Algebraic Geometric codes on minimal Hirzebruch surfaces

Abstract

We define a linear code $C_{\eta }({\delta }_T,{\delta }_X)$ by evaluating polynomials of bidegree $({\delta }_T,{\delta }_X)$ in the Cox ring on ${\mathbb{F}}_q$-rational points of the Hirzebruch surface of parameter $\eta$ on the finite field ${\mathbb{F}}_q$. We give explicit parameters of the code, notably using Gröbner bases. The minimum distance provides an upper bound of the number of ${\mathbb{F}}_q$-rational points of a non-filling curve on a Hirzebruch surface.

arXiv commentRevised version [11/2018]