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
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
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]).
|[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://www.encyclopediaofmath.org/index.php?title=Artin%E2%80%93Schreier_code&oldid=22028