How arithmetic and geometry make error correcting codes better

TL;DR

This paper explores how the integration of arithmetic and geometric principles enhances the design and performance of algebraic geometry codes, emphasizing their multiplicative structure and recent applications.

Contribution

It provides a historical overview and discusses recent advances in algebraic geometry codes, focusing on their multiplicative structure and open research questions.

Findings

Enhanced understanding of the multiplicative structure of algebraic geometry codes

Discussion of recent applications and open problems in the field

Insights into how arithmetic and geometry improve code performance

Abstract

This note completes a talk given at the conference Curves over Finite Fields: past, present and future celebrating the publication the book {\em Rational Points on Curves over Finite Fields by J.-P. Serre and organised at Centro de ciencias de Benasque in june 2021. It discusses a part of the history of algebraic geometry codes together with some of their recent applications. A particular focus is done on the "multiplicative" structure of these codes, i.e. their behaviour with respect to the component wise product. Some open questions are raised and discussed.