arXiv subjects
Publications and source records attributed to Bence Csajbók.
We prove that, for $q$ odd, a set of $q+2$ points in the projective plane over the field with $q$ elements has at least $2q-c$ odd secants, where $c$ is a constant and an odd secant is a line incident with an odd number of points of the set.
The main purpose of this paper is to find double blocking sets in $\mathrm{PG}(2,q)$ of size less than $3q$, in particular when $q$ is prime. To this end, we study double blocking sets in $\mathrm{PG}(2,q)$ of size $3q-1$ admitting at least two $(q-1)$-secants. We derive some structural properties of these and show that they cannot have three $(q-1)$-secants. This yields that one cannot remove six points from a triangle, a double blocking set of size $3q$, and add five new points so that the resulting set is also a double blocking set. Furthermore, we give constructions of minimal double blocking sets of size $3q-1$ in $\mathrm{PG}(2,q)$ for $q=13$, $16$, $19$, $25$, $27$, $31$, $37$ and $43$. If $q>13$ is a prime, these are the first examples of double blocking sets of size less than $3q$. These results resolve two conjectures of Raymond Hill from 1984.
We provide sufficient and necessary conditions for the coefficients of a $q$-polynomial $f$ over $\mathbb{F}_{q^n}$ which ensure that the number of distinct roots of $f$ in $\mathbb{F}_{q^n}$ equals the degree of $f$. We say that these polynomials have maximum kernel. As an application we study in detail $q$-polynomials of degree $q^{n-2}$ over $\mathbb{F}_{q^n}$ which have maximum kernel and for $n\leq 6$ we list all $q$-polynomials with maximum kernel. We also obtain information on the splitting field of an arbitrary $q$-polynomial. Analogous results are proved for $q^s$-polynomials as well, where $\gcd(s,n)=1$.
We investigate punctured maximum rank distance codes in cyclic models for bilinear forms of finite vector spaces. In each of these models we consider an infinite family of linear maximum rank distance codes obtained by puncturing generalized twisted Gabidulin codes. We calculate the automorphism group of such codes and we prove that this family contains many codes which are not equivalent to any generalized Gabidulin code. This solves a problem posed recently by Sheekey in [30].
For an arbitrary $q$-polynomial $f$ over $\mathbb{F}_{q^n}$ we study the problem of finding those $q$-polynomials $g$ over $\mathbb{F}_{q^n}$ for which the image sets of $f(x)/x$ and $g(x)/x$ coincide. For $n\leq 5$ we provide sufficient and necessary conditions and then apply our result to study maximum scattered linear sets of $\mathrm{PG}(1,q^5)$.
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 of $V(2,q^n)$ have been constructed, but the equivalence problem of the corresponding linear sets is left open. Here we show that the $\mathbb{F}_q$-linear sets presented in [23] and in [5], for $n=6,8$, are new. Also, for $q$ odd, $q\equiv \pm 1,\,0 \pmod 5$, we present new examples of maximum scattered $\mathbb{F}_q$-linear sets in $\mathrm{PG}(1,q^6)$, arising from trinomial polynomials, which define new $\mathbb{F}_q$-linear MRD-codes of $\mathbb{F}_q^{6\times 6}$ with dimension $12$, minimum distance 5 and middle nucleus (or left idealiser) isomorphic to $\mathbb{F}_{q^6}$.
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 of $\mathrm{PG}(1,q^{2n})$ are maximum scattered and they yield new MRD-codes with parameters $(6,6,q;5)$ for $q>2$ and with parameters $(8,8,q;7)$ for $q$ odd.
There are two known families of maximum scattered $\mathbb{F}_q$-linear sets in $PG(1,q^t)$: the linear sets of pseudoregulus type and for $t\geq 4$ the scattered linear sets found by Lunardon and Polverino. For $t=4$ we show that these are the only maximum scattered $\mathbb{F}_q$-linear sets and we describe the orbits of these linear sets under the groups $PGL(2,q^4)$ and $PΓL(2,q^4)$.
The rank of a scattered $\mathbb{F}_q$-linear set of $\mathrm{PG}(r-1,q^n)$, $rn$ even, is at most $rn/2$ as it was proved by Blokhuis and Lavrauw. Existence results and explicit constructions were given for infinitely many values of $r$, $n$, $q$ ($rn$ even) for scattered $\mathbb{F}_q$-linear sets of rank $rn/2$. In this paper we prove that the bound $rn/2$ is sharp also in the remaining open cases. Recently Sheekey proved that scattered $\mathbb{F}_q$-linear sets of $\mathrm{PG}(1,q^n)$ of maximum rank $n$ yield $\mathbb{F}_q$-linear MRD-codes with dimension $2n$ and minimum distance $n-1$. We generalize this result and show that scattered $\mathbb{F}_q$-linear sets of $\mathrm{PG}(r-1,q^n)$ of maximum rank $rn/2$ yield $\mathbb{F}_q$-linear MRD-codes with dimension $rn$ and minimum distance $n-1$.
