Paper Info
Title
Partitions of matrix spaces with an application to q-rook polynomials
Abstract
We study the row-space partition and the pivot partition on the matrix space Fqn×m. We show that both these partitions are reflexive and that the row-space partition is self-dual. Moreover, using various combinatorial methods, we explicitly compute the Krawtchouk coefficients associated with these partitions. This establishes MacWilliams-type identities for the row-space and pivot enumerators of linear rank-metric codes. We then generalize the Singleton-like bound for rank-metric codes, and introduce two new concepts of code extremality. Both of them generalize the notion of MRD code and are preserved by trace-duality. Moreover, codes that are extremal according to either notion satisfy strong rigidity properties analogous to those of MRD codes. As an application of our results to combinatorics, we give closed formulas for the q-rook polynomials associated with Ferrers diagram boards. Moreover, we exploit connections between matrices over finite fields and rook placements to prove that the number of matrices of rank r over Fq supported on a Ferrers diagram is a polynomial in q, whose degree is strictly increasing in r. Finally, we investigate the natural analogues of the MacWilliams Extension Theorem for the rank, the row-space, and the pivot partitions.
Year
DOI
Venue
2018
10.1016/j.ejc.2020.103120
European Journal of Combinatorics
DocType
Volume
ISSN
Journal
89
0195-6698
Citations 
PageRank 
References 
0
0.34
0
Authors
2
Name
Order
Citations
PageRank
Heide Gluesing-Luerssen16912.81
Alberto Ravagnani2329.46