Algebraic Geometric codes on Hirzebruch surfaces
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We define a linear code C_η(δ_T,δ_X) by evaluating polynomials of bidegree (δ_T,δ_X) in the Cox ring on F_q-rational points of the Hirzebruch surface of parameter η on the finite field F_q. We give explicit parameters of the code, notably using Gröbner bases. The minimum distance provides an upper bound of the number of F_q-rational points of a non-filling curve on a Hirzebruch surface. We also display some punctured codes having optimal parameters.
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