Algebraic Geometric Codes
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.
The basics of function fields; Goppa codes
Abstracting the notion of a point: valuation, valuation rings, and places.
Artin's approximation theorem
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
The co-norm in extension
Constant field extensionss
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 complementary module
Dedekind's Different Theorem
Hurwitz Genus Formula
Construction of Algebraic-Geometric Codes
Towers of function fields
Part 1 - Introduction
Unit 0 - Welcome to AG codes
Part 2 - Abstracting the notion of a point
Unit 3 - Ordered groups
Unit 4 - Valuations
Unit 5 - Valuation rings
Unit 6 - Places
Part 3 - Function fields
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
Part 7 - Function field extensions
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 28 - Wild towers
Unit 29 - Summary
Unit 30 - List of topics for the seminar
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.
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.
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.
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.
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.