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: Algebraic Geometric code; Hirzebruch surface; Gröbner basis; Rational scroll; Error-correcting codes; AMS classication : 94B27; 14G50; 13P25; 14G15; 14M25 Hirzebruch surface

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

Jade Nardi. Algebraic Geometric codes on minimal Hirzebruch surfaces. Journal of Algebra, 2019, 535, pp.556-597. ⟨10.1016/j.jalgebra.2019.06.022⟩. ⟨hal-01842416v2⟩