# Artin-Schreier code

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Given an algebraic curve $X/K$, where $K$ is a field of characteristic $p > 0$, a covering $Y \rightarrow X$ is called an Artin–Schreier curve over $X$ if the corresponding extension of function fields $K ( Y ) /K ( X )$ is generated by some function $y \in K ( Y )$ such that $y ^ {l} \pm y = f \in K ( X )$( where $l$ is a power of $p$, cf. also Extension of a field). If $K = \mathbf F _ {q}$ is a finite field, it turns out that Artin–Schreier curves often have many rational points.

To be precise, let $N ( Y )$( respectively, $g ( Y )$) denote the number of $\mathbf F _ {q}$- rational points (respectively, the genus) of a curve $Y/ \mathbf F _ {q}$. The Hasse–Weil theorem states that

$$N ( Y ) \leq q + 1 + 2g ( Y ) \sqrt q .$$

If the genus is large with respect to $q$, this bound can be improved as follows. Let $( Y _ {i} ) _ {i \in \mathbf N }$ be a sequence of curves over $\mathbf F _ {q}$ such that $g ( Y _ {i} ) \rightarrow \infty$. Then

$${\lim\limits } \sup { \frac{N ( Y _ {i} ) }{g ( Y _ {i} ) } } \leq \sqrt q - 1$$

Curves over $\mathbf F _ {q}$ 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 $\mathbf F _ {q}$, for $q = l ^ {2}$, is given by the equation $y ^ {l} + y = x ^ {l + 1 }$. It has $N = l ^ {3} + 1$ rational points and its genus is $g = { {l ( l - 1 ) } / 2 }$. Hence, for it the Hasse–Weil bound $N = q + 1 + 2g \sqrt q$ is attained, see [a4].

Again, let $q = l ^ {2}$ be a square. Define a tower of function fields $F _ {1} \subseteq F _ {2} \subseteq \dots$ over $\mathbf F _ {q}$( cf. Tower of fields) by $F _ {1} = \mathbf F _ {q} ( x _ {1} )$, $F _ {n + 1 } = F _ {n} ( z _ {n} )$, where

$$z _ {n + 1 } ^ {l} + z _ {n + 1 } = x _ {n} ^ {l + 1 } \textrm{ and } x _ {n} = { \frac{z _ {n} }{x _ {n -1 } } } ( \textrm{ for } n \geq 2 ) .$$

For the corresponding algebraic curves $Y _ {1} ,Y _ {2} , \dots$, the coverings $Y _ {n + 1 } \rightarrow Y _ {n}$ are Artin–Schreier curves. This sequence $( Y _ {n} ) _ {n \in \mathbf N }$ attains the Drinfel'd–Vladut bound, i.e., ${\lim\limits } _ {i \rightarrow \infty } { {N ( Y _ {i} ) } / {g ( Y _ {i} ) } } = l - 1$( see [a1]).

The geometric Goppa codes constructed using these curves $Y _ {n}$ beat the Gilbert–Varshamov bound (cf. also Error-correcting code; [a3]) for all $q \geq 49$. 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)
How to Cite This Entry:
Artin-Schreier code. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Artin-Schreier_code&oldid=54420
This article was adapted from an original article by H. Stichtenoth (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article