On codes from norm-trace curves
type: Item
instance of: scholarly article
title: On codes from norm-trace curves
zbMATH Open document ID: 1032.94015
DOI: 10.1016/S1071-5797(03)00010-8
author: Olav Geil
published in: Finite Fields and their Applications
publication date: 2003-08-17T00:00:00Z
review text: In [On hyperbolic codes, Proc. AAECC-14, Lect. Notes Comput. Sci. 2227, 159-171 (2001)], \textit{T. Høholdt} and \textit{O. Geil} studied a class of evaluation codes arising from the ring \(\mathbb{F}_q[X_1,\dots, X_m]/\langle X_1^q- X_1,\dots, X_m^q- X_m\rangle\) by using Gröbner basis theory. In the present article, the author uses the techniques of that paper to study evaluation codes coming from a more complicated algebraic structure related to Miura-Kamiya curves. This leads to the ring \(\mathbb{F}_q[X,Y]/\langle X^a- \mu Y^b- F(X,Y), X^q- X, Y^q-Y\rangle\), where \(a\) and \(b\) are coprime, \(\mu\) is nonzero and any monomial \(X^i Y^j\) in the support of \(F\) satisfies \(ib+ ja< ab\). In particular, he concentrates on the family of norm-trace curves, that is, \(\mathbb{F}_{q^r}[X,Y]/ \langle X^{(q^r-1)/(q-1)}- Y^{q^{r-1}}- Y^{q^{r- 2}}-\cdots- Y, X^{q^r}- X, Y^{q^r}- Y\rangle\). The minimum distance of these codes is obtained. Furthermore, it is shown that the new codes are at least as good as the related one-point geometric Goppa codes.
reviewed by: Carlos Munuera
Mathematics Subject Classification ID: 94B27
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 14G50
zbMATH DE Number: 1965049
zbMATH Keywords: algebraic geometry codes
zbMATH Keywords: evaluation codes
zbMATH Keywords: footprint
zbMATH Keywords: Goppa bound
zbMATH Keywords: Gröbner basis
zbMATH Keywords: hyperbolic codes
zbMATH Keywords: minimum distance
zbMATH Keywords: norm
zbMATH Keywords: order bound
zbMATH Keywords: trace
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1965049