Sums of residues on algebraic surfaces and application to coding theory

TL;DR

This paper investigates residues of differential forms on algebraic surfaces, establishing sum properties, and applies these findings to construct and analyze algebraic-geometric codes extending curve-based codes to surfaces.

Contribution

It introduces a novel method for constructing algebraic-geometric codes on surfaces using residue sums, extending classical curve-based coding techniques.

Findings

Residue sum properties on algebraic surfaces are established.

New algebraic-geometric codes on surfaces are constructed.

Properties of surface codes are extended from curve codes.

Abstract

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.