Aart Blokhuis

• Eindhoven University of Technology

A t-intersecting constant dimension subspace code C is a set of k-dimensional subspaces in a projective space \(\mathrm {PG}(n,q)\), where distinct subspaces intersect in exactly a t-dimensional subspace. A classical example of such a code is the sunflower, where all subspaces pass through the same t-space. The sunflower bound states that such a co...

The extended coset leader weight enumerator of the generalized Reed-Solomon $[q + 1, q - 3, 5]_q$ code is computed. The computation is considered as a question in finite geometry. For this we need the classification of the points, lines and planes in the projective three space under projectivities that leave the twisted cubic invariant. A line in t...

In this paper, we study vertex colorings of hypergraphs in which all color class sizes differ by at most one (balanced colorings) and each hyperedge contains at least two vertices of the same color (rainbow-free colorings). For any hypergraph H, the maximum number k for which there is a balanced rainbow-free k-coloring of H is called the balanced u...

A $t$-intersecting constant dimension subspace code $C$ is a set of $k$-dimensional subspaces in a projective space PG(n,q), where distinct subspaces intersect in a $t$-dimensional subspace. A classical example of such a code is the sunflower, where all subspaces pass through the same $t$-space. The sunflower bound states that such a code is a sunf...

In this paper, we study vertex colorings of hypergraphs in which all color class sizes differ by at most one (balanced colorings) and each hyperedge contains at least two vertices of the same color (rainbow-free colorings). For any hypergraph $H$, the maximum number $k$ for which there is a balanced rainbow-free $k$-coloring of $H$ is called the ba...

We show that, for small t, the smallest set that blocks the long secants of the union of t pairwise disjoint Baer subplanes in PG(2,q2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \be...

The existence of ovals and hyperovals is an old question in the theory of non-Desarguesian planes. The aim of this paper is to describe when a conic of PG(2,q) remains an arc in the Hall plane obtained by derivation. Some combinatorial properties of the inherited conics are obtained also in those cases when it is not an arc. The key ingredient of t...

Cameron–Liebler sets of k-spaces were introduced recently in Filmus and Ihringer (J Combin Theory Ser A, 2019). We list several equivalent definitions for these Cameron–Liebler sets, by making a generalization of known results about Cameron–Liebler line sets in \({{\mathrm{PG}}}(n,q)\) and Cameron–Liebler sets of k-spaces in \({{\mathrm{PG}}}(2k+1,...

The existence of ovals and hyperovals is an old question in the theory of non-Desarguesian planes. The aim of this paper is to describe when a conic of ${\rm PG}(2,q)$ remains an arc in the Hall plane obtained by derivation. Some combinatorial properties of the inherited conics are obtained also in those cases when it is not an arc. The key ingredi...

Cameron-Liebler sets of k-spaces were introduced recently by Y. Filmus and F. Ihringer. We list several equivalent definitions for these Cameron-Liebler sets, by making a generalization of known results about Cameron-Liebler line sets in PG(n, q) and Cameron-Liebler sets of k-spaces in PG(2k + 1, q). We also present a classification result.

Let n be a positive integer, q=2ⁿ, and let Fq be the finite field with q elements. For each positive integer m, let Dm(X) be the Dickson polynomial of the first kind of degree m with parameter 1. Assume that m>1 is a divisor of q+1. We study the existence of α∈Fq⁎ such that Dm(α)=Dm(α⁻¹)=0. We also explore the connections of this question to an ope...

We demonstrate that if there exists a real symmetric conference matrix of order n, then there exists a complex symmetric conference matrix of order \(n-1\). A v-set of equi-isoclinic planes in \(\mathbb {R}^{n}\) is a set of v planes spanning \(\mathbb {R}^{n}\), each pair of which has the same non-zero angle \(\arccos \sqrt{\lambda }\). We prove t...

We describe efficient algorithms to search for cases in which binomial coefficients are equal or almost equal, give a conjecturally complete list of all cases where two binomial coefficients differ by 1, and give some identities for binomial coefficients that seem to be new.

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We determine the independence number of the Kneser graph on line-plane flags in the projective space PG(4;q). We also classify the corresponding maximum-size cocliques. © 2016 János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg

A set $U$ of unit vectors is selectively balancing if one can find two disjoint subsets $U^+$ and $U^-$, not both empty, such that the Euclidean distance between the sum of $U^+$ and the sum of $U^-$ is smaller than $1$. We prove that, to guarantee a selectively balancing set, $n \log n$ unit vectors suffice for sufficiently large $n$, but $\tfrac{...

