Modul: MAT076 Arbeitsgemeinschaft in Codierungstheorie und Kryptographie

## Matroidal Approach to Coding Theory

Talk by Dr. Gianira Nicoletta Alfarano

Speaker invited by: Prof. Dr. Joachim Rosenthal

**Date:** 07.12.22 **Time:** 15.00 - 16.00 **Room:** Y27H12

In classical combinatorics, polymatroids have been introduced as an extension of the concept of matroid. There are many known connections between linear codes and matroids and many invariant in coding theory are also matroid invariants. $q$-Matroids and $q$-polymatroids are the $q$-analogue of matroids and polymatroids. These objects have gained a lot of interest among an increasing number of researchers, especially in the last few years, due to their connection with rank-metric codes. In particular, it has been shown that to an $\mathbb{F}_{q}$-linear rank metric code $\mathcal{C}\leq \mathbb{F}_{q}^{n\times m}$ it can be associated a $q$-polymatroid $M_\mathcal{C}$ and when $\mathcal{C}$ is also $\mathbb{F}_{q^m}$-linear, $M_\mathcal{C}$ is a $q$-matroid.
In this talk we will show some recent results on the invariants of $q$-polymatroids and rank-metric codes. One of these results lead to the solution of the $q$-analogue of the classical Critical Problem, proposed by Crapo and Rota. We will make use of the characteristic polynomial of a $q$-polymatroid as a basic tool for this result. Finally, we will provide the coding theoretic interpretation and we will partially solve it for \emph{maximum rank distance} codes.

(**This eSeminar will take place over Zoom, using the same meeting details as previous seminars. If you do not have meeting details, please contact zita.fiquelideabreu@math.uzh.ch **)