Algebraic geometry codes and some applications

TL;DR

This paper surveys the development and applications of algebraic geometry codes, highlighting key theoretical results and their use in cryptography, complexity theory, and distributed storage.

Contribution

It provides a comprehensive overview of algebraic geometry codes, including recent advances in parameters, decoding, and diverse modern applications.

Findings

Summarizes major theoretical results on algebraic geometry codes.

Highlights improved estimates on minimum distance and decoding algorithms.

Discusses modern applications in cryptography and distributed systems.

Abstract

This article surveys the development of the theory of algebraic geometry codes since their discovery in the late 70's. We summarize the major results on various problems such as: asymptotic parameters, improved estimates on the minimum distance, and decoding algorithms. In addition, we present various modern applications of these codes such as public-key cryptography, algebraic complexity theory, multiparty computation or distributed storage.