It is known that the classical unital arising from the Hermitian curve in does not have a 2-coloring without monochromatic lines. Here we show that for the Hermitian curve in does possess 2-colorings without monochromatic lines. We present general constructions and also prove a lower bound on the size of blocking sets in the classical unital.

Let $L$ be a set of lines of an affine space over a field and let $S$ be a set of points with the property that every line of $L$ is incident with at least $N$ points of $S$. Let $D$ be the set of directions of the lines of $L$ considered as points of the projective space at infinity. We give a geometric construction of a set of lines $L$, where $D...

A {\em generalized hyperfocused arc} $\mathcal H $ in $PG(2,q)$ is an arc of size $k$ with the property that the $k(k-1)/2$ secants can be blocked by a set of $k-1$ points not belonging to the arc. We show that if $q$ is a prime and $\mathcal H$ is a generalized hyperfocused arc of size $k$, then $k=1,2$ or 4. Interestingly, this problem is also re...

Kakeya sets in the affine plane $\mathrm AG (2,q)$ are point sets that are the union of lines, one through every point on the line at infinity. The finite field Kakeya problem asks for the size of the smallest Kakeya sets and the classification of these Kakeya sets. In this article we present a new example of a small Kakeya set and we give the cl...

We show for binary Armstrong codes Arm(2, k, n) that asymptotically n/k ≤ 1.224, while such a code is shown to exist whenever n/k ≤ 1.12. We also construct an Arm(2, n − 2, n) and Arm(2, n − 3, n) for all admissible n.

We prove an Erd˝ os-Ko-Rado-type theorem for the Kneser graph on the point-hyperplane flags in a finite projective space.

We determine the maximal cocliques of size >=5q^2+5q+2 in the Kneser graph on point-plane flags in PG(4,q). The maximal size of a coclique in this graph is (q^2+q+1)(q^3+q^2+q+1).

There doesn't exists a finite planar map with all edges having the same length, and each vertex on exactly 5 edges.

Đoković (2006) [3] gave an algorithm for the computation of the Poincaré series of the algebra of invariants of a binary form, where the correctness proof for the algorithm depended on an unproven conjecture. Here we prove this conjecture.

A {\em generalized hyperfocused arc} $\mathcal H $ in $PG(2,q)$ is an arc of size $k$ with the property that the $k(k-1)/2$ secants can be blocked by a set of $k-1$ points not belonging to the arc. We show that if $q$ is a prime and $\mathcal H$ is a generalized hyperfocused arc of size $k$, then $k=1,2$ or 4. Interestingly, this problem is also re...

For non-negative integers $r\ge d$, how small can a subset $C\subset F_2^r$ be, given that for any $v\in F_2^r$ there is a $d$-flat passing through $v$ and contained in $C\cup\{v\}$? Equivalently, how large can a subset $B\subset F_2^r$ be, given that for any $v\in F_2^r$ there is a linear $d$-subspace not blocked non-trivially by the translate $B+...

It is shown that the parameters of a linear code over Fq of length n, dimension k, minimum weight d, and maximum weight m satisfy a certain congruence relation. In the case that q = p is a prime, this leads to the bound m ≤ (n-d)p-e(p-1), where e ε {0, 1,.., k-2} is maximal with the property that (n-de) ≠ 0 (mod pk-1-e). Thus, if C contains a codew...

We characterize a 55-point graph by its spectrum ${4^1, (-2)^{10}, (-1 \pm \sqrt{3})^{10}, ((3 \pm \sqrt{5})/2)^{12}}$ . No interlacing is used: examination of tr A m for m ≤ 7 together with study of the representation in the eigenspace for the eigenvalue −2 suffices.

We show that there is a unique graph with spectrum as in the title. It is a subgraph of the McLaughlin graph. The proof uses a strong form of the eigenvalue interlacing theorem to reduce the problem to one about root lattices.

A fast method is presented for the calculation of the MSD and the MWD of polymers obtained via step-growth polymerization of polyfunctional monomers bearing identical reactive groups (i.e., systems of type Afi). Using this method, the complete distribution can be calculated rapidly, not just the statistical averages of the polymer population such a...

In this survey recent results about q-analogues of some clas-sical theorems in extremal set theory are collected. They are related to determining the chromatic number of the q-analogues of Kneser graphs. For the proof one needs results on the number of 0-secant subspaces of point sets, so in the second part of the paper recent results on the struct...

