DescriptionThis is a project in Pure Mathematics, which should be of interest to students wishing to learn some of the theory of Algebraic Curves (and Algebraic Geometry) and apply this knowledge to the theory of Error-Correcting Codes.Algebraic curves (see also Wikipedia page ) have been in the centre of pure mathematics for centuries. Some of the most important mathematicians have contributed massively to their study, Riemann and Weil just to name a few. Some of the most celebrated theorems in Pure mathematics such as the Riemann-Roch Theorem and the Hasse-Weil Theorem were first established for Algebraic Curves, and later generalised to higher dimensional algebraic varieties. It was a very pleasant surprise to the mathematical community when the Russia mathematician Valerii Goppa used the full force of this powerful theory to define some error-correcting codes (which nowadays are called Goppa Codes or AG Codes), which turned out to have some very strong properties. The main aim of this project is to explore the above mentioned connection between Algebraic Curves 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 Curves are studied as part of Algebraic Geometry, 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 Curves 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 existence of some "good" error-correcting codes. Previous knowledge of Coding Theory is not required even though some basic notions from Codes and Cryptography III could be useful. Basics from the theory of Error-cCorrecting Codes can be easily obtained with some summer reading or during the project. 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 generalisation 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 Geometric Codes derived from Elliptic Curves, Hyperelliptic Curves and Hermitian Curves, 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
[2] M. Tsfasman, S. Vladut, D. Nogin, Algberaic Geometry Codes, Basic Notions ,, Mathematical Surveys and Monographs, AMS, 2007
Prerequisites
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