On codes from norm-trace curves
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.
scientific article; zbMATH DE number 1965049