In this paper we collect results on the possible sizes of k-blocking sets. Sinceprevious surveys focused mainly on blocking sets in the plane, we concentrate ourattention on blocking sets in higher dimensions. Lower bounds on the size of thesmallest non-trivial k-blocking set are surveyed in detail. The linearity conjecture andknown results support...

We show for k 3 that if q 3, n 2k + 1 or q = 2, n 2k + 2, then any intersecting family F of k-subspaces of an n-dimensional vector space over GF(q) with T F2F F = 0 has size at most n 1 k 1 qk(k 1) n k 1 k 1 +q k. This bound is sharp as is shown by Hilton-Milner type families. As an application of this result, we determine the chro- matic number of...

In many point-line geometries, to cover all points except one, more lines are needed than to cover all points. Bounds can be given by looking at the dimension of the space of functions induced by polynomials of bounded degree. KeywordsFinite geometry-Covering-Blocking set

We investigate finite 3-nets embedded in a projective plane over a (fnite or infinite) field of any characteristic p. Such an embedding is regular when each of the three classes of the 3-net comprises con- current lines, and irregular otherwise. It is completely irregular when no class of the 3-net consists of concurrent lines. We are interested in...

We show that the q-Kneser graph qK 2k:k (the graph on the k-subspaces of a 2k-space over GF(q), where two k-spaces are adjacent when they intersect trivially), has chromatic number q k +q k−1 for k=3 and for k < q log q − q. We obtain detailed results on maximal cocliques for k = 3. KeywordsChromatic number– q-analog of Kneser graph

A Besicovitch set in AG(n,q) is a set of points containing a line in every direction. The Kakeya problem is to determine the minimal size of such a set. We solve the Kakeya problem in the plane, and substantially improve the known bounds for n greater than 4. Comment: 13 pages

Let \mathcal{L} be a general linear complex in PG(3, q) for any prime power q. We show that when GF(q) is extended to GF(q2), the extended lines of \mathcal{L} cover ...

Semiovals which are contained in the union of three concurrent lines are studied. The notion of a strong semioval is introduced, and a complete classification of these objects in PG(2,p) and PG(2,p2), p an odd prime, is given. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 491–501, 2007

In this paper we prove that a set of points (in a projective space over a finite field of q elements), which is incident with 0 mod r points of every hyperplane, has at least (r−1)q+(p−1)r points, where 1rq=ph, p prime. An immediate corollary of this theorem is that a linear code whose weights and length have a common divisor rq and whose dual mini...

This article continues the study of multiple blocking sets in PG(2, q). In [3], using lacunary polynomials, it was proven that t-fold blocking sets of PG(2, q), q square, t < q(1/4)/2, of size smaller than t(q + 1) + c(q)q(2/3), with c(q) = 2(-1/3) when q is a power of 2 or 3 and c(q) = 1 otherwise, contain the union of t pairwise disjoint Baer sub...

When Jack van Lint was appointed as full professor at the Eindhoven University of Technology at the age of 26 he combined a PhD in number theory with a very open scientific mind. It took a sabbatical visit to Bell Laboratories in 1966 to make him understand that a new and fascinating field of applied mathematics was emerging: discrete mathematics....

We investigate locally grid graphs. The main results are (i) a characterization of the Johnson graphs (and certain quotients of these) as locally grid graphs such that two points at distance 2 have precisely four common neighbors, and (ii) a complete determination of all graphs that are locally a 4 × 4 grid (it turns out that there are four such gr...

The problem we consider here is based on the following question posed by François Jaeger in 1981 at a combinatorics meeting in Eger (Hungary):

We give some necessary conditions for a graph to be 3-chromatic in terms of the spectrum of the adjacency matrix. For all known distance-regular graphs it is determined whether they are 3-chromatic. A start is made with the classification of 3-chromatic distance-regular graphs, and it is shown that such graphs, if not complete 3-partite, must have...

We determine the parameters of the optimal additive quaternary codes of length at most 12 over . Equivalently, we determine how many lines one can pick in a binary projective space such that any t are independent. Or again, how many lines one can pick in a binary projective space such that no hyperplane contains more than m of them.

A classical lemma of Weil is used to characterise quadratic polynomials f with coefficients GF(qn), q odd, with the property that f(x) is a non-zero square for all x∈GF(q). This characterisation is used to prove the main theorem which states that there are no subplanes of order q contained in the set of internal points of a conic in PG(2,qn) for q⩾...

In this note we prove that the Radon number of the three-dimensional integer lattice is at most 17, that is, any set of 17 points with integral coordinates in the three-dimensional Euclidean space can be partitioned into two sets such that their convex hulls have an integer point in common.

