Vortrag von Benjamin Jany
Datum: 02.11.22 Zeit: 15.00 - 16.00 Raum:
Matroids, originally defined to generalize the notion of linear dependence, were found to capture many invariants of linear block codes such as the the generalized weights and the weight distribution. One of the most celebrated results is the Critical Theorem, established by Crapo and Rota. The theorem states that, given a code, the number of codewords with given support is fully determined by the characteristic polynomial of the matroid associated to that code. In recent years, there has been a large interest in establishing analogous results for rank metric codes. Jurrius and Pelikaan showed that invariants of rank metric code are captured by $q$-matroids, the $q$-analogue of matroids.
In this talk, I will discuss a $q$-analogue of the Critical Theorem for rank metric codes and $q$-matroids. This will be done by considering a linear block code associated to a rank metric code and by comparing the underlying structure of the matroid and $q$-matroid these two codes induce. As we will see, this approach further establishes a strong connection between the $q$-analogue of the Critical Theorem and the classical Critical Theorem of Crapo and Rota.
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