Sums of residues on algebraic surfaces and application to coding theory

In this paper, we study residues of differential 2-forms on a smooth algebraic surface over an arbitrary field and give several statements about sums of residues. Afterwards, using these results, we construct algebraic–geometric codes which are an extension to surfaces of the well-known differential codes on curves. We also study some properties of these codes and extend to them some known properties for codes on curves.