Modul: MAT076 Arbeitsgemeinschaft in Codierungstheorie und Kryptographie

## Partitions of Matrix Spaces and q-Rook Polynomials

Dr. Alberto Ravagnani talk

**Date:** 20.02.19 **Time:** 15.00 - 17.00 **Room:**

In this talk, I will describe the row-space and the pivot partition on the space of n x m matrices over GF(q). Both these partitions are reflexive and yield invertible MacWilliams identities for matrix codes endowed with the row-space and the pivot enumerators, respectively. Moreover, they naturally give rise to notions of extremality. Codes that are extremal with respect to any of these notions satisfy strong rigidity properties, analogous to those of MRD codes. It turns out that the Krawtchouk coefficients of both the row-space and the pivot partition can be explicitly computed using combinatorial methods. For the pivot partition, the computation relies on the properties of the q-rook polynomials associated with Ferrers diagrams, introduced by Garsia/Remmel and Haglund in the 80's. I will describe this connection between codes and rook theory, and present a closed formula for the q-rook polynomial (of any degree) associated to an arbitrary Ferrers board. The new results in this talk are joint work with Heide Gluesing-Luerssen.