Related papers: Linear sets in the projective line over the endomo…

Lunardon and Polverino introduced in 2001 a new family of maximum scattered linear sets in $\mathrm{PG}(1,q^n)$ to construct linear minimal R\'edei blocking sets. This family has been extended first by Lavrauw, Marino, Trombetti and…

We discuss representations of the projective line over a ring $R$ with 1 in a projective space over some (not necessarily commutative) field $K$. Such a representation is based upon a $(K,R)$-bimodule $U$. The points of the projective line…

Saturating sets are combinatorial objects in projective spaces over finite fields that have been intensively investigated in the last three decades. They are related to the so-called covering problem of codes in the Hamming metric. In this…

In this paper, we investigate the Andr\'e/Bruck-Bose representation of certain $\mathbb{F}_q$-linear sets contained in a line of $\text{PG}(2,q^t)$. We show that scattered $\mathbb{F}_q$-linear sets of rank $3$ in $\text{PG}(1,q^3)$…

For a closure space (P,f) with f(\emptyset)=\emptyset, the closures of open subsets of P, called the regular closed subsets, form an ortholattice Reg(P,f), extending the poset Clop(P,f) of all clopen subsets. If (P,f) is a finite convex…

We consider a one-parametric series of left-invariant Lorentzian structures on the universal covering of the Lie group SL(2,R). These structures have SO(1,1)-symmetry and they are deformations of the anti-de Sitter Lorentzian manifold. We…

If an Fq-linear set LU in a projective space is defined by a vector subspace U which is linear over a proper superfield of Fq, then all of its points have weight at least 2. It is known that the converse of this statement holds for linear…

In [2] and [19] are presented the first two families of maximum scattered $\mathbb{F}_q$-linear sets of the projective line $\mathrm{PG}(1,q^n)$. More recently in [23] and in [5], new examples of maximum scattered $\mathbb{F}_q$-subspaces…

We introduce a family of linear sets of $\mathrm{PG}(1,q^{2n})$ arising from maximum scattered linear sets of pseudoregulus type of $\mathrm{PG}(3,q^{n})$. For $n=3,4$ and for certain values of the parameters we show that these linear sets…

In this paper, we show that a small minimal blocking set with exponent e in PG(n,p^t), p prime, spanning a (t/e-1)-dimensional space, is an F_p^e-linear set, provided that p>5(t/e)-11. As a corollary, we get that all small minimal blocking…

The linear representation $T_n^*(\mathcal{K})$ of a point set $\mathcal{K}$ in a hyperplane of $\mathrm{PG}(n+1,q)$ is a point-line geometry embedded in this projective space. In this paper, we will determine the isomorphisms between two…

A recent article of J.P. Bell, K. Huang, W. Peng and T.J. Tucker establishes an analog of the Tits alternative for semigroups of endomorphisms of the projective line. The proof involves a ping-pong argument on arithmetic height functions.…

We introduce a new method to study mixed characteristic deformation of line bundles. In particular, for sufficiently large smooth projective families $f : \mathscr{X} \to \mathscr{S}$ defined over the ring of $N$-integers…

The concept of linear set in projective spaces over finite fields was introduced by Lunardon in 1999 and it plays central roles in the study of blocking sets, semifields, rank-distance codes and etc. A linear set with the largest possible…

The maximum scattered linear sets in $PG(1,q^n)$ have been completely classified for $n \le 4$ by Csajb\'ok-Zanella and Lavrauw-Van de Voorde. Here a wide class of linear sets in $PG(1,q^5)$ is studied which depends on two parameters.…

We study $t$-designs of parameters $(n,k,\lambda)$ over finite fields as group divisible designs and set systems admitting a transitive action of a linear group encoded in an hypergraph $G$ whose vertex set of size $n$ is partitioned into…

Let $\mathcal S$ be a Desarguesian $(t-1)$--spread of $PG(rt-1,q)$, $\Pi$ a $m$-dimensional subspace of $PG(rt-1,q)$ and $\Lambda$ the linear set consisting of the elements of $\mathcal S$ with non-empty intersection with $\Pi$. It is known…

Let $L=\mathbb F_{q^n}$ be a finite field and let $F=\mathbb F_q$ be a subfield of $L$. Consider $L$ as a vector space over $F$ and the associated projective space that is isomorphic to ${\mathrm{PG}}(n-1,q)$. The properties of the…

In this paper, we develop a representation-theoretic formulation of discrete-time linear systems. We show that such systems are naturally viewed as representations of time groups acting on vector spaces, thereby endowing the state space…

Let A be a finite-dimensional associative algebra and $\phi$ a symmetric linear function on $A$. In this note, we will show that the pseudotrace maps are obtained as special cases of well-known symmetric linear functions on the endomorphism…