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View article: Representability of the direct sum of uniform q-matroids
Representability of the direct sum of uniform q-matroids Open
There are many similarities between the theories of matroids and $q$-matroids. However, when dealing with the direct sum of $q$-matroids many differences arise. Most notably, it has recently been shown that the direct sum of representable …
View article: A q-analogue of delta-matroids and related concepts
A q-analogue of delta-matroids and related concepts Open
We define and study q-delta-matroids, and q-g-matroids. These objects are analogues, for finite-dimensional vector spaces over finite fields, of delta-matroids and g-matroids arising from finite sets. We compare axiomatic descriptions with…
View article: The direct sum of q-matroids
The direct sum of q-matroids Open
For classical matroids, the direct sum is one of the most straightforward methods to make a new matroid out of existing ones. This paper defines a direct sum for q -matroids, the q -analogue of matroids. This is a lot less straightforward …
View article: Combinatorial Derived Matroids
Combinatorial Derived Matroids Open
Let $M$ be an arbitrary matroid with circuits $\mathcal{C}(M)$. We propose a definition of a derived matroid $\delta M$ that has as its ground set $\mathcal{C}(M)$. Unlike previous attempts of such a definition, our definition applies to a…
View article: Constructions of new matroids and designs over $${\mathbb {F}}_q$$
Constructions of new matroids and designs over $${\mathbb {F}}_q$$ Open
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 so, we first establish a new cryptomorphic definition f…
View article: Alternatives for the $q$-matroid axioms of independent spaces, bases, and spanning spaces
Alternatives for the $q$-matroid axioms of independent spaces, bases, and spanning spaces Open
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 rea…
View article: Combinatorial Derived Matroids
Combinatorial Derived Matroids Open
Let $M$ be an arbitrary matroid with circuits $\mathcal{C}(M)$. We propose a definition of a derived matroid $δM$ that has as its ground set $\mathcal{C}(M)$. Unlike previous attempts of such a definition, our definition applies to arbitra…
View article: Weighted Subspace Designs from $q$-Polymatroids
Weighted Subspace Designs from $q$-Polymatroids Open
The Assmus-Mattson theorem gives a way to identify block designs arising from codes. This result was broadened to matroids and weighted designs. In this work we present a further two-fold generalisation: first from matroids to polymatroids…
View article: Constructions of new matroids and designs over GF(q)
Constructions of new matroids and designs over GF(q) Open
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 so, we first establish a new cryptomorphic definition fo…
View article: Codes, Cryptology and Curves with Computer Algebra
Codes, Cryptology and Curves with Computer Algebra Open
This well-balanced text touches on theoretical and applied aspects of protecting digital data. The reader is provided with the basic theory and is then shown deeper fascinating detail, including the current state of the art. Readers will s…
View article: The Tutte q-Polynomial
The Tutte q-Polynomial Open
$q$-Matroids are defined on complemented modular support lattices. Minors of length 2 are of four types as in a "classical" matroid. Tutte polynomials $τ(x,y)$ of matroids are calculated either by recursion over deletion/contraction of sin…