The incidence matrix of a $q$-ary graph
In this preprint we discuss a definition of a $q$-ary graph. Furthermore, we describe how to make an incidence matrix for it, with an eye on the corresponding $q$-matroid.
math.CO↗
arXiv subjects
Michela Ceria
Publications and source records attributed to Michela Ceria.
A q-analogue of delta-matroids and related concepts
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 definitions by means of strong maps of q-matroids.
math.CO↗
Alternatives for the $q$-matroid axioms of independent spaces, bases, and spanning spaces
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 has been proposed. In this paper we show how we can describe these spaces with only three axioms, providing two alternative ways to do that. As an application, we show direct cryptomorphisms between independent spaces and circuits and between independent spaces and bases.
math.CO↗
The complexity of solving Weil restriction systems
The solving degree of a system of multivariate polynomial equations provides an upper bound for the complexity of computing the solutions of the system via Groebner bases methods. In this paper, we consider polynomial systems that are obtained via Weil restriction of scalars. The latter is an arithmetic construction which, given a finite Galois field extension $k\hookrightarrow K$, associates to a system $\mathcal{F}$ defined over $K$ a system $\mathrm{Weil}(\mathcal{F})$ defined over $k$, in such a way that the solutions of $\mathcal{F}$ over $K$ and those of $\mathrm{Weil}(\mathcal{F})$ over $k$ are in natural bijection. In this paper, we find upper bounds for the complexity of solving a polynomial system $\mathrm{Weil}(\mathcal{F})$ obtained via Weil restriction in terms of algebraic invariants of the system $\mathcal{F}$.
cs.CR↗
Weighted Subspace Designs from $q$-Polymatroids
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 and also from sets to vector spaces. To achieve this, we introduce the characteristic polynomial of a $q$-polymatroid and outline several of its properties.
math.CO↗
The direct sum of $q$-matroids
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 than in the classical case, as we will try to convince the reader. With the use of submodular functions and the $q$-analogue of matroid union we come to a definition of the direct sum of $q$-matroids. As a motivation for this definition, we show it has some desirable properties.
math.CO↗
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 so, 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 the only known q-Steiner system, which has parameters S(2,3,13;2), and hence establish the existence of a new subspace design with parameters 2-(13,4,5115;2).
math.CO↗
On the geometry of a $(q + 1)$-arc of $\mathrm{PG}(3, q)$, q even
In $\mathrm{PG}(3, q)$, $q = 2^n$, $n \ge 3$, let ${\cal A} = \{(1,t,t^{2^h},t^{2^h+1}) \mid t \in \mathbb{F}_q\} \cup \{(0,0,0,1)\}$, with $\mathrm{gcd}(n,h) = 1$, be a $(q+1)$-arc and let $G_h \simeq \mathrm{PGL}(2, q)$ be the stabilizer of $\cal A$ in $\mathrm{PGL}(4, q)$. The $G_h$-orbits on points, lines and planes of $\mathrm{PG}(3, q)$, together with the point-plane incidence matrix with respect to the $G_h$-orbits on points and planes of $\mathrm{PG}(3, q)$ are determined. The point-line incidence matrix with respect to the $G_1$-orbits on points and lines of $\mathrm{PG}(3, q)$ is also considered. In particular, for a line $\ell$ belonging to a given line $G_1$-orbits, say $\cal L$, the point $G_1$-orbit distribution of $\ell$ is either explicitly computed or it is shown to depend on the number of elements $x$ in $\mathbb{F}_q$ (or in a subset of $\mathbb{F}_q$) such that $\mathrm{Tr}_{q|2}(g(x)) = 0$, where $g$ is an $\mathbb{F}_q$-map determined by $\cal L$.
math.CO↗
The $m$-ovoids of ${\cal W}(5,2)$
In this paper we are concerned with $m$-ovoids of the symplectic polar space ${\cal W}(2n+1, q)$, $q$ even. In particular we show the existence of an elliptic quadric of ${\rm PG}(2n+1, q)$ not polarizing to ${\cal W}(2n+1, q)$ forming a $\left(\frac{q^n-1}{q-1}\right)$-ovoid of ${\cal W}(2n+1, q)$. A further class of $(q+1)$-ovoids of ${\cal W}(5, q)$ is exhibited. It arises by glueing together two orbits of a subgroup of ${\rm PSp}(6, q)$ isomorphic to ${\rm PSL}(2, q^2)$. We also show that the obtained $m$-ovoids do not fall in any of the examples known so far in the literature. Moreover, a computer classification of the $m$-ovoids of ${\cal W}(5, 2)$ is acquired. It turns out that ${\cal W}(5, 2)$ has $m$-ovoids if and only if $m = 3$ and that there are exactly three pairwise non-isomorphic examples. The first example comes from an elliptic quadric ${\cal Q}^-(5, 2)$ polarizing to ${\cal W}(5, 2)$, whereas the other two are the $3$-ovoids previously mentioned.
math.CO↗
Some hypersurfaces over finite fields, minimal codes and secret sharing schemes
Linear error-correcting codes can be used for constructing secret sharing schemes; however finding in general the access structures of these secret sharing schemes and, in particular, determining efficient access structures is difficult. Here we investigate the properties of certain algebraic hypersurfaces over finite fields, whose intersection numbers with any hyperplane only takes a few values; these varieties give rise to $q$-divisible linear codes with at most $5$ weights. Furthermore, for $q$ odd these codes turn out to be minimal and we characterize the access structures of the secret sharing schemes based on their dual codes. Indeed, the secret sharing schemes thus obtained are democratic, that is each participant belongs to the same number of minimal access sets and can easily be described.
