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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 a conic. 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. We also investigate and obtain some results for codes on blow-ups of Pr, where r is at least 3. We include tables of code parameters as well as Magma functions which can be used to generate the codes.