''</sub>(''P''<sub>n</sub>)).
Therefore
: <math>
\begin{bmatrix}
f_1(P_1) & ... & f_1(P_n) \\
... & ... & ... \\
f_k(P_1) & ... & f_k(P_n) \end{bmatrix}
</math>
 
is a generator matrix for '''C'''('''D''','''G''')
 
Equivalently, it is defined as the image of
 
:<math>\alpha : L(G) \longrightarrow \mathbb{F}^n</math>,
 
where ''f'' is defined by <math>f \longmapsto (f(P_1), \dots ,f(P_n))</math>.
 
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'').
 
'''Proposition A''' The dimension of the Goppa code ''C''(''D'',''G'') is
 
:<math>k = l(G) - l(G-D)</math>,
 
'''Proposition B''' The minimal distance between two code words is
 
:<math>d \geq n - \deg(G)</math>.
 
'''Proof A'''
 
Since
 
:<math>C(D,G) \cong L(G)/\ker(\alpha), </math>
 
we must show that
 
:<math>\ker(\alpha)=L(G-D) </math>.
 
Suppose <math>f \in \ker(\alpha)</math>. Then <math>f(P_i)=0,
i=1, \dots ,n</math>, so <math>\mathrm{div}(f) > D </math>. Thus, <math>f \in
L(G-D)</math>.<br /> Conversely, suppose <math>f \in L(G-D)</math>.<br /> Then
:<math>\mathrm{div}(f)> D</math>
 
since
 
:<math>P_i < G, i=1, \dots ,n</math>.
 
(''G'' doesn't “fix”
the problems with the <math>-D</math>, so ''f'' must do that instead.) It follows
that
 
:<math>f(P_i)=0, i=1, \dots ,n</math>.
'''Proof B'''<br />
To show that <math>d \geq n - \deg(G)</math>, suppose the [[Hamming weight]] of
<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
<math>P_{i_1}, \dots ,P_{i_{n-d}}</math>. Then <math>f \in L(G-P_{i_1} - \dots
- P_{i_{n-d}})</math>, and
 
:<math>\mathrm{div}(f)+G-P_{i_1} - \dots - P_{i_{n-d}}> 0</math>.
 
Taking degrees on both sides and noting that
 
:<math>\deg(\mathrm{div}(f))=0</math>,
 
we get
 
:<math>\deg(G)-(n-d) \geq 0</math>,
 
so
 
:<math>d \geq n - \deg(G)</math>. Q.E.D.
 
== Residue code ==
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.
 
== References ==
* Key One Chung, ''Goppa Codes'', December 2004, Department of Mathematics, Iowa State University.
 
==External links==
* [http://commons.wikimedia.org/wiki/File:Algebraic_Geometric_Coding_Theory.pdf An undergraduate thesis on Algebraic Geometric Coding Theory]
* [http://orion.math.iastate.edu/linglong/Math690F04/Goppa%20codes.pdf Goppa Codes by Key One Chung]
 
[[Category:Coding theory]]
[[Category:Algebraic curves]]
[[Category:Finite fields]]
[[Category:Articles containing proofs]]

Latest revision as of 12:50, 25 November 2014

- The herb gingko will boost circulation levels in 6 to 8 weeks. There are a number of excellent personalised books for children available online at various specialised publishing houses and retailers. It may have ruptured in the male organ itself, or in the perineum; this obstruction causes the blood to stay in the male organ, rather than returning to circulation. If you are going to step of the residences for purchasing or desire to hangout with your good friend, then these hats would be the excellent choice for you. This erection obtained can be maintained for the amount of time desired by the user.

The best place to get Propecia class action lawsuit lawyers is over the Internet. According to Massachusetts Male Aging Study, chances of full dysfunction increases from 5% in males nearing their 40s to 15% in males reaching in their 70s. Those concerned that this source of their impotence is medical can turn to experts just like the ones at St. Let’s start by getting a little familiar with the mechanism of how penis achieves an erection. The important thing when it comes to increasing testosterone levels is to take action.

Did you know that the way you breathe also has a major impact upon your impotency problems. For a male member to be total and erect, it will get lots of blood stream which makes growing the erection tissue inside it. If we talk about an effective and reliable treatment for erectile dysfunction, then the use of herbs would be the most appropriate. She recently wrote the best post on Tech - Crunch for months, a fascinating survey of Silicon Valley "mafias", and her latest book, Brilliant, Crazy, Cocky, is the result of forty weeks travelling around Asia, Africa and South America, conducting an exhaustive survey of entrepreneurialism in emerging economies. British researchers said pelvic floor exercises or Kegels, named after the South Californian gynecologist Dr.

The Men's Health Forum and the Impotence Association said one in five men suffering from impotence believe it ruined their relationship. There are other supplements such as calcium and potassium that may also give you a benefit. Finally, another great tip would be to exercise at least 30 minutes a day. This drug is used to treat hypertension (high blood pressure). The horns are made out of narrow, tightly matted together tubes of keratin (the same substance human hair and nails are composed of).

It can ruin relationships between people, making it extremely difficult for them to manage sexual relationships. The insertion process is painless and does not have any side effects on your body. However, without proper knowledge, they can also be used improperly. An ability to maintain a penile erection requires a healthy & sound nervous system, healthy arteries in and near corpora cavernosa, smooth muscles & fibrous tissues in corpora cavernosa and abundant levels of nitric oxide. The blood is kept within the penis to maintain erection.

In case you have any questions about where as well as how you can use natural ed pills, it is possible to contact us in the web-site.

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