We construct algebraic geometric codes from del Pezzo surfaces and focus on the ones having Picard rank one and the codes associated to the anticanonical class. We give explicit constructions of del Pezzo surfaces of degree 4, 5 and 6, compute the parameters of the associated anticanonical codes and study their isomorphisms arising from the automorphisms of the surface. We obtain codes with excellent parameters and some of them turn out to beat the best known codes listed on the database codetable.
Let us finally mention some works on codes on other blowups of the plane @cite_8 @cite_2 . These blowups are no longer del Pezzo surfaces: since the blown up points lie on one or two lines, they are not in general position. Moreover the evaluation set is not the set of rational points of the surface, but the torus @math .
@cite_2 @cite_8
2030750486 2099044707
Here we improve the lower bound on the minimum distance for the evaluation codes obtained by Davis from certain blowing-ups of the planes (anticanonical surfaces). Algebraic geometric codes (or AG codes) provide a way to correct errors that occur during the transmission of digital information. AG codes on curves have been studied extensively, but much less work has been done for AG codes on higher dimensional varieties. In particular, we seek good bounds for the minimum distance. We study AG codes on anticanonical surfaces coming from blow-ups of P2 at points on a line and points on the union of two lines. We can compute the dimension of such codes exactly due to known results. For certain families of these codes, we prove an exact result on the minimum distance. For other families, we obtain lower bounds on the minimum distance.