The MacWilliams Equivalence Theorem for Codes over Rings
Prof. Jens Zumbrägel
One of the central results on which coding theory is based is
MacWilliams' Equivalence Theorem. It states that every linear Hamming
isometry of codes extends to a monomial transformation of the ambient
space. Originally established for codes over finite fields, it has
been generalized to finite ring alphabets by work of Wood, which
sparked a renewed interest in ring-linear coding theory.
In this talk I provide an overview of the various ring properties and
weight properties that allow for such an Extension Theorem. In
particular, I will discuss the cases of chain rings, principal ideal
rings and Frobenius rings, as well as some weights including the
homogeneous weight. The results are based on joint works with
O. W. Gnilke, M. Greferath, T. Honold, C. Mc Fadden, F. M. Schneider
and J. A. Wood.
From April 2017 Jens Zumbrägel holds the professorship in mathematics/cryptography at the Faculty of Computer Science and Mathematics at University Passau.
Jens Zumbrägel has been a substitute professor at the Algebra Institute of TU Dresden, as well as a scientist at the Laboratory for Cryptologic Algorithms, EPFL. Before that he was a Marie Curie fellow at TU Dresden and a postdoctoral researcher at the School of Mathematical Sciences, University College Dublin. Jens Zumbrägel obtained his doctoral degree under supervision of Joachim Rosenthal at the Mathematics Institute of University of Zurich.