arXiv subjects

Zsuzsa Weiner

Publications and source records attributed to Zsuzsa Weiner.

12 recordsLinked to original sources

arXiv2026

Balanced intersection size distributions in finite geometries and vector spaces

We study intersection size distributions arising in finite geometries and vector spaces. For a projective plane $Π_q$ of order $q$ with line set $\mathcal L$, we show that \[ \min_{S\subseteqΠ_q}\max_k |\{\ell\in\mathcal L:|S\cap\ell|=k\}|=Θ(q^{3/2}). \] The arguments behind these bounds admit a more general formulation. We prove a similar lower bound for balanced incomplete block designs, together with probabilistic upper bounds for uniform block systems: one for fixed block size under a weak dependency condition and another for growing block size under bounded pairwise intersections. This yields corresponding results for subspaces of vector spaces over finite fields, equivalently, for affine subspaces of an affine space. We also study explicit constructions in affine planes of prime order and relate their intersection size distributions to character sum estimates; an elliptic curve construction gives a bound within a polylogarithmic factor of the optimal order. Finally, we relate balanced intersection size distributions to legitimate colorings and prove that every n-uniform linear hypergraph with $n$ edges admits a legitimate 2-coloring.

math.CO↗

arXiv2026

A note on the chromatic number of Kneser graphs on chambers of projective planes and incidence-free sets

Let $D=(\mathcal{P},\mathcal{B})$ be a symmetric $(v,k,λ)$-design and let $(X,Y)$ be an equinumerous incidence-free pair, with $X\subseteq \mathcal{P}$ and $Y\subseteq \mathcal{B}$. In this note, we give an elementary proof which shows the existence of a perfect matching between $\mathcal{P} \setminus X$ and $\mathcal{B}\setminus Y$ in the incidence graph of $D$. This recovers a result of Spiro, Adriaensen and Mattheus, who already showed this using different arguments for $k\geq 36$. We use this to connect some dots in the literature and prove that finding the chromatic number of the Kneser graph on chambers of a projective plane is equivalent to finding the incidence-free number of the incidence graph of the plane. Furthermore, we construct an incidence-free pair for PG$(2,q^2)$ of size roughly $q^3/2+3q^2/4$.

math.CO↗

arXiv2025

A note on short minimal codes from subgeometries

In a 2022, Bartoli, Cossidente, Marino, and Pavese proved that in the projective space ${\rm PG}(3,q^3)$, one can find three $\mathbb F_q$-subgeometries such that the union of their point sets is a strong blocking set. This proves the existence of linear minimal codes with parameters $[3(q^2+1)(q+1),4]_{q^3}$ for every prime power $q$. We give a short proof of this result for odd values of $q > 9$, using the theory of small blocking sets in projective planes.

math.CO↗

arXiv2024

Multisets with few special directions and small weight codewords in Desarguesian planes

In this paper, we tie together two well studied topics related to finite Desarguesian affine and projective planes. The first topic concerns directions determined by a set, or even a multiset, of points in an affine plane. The second topic concerns the linear code generated by the incidence matrix of a projective plane. We show how a multiset determining only $k$ special directions, in a modular sense, gives rise to a codeword whose support can be covered by $k$ concurrent lines. The reverse operation of going from a codeword to a multiset of points is trickier, but we describe a possible strategy and show some fruitful applications. Given a multiset of affine points, we use a bound on the degree of its so-called projection function to yield lower bounds on the number of special directions, both in an ordinary and in a modular sense. In the codes related to projective planes of prime order $p$, there exists an odd codeword, whose support is covered by 3 concurrent lines, but which is not a linear combination of these 3 lines. We generalise this codeword to codewords whose support is contained in an arbitrary number of concurrent lines. In case $p$ is large enough, this allows us to extend the classification of codewords from weight at most $4p-22$ to weight at most $5p-36$.

math.CO↗

arXiv2024

Points below a parabola in affine planes of prime order

The elements of a finite field of prime order canonically correspond to the integers in an interval. This induces an ordering on the elements of the field. Using this ordering, Kiss and Somlai recently proved interesting properties of the set of points below the diagonal line. In this paper, we investigate the set of points lying below a parabola. We prove that in some sense, this set of points looks the same from all but two directions, despite having only one non-trivial automorphism. In addition, we study the sizes of these sets, and their intersection numbers with respect to lines.

