# Artin-Schreier code

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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 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]).