20 research outputs found
An alternative to the q-matroid axioms (I4), (B4) and (S4)
It is well known that in q-matroids, axioms for independent spaces, bases,
and spanning spaces differ from the classical case of matroids, since the
straightforward q-analogue of the classical axioms does not give a q-matroid.
For this reason, a fourth axiom have been proposed. In this paper we show how
we can describe these spaces with only three axioms, providing two alternative
ways to do that.Comment: Preliminary version. Comments welcome
Representability of the Direct Sum of -Matroids
-Matroids are the -analogue of matroids. In this short note we consider
the direct sum of -matroids, as introduced recently in the literature. We
show that the direct sum of representable -matroids may not be
representable. It remains an open question whether representability of the
direct sum can be characterized by the given -matroids
A Polymatroid Approach to Generalized Weights of Rank Metric Codes
We consider the notion of a -polymatroid, due to Shiromoto, and the
more general notion of -demi-polymatroid, and show how generalized
weights can be defined for them. Further, we establish a duality for these
weights analogous to Wei duality for generalized Hamming weights of linear
codes. The corresponding results of Ravagnani for Delsarte rank metric codes,
and Martinez-Penas and Matsumoto for relative generalized rank weights are
derived as a consequence.Comment: 22 pages; with minor revisions in the previous versio
Constructions of new matroids and designs over GF(q)
A perfect matroid design (PMD) is a matroid whose flats of the same rank all
have the same size. In this paper we introduce the q-analogue of a PMD and its
properties. In order to do that, we first establish a new cryptomorphic
definition for q-matroids. We show that q-Steiner systems are examples of
q-PMD's and we use this q-matroid structure to construct subspace designs from
q-Steiner systems. We apply this construction to S(2, 3, 13; q) q-Steiner
systems and hence establish the existence of subspace designs with previously
unknown parameters
Partitions of Matrix Spaces With an Application to -Rook Polynomials
We study the row-space partition and the pivot partition on the matrix space
. 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
codes 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 -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 over
supported on a Ferrers diagram is a polynomial in , whose
degree is strictly increasing in . Finally, we investigate the natural
analogues of the MacWilliams Extension Theorem for the rank, the row-space, and
the pivot partitions
The Cyclic Flats of a -Matroid
In this paper we develop the theory of cyclic flats of -matroids. We show
that the lattice of cyclic flats, together with their ranks, uniquely
determines a -matroid and hence derive a new -cryptomorphism. We
introduce the notion of -independence of an
-subspace of and we show that -matroids
generalize this concept, in the same way that matroids generalize the notion of
linear independence of vectors over a given field
The Projectivization Matroid of a -Matroid
In this paper, we investigate the relation between a -matroid and its
associated matroid called the projectivization matroid. The latter arises by
projectivizing the groundspace of the -matroid, and considering the
projective space as the groundset of the associated matroid, on which is
defined a rank function compatible with that of the -matroid. We show that
the projectivization map is a functor from categories of -matroids to
categories of matroids. This relation is used to prove new results about maps
of -matroids. Furthermore, we show the characteristic polynomial of a
-matroid is equal to that of the projectivization matroid, which we use to
establish a recursive formula for the characteristic polynomial of a
-matroid in terms of the characteristic polynomial of its minors. Finally we
use the projectivization matroid to prove a -analogue of the critical
theorem in terms of -linear rank metric codes and
-matroids