Algebraic Geometric codes on minimal Hirzebruch surfaces

TL;DR

This paper introduces a new class of algebraic geometric codes constructed from Hirzebruch surfaces over finite fields, providing explicit parameters and bounds on minimum distance using Gröbner bases.

Contribution

It defines a novel family of codes on Hirzebruch surfaces, detailing their parameters and employing Gröbner bases for analysis, which advances algebraic geometric coding theory.

Findings

Explicit code parameters are derived.

Minimum distance bounds are established.

Gröbner bases are used for parameter computation.

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 Gr\"obner 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.