Linear and Cyclic Codes over Rings – ICMS Seminar talk by Maryam Bajalan

The study of codes over the rings (ring-linear codes) attracted great interest after the work of Calderbank, Hammons, Kumar, Sloane, and Sole in the early 1990s. In this seminar, the basic theory of linear codes over finite commutative rings will be presented including the importance of codes over rings, various kinds of rings for ring-linear coding theory, the weight functions on finite rings, MacWilliams equivalence theorem and the connection between these codes and codes over fields via the Gray maps. Moreover, the cyclic codes over finite commutative rings will be considered. Finally, some well-known generalizations of cyclic codes such as negacyclic, quasi-cyclic, polycyclic, multivariable, polynomial and Abelian codes will be introduced.