Codes and modular curves

TL;DR

This paper provides an in-depth presentation of the proof of the Tsfasman-Vlub7t Zink Theorem, demonstrating the existence of code sequences over finite fields with superior asymptotic parameters.

Contribution

It offers a detailed exposition of a key theorem in algebraic coding theory, connecting modular curves to code construction.

Findings

Proof of Tsfasman-Vlub7t Zink Theorem

Existence of asymptotically good codes over finite fields

Connection between modular curves and coding theory

Abstract

These lecture notes have been written for a course at the Algebraic Coding Theory (ACT) summer school 2022 that took place in the university of Zurich. The objective of the course propose an in-depth presentation of the proof of one of the most striking results of coding theory: Tsfasman Vl\u{a}du\c{t} Zink Theorem, which asserts that for some prime power $q$, there exist sequences of codes over $\mathbb{F}_q$ whose asymptotic parameters beat random codes.