We use polynomial techniques to derive structural results on Rédei type blocking sets from information on their bisecants. We apply our results to point sets of $PG(2,q)$ with few odd-secants. In particular, we improve the lower bound of Balister, Bollobás, Füredi and Thompson on the number of odd-secants of a $(q+2)$-set in $PG(2,q)$ and we answer a related open question of Vandendriessche. We prove structural results for semiovals and derive the non existence of semiovals of size $q+3$ when 3 does not divide $q$ and $q>5$. This extends a result of Blokhuis who classified semiovals of size $q+2$, and a result of Bartoli who classified semiovals of size $q+3$ when $q\leq 17$. In the $q$ even case we can say more applying a result of Szőnyi and Weiner about the stability of sets of even type. We also obtain new proof to a result of Gács and Weiner about $(q+t,t)$-arcs of type $(0,2,t)$ and to one part of a result of Ball, Blokhuis, Brouwer, Storme and Szőnyi about functions over $GF(q)$ determining less than $(q+3)/2$ directions.
The equivalence problem of $\mathbb{F}_q$-linear sets of rank n of $PG(1,q^n)$ is investigated, also in terms of the associated variety, projecting configurations, $\mathbb{F}_q$-linear blocking sets of Rédei type and MRD-codes.
Let $L$ be a linear set of pseudoregulus type in a line $\ell$ in $Σ^*=\mathrm{PG}(t-1,q^t)$, $t=5$ or $t>6$. We provide examples of $q$-order canonical subgeometries $Σ_1,\, Σ_2 \subset Σ^*$ such that there is a $(t-3)$-space $Γ\subset Σ^*\setminus (Σ_1 \cup Σ_2 \cup \ell)$ with the property that for $i=1,2$, $L$ is the projection of $Σ_i$ from center $Γ$ and there exists no collineation $ϕ$ of $Σ^*$ such that $Γ^ϕ=Γ$ and $Σ_1^ϕ=Σ_2$. Condition (ii) given in Theorem 3 in Lavrauw and Van de Voorde (Des. Codes Cryptogr. 56:89-104, 2010) states the existence of a collineation between the projecting configurations (each of them consisting of a center and a subgeometry), which give rise by means of projections to two linear sets. It follows from our examples that this condition is not necessary for the equivalence of two linear sets as stated there. We characterize the linear sets for which the condition above is actually necessary.
Scattered linear sets of pseudoregulus type in $\mathrm{PG}(1,q^t)$ have been defined and investigated in [G. Lunardon, G. Marino, O. Polverino, R. Trombetti: Maximum scattered linear sets of pseudoregulus type and the Segre Variety ${\cal S}_{n,n}$. J. Algebr. Comb. 39 (2014), 807--831.; G. Donati, N. Durante: Scattered linear sets generated by collineations between pencils of lines. J. Algebr. Comb. 40 (2014), 1121-1134]. The aim of this paper is to continue such an investigation. Properties of a scattered linear set of pseudoregulus type, say $L$, are proved by means of three different ways to obtain $L$: (i) as projection of a $q$-order canonical subgeometry [G. Lunardon, O. Polverino: Translation ovoids of orthogonal polar spaces. Forum Math. 16 (2004), 663-669], (ii) as a set whose image under the field reduction map is the hypersurface of degree $t$ in $\mathrm{PG}(2t-1,q)$ studied in [M. Lavrauw, J. Sheekey, C. Zanella: On embeddings of minimum dimension of $\mathrm{PG}(n,q)\times \mathrm{PG}(n,q)$. Des. Codes Cryptogr. 74 (2015), 427-440], (iii) as exterior splash, by the correspondence described in [M. Lavrauw, J. Sheekey, C. Zanella: On embeddings of minimum dimension of $\mathrm{PG}(n,q)\times \mathrm{PG}(n,q)$. Des. Codes Cryptogr. 74 (2015), 427-440]. In particular, given a canonical subgeometry $Σ$ of $\mathrm{PG}(t-1,q^t)$, necessary and sufficient conditions are given for the projection of $Σ$ with center a $(t-3)$-subspace to be a linear set of pseudoregulus type. Furthermore, the $q$-order sublines are counted and geometrically described.
A $t$-semiarc is a pointset ${\cal S}_t$ with the property that the number of tangent lines to ${\cal S}_t$ at each of its points is $t$. We show that if a small $t$-semiarc ${\cal S}_t$ in $\mathrm{PG}(2,q)$ has a large collinear subset ${\cal K}$, then the tangents to ${\cal S}_t$ at the points of ${\cal K}$ can be blocked by $t$ points not in ${\cal K}$. We also show that small $t$-semiarcs are related to certain small blocking sets. Some characterization theorems for small semiarcs with large collinear subsets in $\mathrm{PG}(2,q)$ are given.
In a projective plane $Π_q$ of order $q$, a non-empty point set ${\cal S}_t$ is a $t$-semiarc if the number of tangent lines to ${\cal S}_t$ at each of its points is $t$. If ${\cal S}_t$ is a $t$-semiarc in $Π_q$, $t 1$, then $t$-semiarcs with $q+1-t$ collinear points exist only if $t\geq \sqrt{q-1}$. In $\mathrm{PG}(2,q)$ we prove the lower bound $t\geq(q-1)/2$, with equality only if ${\cal S}_t$ is a blocking set of Rédei type of size $3(q+1)/2$. We call the symmetric difference of two lines, with $t$ further points removed from each line, a $V_t$-configuration. We give conditions ensuring a $t$-semiarc to contain a $V_t$-configuration and give the complete characterization of such $t$-semiarcs in $\mathrm{PG}(2,q)$.