View source for Artin-Schreier code

Given an [[Algebraic curve|algebraic curve]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a1107301.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a1107302.png" /> is a [[Field|field]] of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a1107303.png" />, a [[Covering|covering]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a1107304.png" /> is called an Artin–Schreier curve over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a1107305.png" /> if the corresponding extension of function fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a1107306.png" /> is generated by some function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a1107307.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a1107308.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a1107309.png" /> is a power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073010.png" />, cf. also [[Extension of a field|Extension of a field]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073011.png" /> is a [[Finite field|finite field]], it turns out that Artin–Schreier curves often have many rational points.

To be precise, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073012.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073013.png" />) denote the number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073014.png" />-rational points (respectively, the genus) of a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073015.png" />. The Hasse–Weil theorem states that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073016.png" /></td> </tr></table>

If the genus is large with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073017.png" />, this bound can be improved as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073018.png" /> be a sequence of curves over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073019.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073020.png" />. Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073021.png" /></td> </tr></table>

(the Drinfel'd–Vladut bound).

Curves over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073022.png" /> can be used to construct error-correcting linear codes, so-called geometric Goppa codes or algebraic-geometric codes (cf. [[Error-correcting code|Error-correcting code]]; [[Goppa code|Goppa code]]; [[Algebraic-geometric code|Algebraic-geometric code]]; [[#References|[a4]]], [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073023.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073024.png" />, is given by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073025.png" />. It has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073026.png" /> rational points and its genus is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073027.png" />. Hence, for it the Hasse–Weil bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073028.png" /> is attained, see [[#References|[a4]]].

Again, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073029.png" /> be a square. Define a tower of function fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073030.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073031.png" /> (cf. [[Tower of fields|Tower of fields]]) by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073033.png" />, where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073034.png" /></td> </tr></table>

For the corresponding algebraic curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073035.png" />, the coverings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073036.png" /> are Artin–Schreier curves. This sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073037.png" /> attains the Drinfel'd–Vladut bound, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073038.png" /> (see [[#References|[a1]]]).

The geometric Goppa codes constructed using these curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073039.png" /> beat the Gilbert–Varshamov bound (cf. also [[Error-correcting code|Error-correcting code]]; [[#References|[a3]]]) for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110730/a11073040.png" />. This construction is simpler and more explicit than the construction based on modular curves (the Tsfasman–Vladut–Zink theorem, [[#References|[a5]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.H. van Lint,   "Introduction to coding theory" , Springer  (1992)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Stichtenoth,   "Algebraic function fields and codes" , Springer  (1993)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M.A. Tsfasman,   S.G. Vladut,   "Algebraic geometric codes" , Kluwer Acad. Publ.  (1991)</TD></TR></table>