cs.IT↗
On large partial ovoids of symplectic and Hermitian polar spaces
In this paper we provide constructive lower bounds on the sizes of the largest partial ovoids of the symplectic polar spaces ${\cal W}(3, q)$, $q$ odd square, $q \not\equiv 0 \pmod{3}$, ${\cal W}(5, q)$ and of the Hermitian polar spaces ${\cal H}(4, q^2)$, $q$ even or $q$ odd square, $q \not\equiv 0 \pmod{3}$, ${\cal H}(6, q^2)$, ${\cal H}(8, q^2)$.
math.CO↗
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.
math.CO↗
On near-MDS codes and caps
Several classes of near-MDS codes of ${\rm PG}(3,q)$ are described. They are obtained either by considering the intersection of an elliptic quadric ovoid and a Suzuki-Tits ovoid of a symplectic polar space ${\cal W}(3, q)$ or starting from the $q+1$ points of a twisted cubic of ${\rm PG}(3, q)$. As a by-product two classes of complete caps of ${\rm PG}(4,q)$ of size $2q^2-q \pm \sqrt{2q} + 2$ are exhibited.
math.CO↗
Macaulay, Lazard and the Syndrome Variety
In this paper we consider the four syndrom varieties ${\sf Z}_e^\times$, i.e. the set of all error locations corresponding to errors of weight $w, 0\leq w\leq 2$, ${\sf Z}_{ns}^\times$ , the set of all {\em non spurious} error locations corresponding to errors of weight $w, 0\leq w\leq 2$, ${\sf Z}_+^\times $, the set of all non-spurious error locations corresponding to errors of weight $w, 1\leq w\leq 2$, ${\sf Z}_2^\times $, the set of all non-spurious error locations corresponding to errors of weight $w= 2$, associated to an up-to-two errors correcting binary cyclic codes. Denoting $J_\ast:=\mathcal{I}({\sf Z}_\ast)$, the ideal of these syndrome varieties, ${\sf N}_\ast := {\bf N}(J_\ast)$ the \GR\ escalier of $J_\ast$ w.r.t. the lex ordering with $x_1<x_2<z_1<z_2$, $Φ_\ast : {\sf Z}_\ast \to {\sf N}_\ast$ a Cerlienco-Mureddu correspondence, and $G_*$ a minimal Groebner basis of the ideal $J_\ast$, the aim of the paper is, assuming to know the structure of the order ideal ${\sf N}_2$ and a Cerlienco Mureddu Correspondence to deduce with elementary arguments ${\sf N}_\ast$, $G_\ast$ and $Φ_\ast$ for $\ast\in\{e,ns,+\}$. The tools are Macaulay's trick and Lazard's formulation of Cerlienco-Mureddu correspondence.
math.CO↗
Applications of Bar Code to involutive divisions and a greedy algorithm for complete sets
In this paper, we describe how to get Janet decomposition for a finite set of terms and detect completeness of that set by means of the associated Bar Code. Moreover, we explain an algorithm to find a variable ordering (if it exists) s.t. a given set of terms is complete according to that ordering. The algorithm is greedy and constructs a Bar Code from the maximal to the minimal variable, adjusting the variable ordering with a sort of backtracking technique, thus allowing to construct the desired ordering without trying all the n! possible orderings
math.CO↗
Bar Code and Janet-like division
Bar Codes are combinatorial objects encoding many properties of monomial ideals. In this paper we employ these objects to study Janet-like divisions. Given a finite set of terms U, from its Bar Code we can compute the Janet-like nonmultiplicative power of its elements and detect completeness of the set. Some observation on the computation of Janet-like bases conclude the work.
math.CO↗
A general framework for Noetherian well ordered polynomial reductions
Polynomial reduction is one of the main tools in computational algebra with innumerable applications in many areas, both pure and applied. Since many years both the theory and an efficient design of the related algorithm have been solidly established. This paper presents a general definition of polynomial reduction structure, studies its features and highlights the aspects needed in order to grant and to efficiently test the main properties (noetherianity, confluence, ideal membership). The most significant aspect of this analysis is a negative reappraisal of the role of the notion of term order which is usually considered a central and crucial tool in the theory. In fact, as it was already established in the computer science context in relation with termination of algorithms, most of the properties can be obtained simply considering a well-founded ordering, while the classical requirement that it be preserved by multiplication is irrelevant. The last part of the paper shows how the polynomial basis concepts present in literature are interpreted in our language and their properties are consequences of the general results established in the first part of the paper.
math.AC↗
Combinatorics of involutive divisions
The classical involutive division theory by Janet decomposes in the same way both the ideal and the escalier. The aim of this paper, following Janet's approach, is to discuss the combinatorial properties of involutive divisions, when defined on the set of all terms in a fixed degree D, postponing the discussion of ideal membership and related test. We adapt the theory by Gerdt and Blinkov, introducing relative involutive divisions and then, given a complete description of the combinatorial structure of a relative involutive division, we turn our attention to the problem of membership. In order to deal with this problem, we introduce two graphs as tools, one is strictly related to Seiler's L-graph, whereas the second generalizes it, to cover the case of "non-continuous" (in the sense of Gerdt-Blinkov) relative involutive divisions. Indeed, given an element in the ideal (resp. escalier), walking backwards (resp. forward) in the graph, we can identify all the other generators of the ideal (resp. elements of degree D in the escalier).
math.AC↗