18 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
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
Constructions of new q-cryptomorphisms
In the theory of classical matroids, there are several known equivalent axiomatic systems that define a matroid, which are described as matroid cryptomorphisms. A q-matroid is a q-analogue of a matroid where subspaces play the role of the subsets in the classical theory. In this article we establish cryptomorphisms of q-matroids. In doing so we highlight the difference between classical theory and its q-analogue. We introduce a comprehensive set of q-matroid axiom systems and show cryptomorphisms between them and existing axiom systems of a q-matroid. These axioms are described as the rank, closure, basis, independence, dependence, circuit, hyperplane, flat, open space, spanning space, non-spanning space, and bi-colouring axioms
Relations between M\"obius and coboundary polynomial
It is known that, in general, the coboundary polynomial and the M\"obius
polynomial of a matroid do not determine each other. Less is known about more
specific cases. In this paper, we will try to answer if it is possible that the
M\"obius polynomial of a matroid, together with the M\"obius polynomial of the
dual matroid, define the coboundary polynomial of the matroid. In some cases,
the answer is affirmative, and we will give two constructions to determine the
coboundary polynomial in these cases.Comment: 12 page
The extended and generalized rank weight enumerator
Non UBCUnreviewedAuthor affiliation: University of NeuchâtelPostdoctora
Weight enumeration of codes from finite spaces
We study the generalized and extended weight enumerator of the q-ary Simplex code and the q-ary first order Reed-Muller code. For our calculations we use that these codes correspond to a projective system containing all the points in a finite projective or affine space. As a result from the geometric method we use for the weight enumeration, we also completely determine the set of supports of subcodes and words in an extension code