The dual minimum distance of arbitrary dimensional algebraic--geometric codes

TL;DR

This paper investigates the minimum distance of dual algebraic-geometric codes on varieties over finite fields, introducing a new approach based on minimal point configurations, with improvements for curve cases.

Contribution

It presents a novel method to determine the dual code's minimum distance by analyzing point configurations, extending previous results for algebraic-geometric codes.

Findings

Provides new bounds for the minimum distance of dual codes on varieties.

Improves known results for algebraic-geometric codes on curves.

Introduces a geometric approach based on point configurations.

Abstract

In this article, the minimum distance of the dual $C^{\bot}$ of a functional code $C$ on an arbitrary dimensional variety $X$ over a finite field $\F_q$ is studied. The approach consists in finding minimal configurations of points on $X$ which are not in "general position". If $X$ is a curve, the result improves in some situations the well-known Goppa designed distance.