Algebraic Geometric codes on Hirzebruch surfaces

We define a linear code Cη(δT , δX) by evaluating polynomials of bidegree (δT , δX) in the Cox ring on Fq-rational points of the Hirzebruch surface on 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. We also display some punctured codes having optimal parameters. AMS classification : 94B27, 14G50, 13P25, 14G15, 14M25