Algebraic Geometric Codes

Spring 2022

About the course

This course is devoted to the study of algebraic geometric codes (aka Goppa codes). AG codes, like the Reed-Solomon or Reed-Muller codes, involve the evaluation of polynomials. The difference is that the set of evaluation points is a carefully chosen algebraic curve. This perhaps seemingly minor difference entails a significant leap in the mathematics that is required for constructing and analyzing such codes. Indeed, as part of the analysis, one invokes the Riemann Hypothesis over algebraic function fields (which is known to hold and goes by the name The Hasse-Weil Theorem).

In this course, we will thoroughly develop the deep and fascinating mathematics underlying AG codes, and give explicit constructions of codes that beat the Gilbert-Varshamov bound over a sufficiently large constant size alphabet.

The course is algebraic in nature. We will be working with rings and groups. Mostly, though, we will deal with field extensions and Galois theory. For the proof of the Riemann hypothesis over function fields, we will work a bit with complex functions.

Video lectures & recitations

The course lectures and recitations that were already given are available on YouTube (in Hebrew).

Syllabus (tentative)

The basics of function fields; Goppa codes

Introduction

Abstracting the notion of a point: valuation, valuation rings, and places.

Function fields

Artin's approximation theorem

Divisors

Riemann's theorem

The genus

Goppa codes

The Riemann-Roch Theorem

Adeles and Weil differentials

The Riemann-Roch Theorem

Elliptic and hyperelliptic function fields

Function field extensions

Places in extensions

The ramification index and relative degree

The fundamental equality

Normal extensions

The co-norm in extension

Constant field extensionss

Integral elements

Kummer's theorem

The Hasse-Weil Theorem (in the seminar part of the course)

The Zeta Function of a function field

The Zeta function under constant field extensions

The Hasse-Weil theorem

The Different

The complementary module

The different

Dedekind's Different Theorem

Hurwitz Genus Formula

Cyclic extensions

Kummer extensions

Artin-Schreier extensions

Construction of Algebraic-Geometric Codes

Towers of function fields

Several constructions

Slides

Part 1 - Introduction

Unit 0 - Welcome to AG codes

Unit 1 - A quick introduction to coding theory

Part 2 - Abstracting the notion of a point

Unit 2 - Abstracting the notion of a point - informal discussion

Unit 3 - Ordered groups

Unit 4 - Valuations

Unit 5 - Valuation rings

Unit 6 - Places

Part 3 - Function fields

Unit 7 - The ramification and residual indices

Unit 8 - Artin's approximation theorem

Unit 9 - Function fields

Part 4 - Riemann's Theorem

Unit 10 - Divisors and Riemann-Roch spaces

Unit 11 - Principal divisors and Riemann's Theorem

Part 5 - Goppa codes

Unit 12 - Goppa codes

Part 6 - Riemann-Roch Theorem

Unit 13 - Adeles

Unit 14 - Weil differentials

Unit 15 - The Riemann-Roch Theorem and consequences

Part 7 - Function field extensions

Unit 16 - Function field extensions and the fundamental equality

Unit 17 - Normal extensions

Unit 18 - The co-norm

Unit 19 - Constant field extensions

Unit 20 - Integrality and valuation rings

Unit 21 - The different

Unit 22 - Hurwitz genus formula

Unit 23 - Kummer's Theorem

Unit 24 - More on the different

Unit 25 - Kummer extensions

Unit 26 - Our elliptic curve example

Part 8 - AG Codes

Unit 27 - Towers of function fields and example of optimal tame towers

Unit 28 - Wild towers

Unit 29 - Summary

Unit 30 - List of topics for the seminar

Recitations

Technicalities

When and where

Lectures are on Mondays 9:10-12:00 and recitations by Shir Peleg on Tuesdays 15:10-16:00, both will be held in Shenkar 104 (the Physics building). For those who cannot attend physically, the Zoom link is https://tau-ac-il.zoom.us/j/85781709142.

Grade

The grade will be determined by a two hours presentation to be given by each student at the end of the semester covering material we do not cover in the lectures and recitations. In particular, the proof of the Hasse-Weil bound (Part 4 above), connection between Weil differentials and "ordinary" differentials, and other topics.

Problem sets

We will publish about 5 or 6 problem sets throughout the semester. These are not to be submitted and will not affect the grade. Nonetheless, we strongly encourage you to complete them. Upon request we will make time to discuss them.

Literature

We will mostly follow Stichtenoth's Algebraic Function Fields and Codes and Dan Haran's lecture notes (available online in Hebrew; relevant for most of parts 1-5). However, during the semester, I will publish lecture notes that will contain all the material in a more elaborated form.

Some recommended literature that we will not directly follow is Lorenzini's marvellous An Invitation to Arithmetic Geometry, and Morandi's lecture notes.

Online video courses

Peter Beelen has a very nice crash course on function fields with some geometric flavour. For a refresher on ring theory and Galois theory see Richard Borcherds fantastic courses on YouTube (though beware, his channel is addictive). If you want to learn more about algebraic number theory, I recommend U.K. Anandavardhanan's course on YouTube.