Artin-Schreier code

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

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

(the Drinfel'd–Vladut bound).

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)