Artin-Schreier curves, exponential sums, and coding theory
Let \(q\) be a power of a prime \(p\) and let \(\mathbb{F}_ q\) be a finite field with \(q\) elements. It is well known that \(\alpha\in\mathbb{F}_ q\) has \(\mathbb{F}_ q/\mathbb{F}_ p\)-trace zero if and only if it is of the form \(\beta^ p-\beta\) for some \(\beta\in\mathbb{F}_ q\). Therefore there is, for a family \({\mathfrak F}\) of rational functions \(f(X)\in\mathbb{F}_ q(X)\), a clear relation between the \(\mathbb{F}_ q\)-points of the ``Artin-Schreier''- curves \(Y_ p-Y=f(X)\) and the weights of the \(\mathbb{F}_ p\)-code \(\{Tr(f(x))_{x\in\mathbb{F}_ q}:f\in{\mathcal F}\}\). In this survey paper these curves are studied and their number of \(\mathbb{F}_ q\)-rational points are estimated. The main tool is the analogue of the Riemann Hypothesis. As a corollary the author can bound the possible weights that occur in the corresponding codes. Special attention is paid to the ``Kloosterman'' curves \(Y^ p- Y=aX+bX^{-1}\), \(a,b\in\mathbb{F}_ q\). If \(p=2\) this is a family of curves of genus 1 and the author describes completely the weights and the frequencies with which they occur. This involves certain class numbers of positive definite integral binary quadratic forms. As a result the author obtains the weight distribution of the associated code. If \(p=3\), the curves have genus 2 and their Jacobians are isogenous to a product of two elliptic curves. In this case there are similar results. In the final section the author obtains bounds for the weights of ``geometric BCH-codes''; these are restrictions to \(\mathbb{F}_ p\) of Goppa codes asscociated to divisors of algebraic curves over \(\mathbb{F}_ q\).
scientific article; zbMATH DE number 55536
scientific article