We describe some combinatorial problems in finite projective planes and indicate how R\'edei's theory of lacunary polynomials can be applied to them.

We show that the cardinality of a nonempty set of points without tangents in the desarguesian projective plane PG(2, q), q even, is at least q + 1 + Ö{q/6} \sqrt {q/6} provided that the set is not of even type.

The universal embedding dimension of the binary symplectic dual polar space DSp(2d,2) equals (2d+1)(2d−1+1)/3. This settles an old conjecture.

We find all minimal blocking sets of size 3/2 (p + 1) in PG(2, p) for p < 41. There is one new sporadic example, for p = 13. We find all maximal partial spreads of size 45 in PG (3, 7).

In [Blokhuis and Lavrauw (Geom. Dedicata81 (2000), 231–243)] a construction of a class of two-intersection sets with respect to hyperplanes in PG(r−1,qt), rt even, is given, with the same parameters as the union of (qt/2−1)/(q−1) disjoint Baer subgeometries if t is even and the union of (qt−1)/(q−1) elements of an (r/2−1)-spread in PG(r−1,qt) if t...

There are three types of maximal arcs in the planes of order 16, the hyperovals of degree 2, the dual hyperovals of degree 8 and the maximal arcs of degree 4. The hyperovals and dual hyperovals of the Desarguesian projective plane PG(2, q) have been classified for q ≤ 32. This article completes the classification of maximal arcs in PG(2, 16). The i...

Let G be an abelian collineation group of order n2 of a projective plane of order n. We show that n must be a prime power, and that the p-rank of G is at least b +1 ifn = pb for an odd prime p.

Let G G be an abelian collineation group of order n 2 n^2 of a projective plane of order n n . We show that n n must be a prime power, and that the p p -rank of G G is at least b + 1 b+1 if n = p b n=p^b for an odd prime p p .

We show that a code C of length n over an alphabet Q of size q with minimum distance 2 and covering radius 1 satisfies |C| ≥ qn−1/(n − 1). For the special case n = q = 4 the smallest known example has |C| = 31. We give a construction for such a code C with |C| = 28.

A t-round χ-coloring is defined as a sequence ψ1, …, ψt of t (not necessarily distinct) edge colorings of a complete graph, using at most χ colors in each of the colorings. For positive integers k⩽n and t let χt(k, n) denote the minimum number χ of colors for which there exists a t-round χ-coloring of Kn such that all (k2) edges of each Kk⊆Kn get d...

Presented is a construction of quasi-symmetric 2-(q^3,q^2(q−1)/2,q(q^3−^2−2)/4) designs with block intersection numbers q^2(q−2)/4 and q^2(q−1)/4, where q is a power of 2. The framework is given by the three-dimensional affine space over F_2 .

We consider a class of symmetric divisible designs D which are almost projective planes in the following sense: Given any point p, there is a unique point p’ such that p and p’ are on two lines, whereas any other point is joined to p by exactly one line; and dually. We note that either the block size k or k — 2 is a perfect square, and exhibit exam...

When? These are the proceedings of Finite Geometries, the Fourth Isle of Thorns Conference, which took place from Sunday 16 to Friday 21 July, 2000. It was organised by the editors of this volume. The Third Conference in 1990 was published as Advances in Finite Geometries and Designs by Oxford University Press and the Second Conference in 1980 was...

We note that certain Dembowski-Ostrom polynomials can be obtained from the product of two linearised polynomials. We examine this subclass for permutation behaviour over finite fields. In particular, a new infinite class of permutation polynomials is identified.

We obtain new lower bounds on the size of a t-fold blocking set in AG(2,q), in the case that (t, q) = 1. As a consequence, we get that the Lunelli-Sce conjecture on the maximal size of a (k, n)-arc is true in the a#ne plane. 1 Introduction Let A =AG(2,q) be the desarguesian a#ne plane of order q.Anucleus of a set S of q +1pointsofA,isapointP ## S,...

A (q+1)-fold blocking set of size (q+1)(q4+q2+1) in PG(2, q4) which is not the union of q+1 disjoint Baer subplanes, is constructed

After Gleason's result, in the late fifties the following conjecture appeared: if in a finite projective plane every quadrangle is contained in a unique Desarguesian proper subplane of order p, then the plane is Desarguesian (and its order is p d for some d). In this paper we prove the conjecture in the case when the plane is of order p 2 and p is...

