Construction of good codes from weak Del Pezzo surfaces

We construct algebraic geometric codes from weak del Pezzo surfaces. The codes are associated to the anti-canonical class of the anti-canonical model and to the set of rational points of these models. Since we consider weak Del Pezzo surfaces, the anti canonical model is not smooth any more. This complicates the computation of the parameters of the codes; in particular we need to distinguish the Cartier divisors from the Weil ones.