DescriptionThe main aim of this project is to explore the connections between Algebraic Geometry and Error-Correcting Codes. Error-Correcting Codes have very important applications to every day life such as data transmission and data storage to name a few. On the other hand Algebraic Geomtery is one of the most important branches of Pure Mathematics, with the aim of studying the zero locus of polynomial equations. In particular we will see that some classical questions of Algebraic Geometry such as the growth of the number of points of a curve, defined over a finite field, in relation to its genus has important applications to the existense of some "good" error-correcting codes.
This project should be of interest to students wishing to learn more about Algberaic Geometry (beyond the Algebraic Geometry IV course) with an important application in mind. Previous knowledge of Coding Theory is not required even though some basic notions from Codes and Cryptography III could be useful. Of course some basics in Coding Theory can be obtained with some summer reading.
The starting point of the project will be the study of Algebraic Function Fields of one variable, that is, the set of functions on a curve. We will introduce some fundamental notions of Algebraic Geometry such as divisors, differentials and the genus of a curve. The main aim will be Andre Weil's proof of the Riemann-Roch theorem for algebraic curves, a powerful tool for studying functions on algebraic curves. Actually it is a vast generalization of the fundamental theorem of algebra that a polynomial of degree d can have at most d-many zeros.
The next step will be to define the so-called Goppa Codes or Algebraic Geometric Codes, and using the Riemann-Roch theorem we will study their parameters. Then the project, depending on interest may take various directions. The following list is by all means not complete.
1) Study of special classes of Algebraic Geomtric Codes such as Elliptic, Hyperelliptic or Hermitian Codes,
2) Decoding and List Decoding of Algebraic Geometric Codes,
3) Modifications of the basic construction (Trace codes, Partial Algebraic Geometric Codes, Elkies Codes, Xing Codes etc),
4) Trace Codes and the Hasse-Weil Bound.
5) Asymptotic bounds.
ResourcesThere are many reference for Algebraic Geometric Codes. A good starting point is the following two
 M. Tsfasman, S. Vladut, D. Nogin, Algberaic Geometry Codes, Basic Notions ,, Mathematical Surveys and Monographs, AMS, 2007