Algebraic Geometric codes on minimal Hirzebruch surfaces

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.