DescriptionError correcting codes have not only important applications in data transmission and storage (for example they are used for reliable storage of data in CD's) but also interesting connections with other branches of mathematics such as Invariant Theory and Algebraic Geometry. The main goal of this project is to go beyond the study of error correcting codes as done in Codes II, and depending on the interest, to explore the connections of codes with other objects of mathematical interest.There are at least three different directions that we could take: Study more codes: We will study further constructions of codes beyond the ones already seen in Codes II. This inlcudes the Quadratic Residue Codes, the Group Codes, the BCH Codes, the Quaternary Codes and others. This direction requires the least prerequisites. There are many references for this direction as for example [2]. Goppa Codes: Here we will study the theory of Goppa Codes (often called Algebraic Geometric Codes). Although the underlying construction is geometric, our study will be purely algebraic. This direction could also be seen as an introduction to the theory of algebraic function fields of one variable. The main reference for this direction is [4]. Self-dual Codes, Lattices and Invariant Theory: Here the aim will be to explore connections between the theory of error correcting codes with other interesting branches of mathematics, and in particular with Invariant Theory. We will concentrate on the so-called self-dual codes and their various generalizations, and study the connections with the theory of lattices, the sphere packing problem and t-designs. The main references for this direction is [1] and [3]. ResourcesThe following books will be used as references
[2] J.H. van Lint, Introduction to Coding Theory, Graduate Texts in Mathematics, Third Edition, Springer 1999 [3] G. Nebe, E.M. Rains and N. J.A. Sloane, Self-Dual Codes and Invariant Theory, Algorithms and Computations in Mathematics, Volume 17, Springer 2006 [4] H. Stichtenoth, Algebraic Function Fields and Codes, Universitext , Springer-Verlag 1993
Prerequisites
|