We determine the p-rank of the incidence matrix of hyperplanes of PG(n, p e) and points of a nondegenerate quadric. This yields new bounds for ovoids and the size of caps in finite orthogonal spaces. In particular, we show the nonexistence of ovoids in \(O_{10}^ + (2^e ),O_{10}^ + (3^e ),O_9 (5^e ),O_{12}^ + (5^e )\) and \(O_{12}^ + (7^e )\). We al...

New lower bounds are given for the size of a point set in a Desarguesian projective plane over, a finite field that contains at least a prescribed number s of points on every line. These bounds are best possible when q is square and s is small compared with q. In this case the smallest set is shown to be the union of disjoint Baer subplanes. The re...

A (q + 1)-fold blocking set of size (q + 1)(q 4 + q 2 + 1) in PG(2; q 4 ) is constructed, which is not the union of q + 1 disjoint Baer subplanes. 1 Introduction Let PG(2; q) and AG(2; q), where q = p h and p is prime, be the Desarguesian projective and affine planes over GF (q), the finite field of order q. An s-fold blocking set B in PG(2; q) is...

We sho that the universal embedding dimensions (over F 2) of the near polygons associated ith Sym(2n) (vieed as subgroup of Sp1(2n - 2, 2)) are the Catalan numbers.

We identify the points of PG(2, q) ith the directions of lines in GF(q 3), viewed as a 3-dimensional affine space over GF(q). Within this frameork we associate to a unital in PG(2, q) a certain polynomial in to variables, and show that the combinatorial properties of the unital force certain restrictions on the coefficients of this polynomial. In p...

A scattered subspace of PG(n 1; q) with respect to a (t 1)-spread S is a subspace intersecting every spread element in at most a point. Upper and lower bounds for the dimension of a maximum scattered space are given. In the case of a normal spread new classes of two intersection sets with respect to hyperplanes in a projective space are obtained us...

Abstract Given a set U of size q in an ane,plane of order q, we determine the possibilities for the number of directions of secants of U, and in many cases characterize the sets U with given number,of secant directions.

We consider a problem mentioned in [1], which is in partitioning the n-cube in as many sets as possible, such that two different sets always have distance one. 1 Introduction and approach Assume that the vertices of the n-cube are partitioned in such a way that the Hamming distance between each pair of the subsets in the partition is 1. What is the...

We provide short proofs that suitable unitals in derivable projective planes give rise to unitals in the derived planes. Some known constructions of unitals in Hall planes are immediate corollaries.

We present an analog of the well-known Kruskal-Katona theorem for the poset of subspaces of PG (n; 2) ordered by inclusion. For given k; ` (k ! `) and m the problem is to find a family of size m in the set of `-subspaces of PG (n; 2), containing the minimal number of k-subspaces. We introduce two lexicographic type orders O 1 and O 2 on the set of...

We investigate the completeness of an ( nq – q + n – , n)-arc in the Desarguesian plane of order q where n divides q. It is shown that such arcs are incomplete for 0 n/2 if q/n3. For q = 2n they are incomplete for 0 < 0.381n="" and="" for="" q="3n" they="" are="" incomplete="" for="" 0=""> < 0.476n.="" for="" q="" odd="" it="" is="" known="" that="...

We present an analog of the well-known Kruskal-Katona theorem for the poset of subspaces of PG (n; 2) ordered by inclusion. For given k; ` (k ! `) and m the problem is to find a family of size m in the set of `-subspaces of PG (n; 2), containing the minimal number of k-subspaces. We introduce two lexicographic type orders O 1 and O 2 on the set of...

In this paper it is shown that given a non-degenerate elliptic quadric in the projective spacePG(2n−1,q),qodd, then there does not exist a spread ofPG(2n−1,q) such that each element of the spread meets the quadric in a maximal totally singular subspace. An immediate consequence is that the construction of [9’ does not give maximal arcs in projectiv...

We give a lower bound for the size of a cap being contained and complete in the Klein quadric in PG(5, q), or equivalently, for the size of a set ℒ of lines in PG(3, q) such that no three of them are concurrent and coplanar in the same time, and being also maximal for this property.

This article studies covers in PG(3; q) and in generalized quadrangles. The excess of a cover is defined to be the difference between the number of lines in the cover and the number of lines in a spread. In contrast with the theory of partial spreads which tells us that large partial spreads can be extended to spreads, in PG(3; q) and in some gener...

It was a long-standing conjecture in nite geometry that a Desar- guesian plane of odd order contains no maximal arcs. A rather inaccessible and long proof was given recently by the authors in collaboration with Maz- zocca. In this paper a new observation leads to a greatly simplied proof of the conjecture.