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 Modular Forms, Invariant Theory and Association Schemes to name a few. The main goal of this project is to go beyond the study of error correcting codes as done in Codes and Crypto III, and depending on interest, to either study in depth a family of codes or explore connections of codes to other objects of mathematical interest.We will start the project with learning the basics of codes (generator matrix, parity check matrix, Hamming distance etc), and then, depending on interest, the project could take various directions. Here is a short list of possible directions, which is by all means not complete. Study in depth a family of codes: There are plenty of families of codes with very rich structure and interesting mathematics involved. In this direction one can pick a family of such codes for an in depth study. Here is a list of some families: Quadratic Residue Codes, Cyclic (especially BCH) Codes, the Golay Codes, Quaternary Codes, Reed-Muller codes, to name a few. One can have a look at [2] and [7] as first references. Error-Correcting Codes and Modular Forms: Modular forms is a central object of study of modern number theory. Via the theory of lattices (Construction A for example) there is a connection between modular forms and error-correcting codes. The aim of this direction is to explore this connection. References for this direction are [5] and the more advanced [3]. Self-dual Codes and Invariant Theory: Here the aim will be to explore connections between the theory of error correcting codes and Invariant Theory. The first aim in this direction is to understand Gleason's theorem and then study its (many) generalisations. See for example Chapter 19 of [2] below for a first reading. For the most up to date see [3]. Error-Correcting Codes and Association Schemes: After the fundamental work of Delsarte, it is now understood that error-correcting codes have a very rich algebraic structure with connections to graph theory and algebraic combinatorics. A starting point for this direction is Chapter 21 of [2]. For a more up to date reference see [6]. Resources
[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. [5] W. Ebeling, Lattices and Codes, Springer, 2013. [6] Ei. Bannai, Et. Bannai, T. Ito, R. Tanaka, Algebraic Combinatorics, de Gruyter, 2020. [7] W. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, CUP 2003.
Prerequisites
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