Algebraic Geometric codes on minimal Hirzebruch surfaces

Jade Nardi (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

We define a linear code by evaluating polynomials of bidegree in the Cox ring on Fq-rational points of a minimal Hirzebruch surface over the finite field Fq. We give explicit parameters of the code, notably using Gröbner bases. The minimum distance provides an upper bound of the number of Fq-rational points of a non-filling curve on a Hirzebruch surface.

Mots clés: Hirzebruch surface; Algebraic Geometric code; Gröbner basis; Rational scroll; Error-correcting codes

Domaines: Géométrie algébrique [math.AG]; Théorie de l'information et codage [math.IT]

Jade Nardi. Algebraic Geometric codes on minimal Hirzebruch surfaces. 2018. ⟨hal-01842416v1⟩