Differential approach for the study of duals of algebraic-geometric codes on surfaces

The purpose of the present article is the study of duals of functional codes on algebraic surfaces. We give a direct geometrical description of them, using differentials. Even if this description is less trivial, it can be regarded as a natural extension to surfaces of the result asserting that the dual of a functional code CL(D,G) on a curve is the differential code C(D,G). We study the parameters of such codes and state a lower bound for their minimum distance. Using this bound, one can study some examples of codes on surfaces, and in particular surfaces with Picard number 1 like elliptic quadrics or some particular cubic surfaces. The parameters of some of the studied codes reach those of the best known codes up to now.