Algebraic Geometric Codes
Fall 2024/5
About the course
This course focuses on the study of algebraic geometric codes, also known as Goppa codes. Like Reed-Solomon or Reed-Muller codes, AG codes involve polynomial (or rather rational function) evaluations. However, AG codes differ in that the evaluation points are selected from a specifically chosen algebraic curve. This seemingly minor distinction leads to a substantial increase in the complexity of the mathematics required to construct, understand, and analyze these codes. Indeed, the analysis includes deep results such as the Riemann-Roch Theorem and the Riemann Hypothesis over algebraic function fields, which is, in fact, a famous theorem known as the Hasse-Weil Theorem.
Throughout this course, we will develop the intricate and beautiful mathematics that underpin AG codes and provide constructions of codes that surpass the Gilbert-Varshamov bound (for sufficiently large constant size alphabets). The course content is fundamentally algebraic. We will work with rings and groups, focusing primarily on field extensions and Galois theory.
My goal is to make this course accessible to students of theoretical computer science, even though it involves a significant amount of mathematics. Specifically, I will not assume any prior knowledge in topology, complex analysis, or commutative algebra. However, I will occasionally point out connections between the course material and these areas to offer some insight and help those familiar with these topics connect the dots.
The course lectures and recitations will be conducted in English and will be regularly uploaded to a dedicated YouTube channel.
Syllabus (tentative)
Introduction
Introduction to coding theory
The geometric picture: affine & projective curves
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
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
Technicalities
When and where
Lectures are on Wednesdays 10:00-13:00 and recitations by Tomer Manket are an hour afterwards, Wednesdays 14:00-15:00. Location TBD.
Grade
Fifty percent of your grade will be based on a one-hour presentation at the end of the semester, covering material not addressed in the lectures or recitations. The remaining 50% will be determined by your performance on the problem sets (submissions in pairs). We will publish 5 or 6 problem sets throughout the semester.
Literature
We will not follow a particular text but most of what we will cover appears in Stichtenoth' Algebraic Function Fields and Codes. Combined with the seminar portion of the course, we will cover a significant part of this book. I will also partially follow Dan Haran's lecture notes, though these are in Hebrew (which, honestly, is easier to learn than algebraic geometry codes).
Some recommended literature that we will not directly follow is Lorenzini's marvellous An Invitation to Arithmetic Geometry, and Morandi's lecture notes.
Video lectures
There are some excellent online courses available that cover some of the mathematics we’ll need. While none focus specifically on the coding aspect, the underlying math is generally the more challenging part of the course compared to its application to coding theory.
Plane algebraic curves by Andreas Gathmann
This is a superb course that covers the theory of plane curves with minimal background in commutative algebra. Its strong emphasis on geometry, alongside the commutative algebra, provides a great complement to our course, which is more algebraically focused. Highly recommended.
Computational Arithmetic - Geometry for Algebraic Curves by Nitin Saxena
This excellent (still ongoing) course places more emphasis on the case where the ground field is finite, which is of particular interest to us, compared to Gathmann's course mentioned earlier. Here too, the prerequisites in commutative algebra are fairly minimal.
Algebraic Geometric Codes by Peter Beelen
This crash course provides an overview of some of the topics we will cover. However, the prerequisites are more advanced. It will likely be more beneficial to you either during or after completing our course.
Algebraic Number Theory by U. K. Anandavardhanan
Since we will be working with finite fields, much of the mathematics we need overlaps with the underlying concepts in algebraic number theory (arguably as much as those in algebraic geometry). While we will develop our theory independently of algebraic number theory, for those interested in the connection between the two fields—and in gaining some "arithmetic" intuition—this is a recommended course.
Basic Algebraic Geometry by Miles Reid
This introductory course in algebraic geometry emphasizes intuitive geometric concepts while gradually developing the necessary commutative algebraic ideas. Although we will focus solely on algebraic curves, which are "one-dimensional," it is highly valuable to understand the broader development of algebraic geometry in greater generality. Like many other courses (such as Gathmann's course mentioned earlier), it concludes with a proof of the Riemann-Roch Theorem, which we will cover about halfway through our course. The author also has an excellent book.
Algebraic Geometry by Richard Borcherds
There’s nothing quite like Richard Brocherd's courses. This one offers a more in-depth exploration of algebraic geometry. There is also a follow-up course on schemes.
Commutative Algebra by Richard Borcherds
Richard Borcherds also offers a fantastic and in-depth course on commutative algebra.
Galois Theory by Richard Borcherds
In addition to commutative algebra, we will also need Galois theory, especially for the second part of the course. While I will assume you are all familiar with Galois theory (as it is part of the formal prerequisites), you're likely to learn something new, as is always the case with Richard Brocherd's courses.
Algebraic Coding Theory by Mary Wootters
Since this is a course on coding theory after all, I feel obligated to recommend a relevant course on the subject. Mary Wootters offers such an excellent course, with a focus on algebraic techniques. While the mathematics is more elementary compared to that used in algebraic geometry codes, their sophisticated application will give you valuable insight into how algebraic methods can be applied in coding theory. Additionally, Mary is an outstanding lecturer.