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

TL;DR

This paper extends the understanding of duals of functional algebraic-geometric codes from curves to surfaces, providing geometric descriptions, parameter analysis, and bounds on minimum distance, with examples on specific surfaces.

Contribution

It introduces a geometric description of dual codes on surfaces using differentials, generalizing known results from curves and analyzing their parameters.

Findings

Lower bounds for minimum distance of dual codes on surfaces

Examples of codes on elliptic quadrics and cubic surfaces

Some codes match the parameters of the best known codes

Abstract

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 geometrical 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 on a curve is a differential code. 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.