math.CO↗

arXiv2023

Renitent lines

There are many examples for point sets in finite geometry, which behave "almost regularly" in some (well-defined) sense, for instance they have "almost regular" line-intersection numbers. In this paper we investigate point sets of a desarguesian affine plane, for which there exist some (sometimes: many) parallel classes of lines, such that almost all lines of one parallel class intersect our set in the same number of points (possibly mod $p$, the characteristic). The lines with exceptional intersection numbers are called renitent, and we prove results on the (regular) behaviour of these renitent lines.

math.CO↗

arXiv2023

Intersecting families of graphs of functions over a finite field

Let $U$ be a set of polynomials of degree at most $k$ over $\mathbb{F}_q$, the finite field of $q$ elements. Assume that $U$ is an intersecting family, that is, the graphs of any two of the polynomials in $U$ share a common point. Adriaensen proved that the size of $U$ is at most $q^k$ with equality if and only if $U$ is the set of all polynomials of degree at most $k$ passing through a common point. In this manuscript, using a different, polynomial approach, we prove a stability version of this result, that is, the same conclusion holds if $|U|>q^k-q^{k-1}$. We prove a stronger result when $k=2$. For our purposes, we also prove the following results. If the set of directions determined by the graph of $f$ is contained in an additive subgroup of $\mathbb{F}_q$, then the graph of $f$ is a line. If the set of directions determined by at least $q-\sqrt{q}/2$ affine points is contained in the set of squares/non-squares plus the common point of either the vertical or the horizontal lines, then up to an affinity the point set is contained in the graph of some polynomial of the form $αx^{p^k}$.

math.CO↗

arXiv2023

Covering all but the low weight vertices of the unit cube

In this paper we discuss a result similar to the polynomial version of the Alon-Füredi theorem. We prove that if you want to cover the vertices of the $n$-dimensional unit cube, except those of weight at most $r$ then you need an algebraic surface of degree at least $n-r$.

math.CO↗

arXiv2020

Generalizing Korchmáros--Mazzocca arcs

In this paper, we generalize the so called Korchmáros--Mazzocca arcs, that is, point sets of size $q+t$ intersecting each line in $0, 2$ or $t$ points in a finite projective plane of order $q$. For $t\neq 2$, this means that each point of the point set is incident with exactly one line meeting the point set in $t$ points. In $\mathrm{PG}(2,p^n)$, we change $2$ in the definition above to any integer $m$ and describe all examples when $m$ or $t$ is not divisible by $p$. We also study mod $p$ variants of these objects, give examples and under some conditions we prove the existence of a nucleus.

math.CO↗

arXiv2019

Small weight code words arising from the incidence of points and hyperplanes in PG($\boldsymbol{n,q}$)

Let $C_{n-1}(n,q)$ be the code arising from the incidence of points and hyperplanes in the Desarguesian projective space PG($n,q$). Recently, Polverino and Zullo proved that within this code, all non-zero code words of weight at most $2q^{n-1}$ are scalar multiples of either the incidence vector of one hyperplane, or the difference of the incidence vectors of two distinct hyperplanes. We improve this result, proving that when $q>17$ and $q\notin\{25,27,29,31,32,49,121\}$, all code words of weight at most $(4q-\sqrt{8q}-\frac{33}{2})q^{n-2}$ are linear combinations of incidence vectors of hyperplanes through a fixed $(n-3)$-space. Depending on the omitted value for $q$, we can lower the bound on the weight of $c$ to obtain the same results.

math.CO↗

arXiv2019

Stability of k mod p multisets and small weight codewords of the code generated by the lines of PG(2, q)

In this paper, we prove a stability result on k mod p multisets of points in PG(2,q), q = p^h. The particular case k=0 is used to describe small weight codewords of the code generated by the lines of PG(2, q), as linear combination of few lines. Earlier results proved this for codewords with weight less than 2.5q, while our result is valid until cq sqrt(q). It is sharp when 27 =4. When q is a prime, De Boeck and Vandendriessche constructed a codeword of weight 3p-3 that is not the linear combination of three lines. We characterise their example.

math.CO↗

arXiv2014

On the representability of the bi-uniform matroid

Every bi-uniform matroid is representable over all sufficiently large fields. But it is not known exactly over which finite fields they are representable, and the existence of efficient methods to find a representation for every given bi-uniform matroid has not been proved. The interest of these problems is due to their implications to secret sharing. The existence of efficient methods to find representations for all bi-uniform matroids is proved here for the first time. The previously known efficient constructions apply only to a particular class of bi-uniform matroids, while the known general constructions were not proved to be efficient. In addition, our constructions provide in many cases representations over smaller finite fields.

math.CO↗