DescriptionError correcting codes have not only important applications (data transmission and storage to name a few) but also interesting connections with other branches of mathematics such as Invariant Theory and Sphere Packing. The main goal of this project is to go beyond the study of error correcting codes as done in Codes and Cryptography III, 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 seen in Codes and Cryptography III. This inlcudes the Quadratic Residue Codes, the Group Codes, the BCH Codes, the Quaternary Codes and others. There are many references for this direction as for example [2]. 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]. Codes and Expander Graphs: In this directions the main focus will be on the connections of error-correcting codes with graph theory. In particular it will be investigated the connection between codes and expander graphs, an important concept of graph theory with many applications. The starting point for this direction will be the book [4].
ResourcesThe following books will be used as references
[2] F.J.Macwilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes , North-Holland Mathematical Library, 1977 [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] R. Roth, Introduction to Coding Theory, Cambridge University Press, 2006
Prerequisites
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