Artin-Schreier code
Given an algebraic curve , where
is a field of characteristic
, a covering
is called an Artin–Schreier curve over
if the corresponding extension of function fields
is generated by some function
such that
(where
is a power of
, cf. also Extension of a field). If
is a finite field, it turns out that Artin–Schreier curves often have many rational points.
To be precise, let (respectively,
) denote the number of
-rational points (respectively, the genus) of a curve
. The Hasse–Weil theorem states that
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If the genus is large with respect to , this bound can be improved as follows. Let
be a sequence of curves over
such that
. Then
![]() |
(the Drinfel'd–Vladut bound).
Curves over can be used to construct error-correcting linear codes, so-called geometric Goppa codes or algebraic-geometric codes (cf. Error-correcting code; Goppa code; Algebraic-geometric code; [a4], [a5]). If the curves have sufficiently may rational points, these codes have very good error-correcting properties. Hence, one is interested in explicit constructions of curves with many rational points.
Examples of Artin–Schreier curves.
The Hermitian curve over , for
, is given by the equation
. It has
rational points and its genus is
. Hence, for it the Hasse–Weil bound
is attained, see [a4].
Again, let be a square. Define a tower of function fields
over
(cf. Tower of fields) by
,
, where
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For the corresponding algebraic curves , the coverings
are Artin–Schreier curves. This sequence
attains the Drinfel'd–Vladut bound, i.e.,
(see [a1]).
The geometric Goppa codes constructed using these curves beat the Gilbert–Varshamov bound (cf. also Error-correcting code; [a3]) for all
. This construction is simpler and more explicit than the construction based on modular curves (the Tsfasman–Vladut–Zink theorem, [a5]).
References
[a1] | A. Garcia, H. Stichtenoth, "A tower of Artin–Schreier extensions of function fields attaining the Drinfeld–Vladut bound" Invent. Math. , 121 (1995) pp. 211–222 |
[a2] | G. van der Geer, M. van der Vlugt, "Curves over finite fields of characteristic two with many rational points" C.R. Acad. Sci. Paris , 317 (1993) pp. 693–697 |
[a3] | J.H. van Lint, "Introduction to coding theory" , Springer (1992) |
[a4] | H. Stichtenoth, "Algebraic function fields and codes" , Springer (1993) ISBN 3-540-58469-6 Zbl 0816.14011 |
[a5] | M.A. Tsfasman, S.G. Vladut, "Algebraic geometric codes" , Kluwer Acad. Publ. (1991) |
Artin-Schreier code. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Artin-Schreier_code&oldid=45227