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A Cooperstein ovoid is a set of q 8 + q 4 + 1 pairwise non-collinear points in the Lie incidence geometry E 6 , 1 (q) . They were introduced by Cooperstein twenty-six years ago, motivated by the fact that possible non-existence of them would imply non-existence of ovoids in hyperbolic quadrics of rank 5. Since then, no progress has been made on their existence question. We prove that Cooperstein ovoids do not exist under some natural additional conditions. In particular, Cooperstein ovoids intersecting every symplecton of E 6 , 1 (q) do not exist, Cooperstein ovoids which are the fixed points of a collineation do not exist, and Cooperstein ovoids which are the absolute points of a polarity of E 6 , 1 (q) do not exist. [ABSTRACT FROM AUTHOR]

A full classification (up to equivalence) of all minimal blocking sets in PG (2 , 9) was obtained by computer. The resulting numbers of minimal blocking sets are tabulated according to size of the set and order of the automorphism group. For the minimal blocking sets with the larger automorphism groups explicit (geometric) descriptions are given. Some of these results can also be generalised to Desarguesian projective planes of higher order. We also give a complete list of all blocking semiovals in PG (2 , 9) (up to equivalence). [ABSTRACT FROM AUTHOR]

Inspired by the connection between ovoids and unitals arising from the Buekenhout construction in the André/Bruck-Bose representation of translation planes of dimension at most two over their kernel, and since eggs of PG (4 m - 1 , q) , m ≥ 1 , are a generalization of ovoids, we explore the relation between eggs and unitals in translation planes of higher dimension over their kernel. By investigating such a relationship, we construct a unital in the Dickson semifield plane of order 3 10 , which is represented in PG (20 , 3) by a cone whose base is a set of points constructed from the dual of the Penttila-Williams egg in PG (19 , 3) . This unital is not polar; so, up to the knowledge of the authors, it seems to be a new unital in such a plane. [ABSTRACT FROM AUTHOR]

In this paper we consider two pointsets in $ \mathrm{PG}(2,q^n) $ arising from a linear set $ L $ of rank $ n $ contained in a line of $ \mathrm{PG}(2,q^n) $: the first one is a linear blocking set of Rédei type, the second one extends the construction of translation KM-arcs. We point out that their intersections pattern with lines is related to the weight distribution of the considered linear set $ L $. We then consider the Hamming metric codes associated with both these constructions, for which we can completely describe their weight distributions. By choosing $ L $ to be an $ {\mathbb F}_{q} $-linear set with a short weight distribution, then the associated codes have few weights. We conclude the paper by providing a connection between the $ \Gamma\mathrm{L} $-class of $ L $ and the number of inequivalent codes we can construct starting from it. [ABSTRACT FROM AUTHOR]

In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, recently introduced by Bonini and Borello, and then completely characterize minimal linear codes as cutting blocking sets. As a direct result, minimal projective codes of dimension 3 and t-fold blocking sets with t ≥ 2 in projective planes are identical objects. Some bounds on the parameters of minimal codes are derived from this characterization. Using this new link between minimal codes and blocking sets, we also present new general primary and secondary constructions of minimal linear codes. As a result, infinite families of minimal linear codes not satisfying the Aschikhmin-Barg’s condition are obtained. In addition to this, open problems on the parameters and the weight distributions of some generated linear codes are presented. [ABSTRACT FROM AUTHOR]

The study of symmetric configurations v3 with block size 3 has a long and rich history. In this paper we consider two colouring problems which arise naturally in the study of these structures. The first of these is weak colouring, in which no block is monochromatic; the second is strong colouring, in which every block is multichromatic. The former has been studied before in relation to blocking sets. Results are proved on the possible sizes of blocking sets and we begin the investigation of strong colourings. We also show that the known 213 and 223 configurations without a blocking set are unique and make a complete enumeration of all nonisomorphic 203 configurations. We discuss the concept of connectivity in relation to symmetric configurations and complete the determination of the spectrum of 2‐connected symmetric configurations without a blocking set. A number of open problems are presented. [ABSTRACT FROM AUTHOR]

For a fixed hyperbolic quadric 𝓗 in PG(3, q), let 𝔼 (respectively 𝕋, 𝕊) denote the set of all lines of PG(3, q) which are external (respectively tangent, secant) with respect to 𝓗. We characterize the minimum size blocking sets of PG(3, q) with respect to each of the line sets 𝕊, 𝕋 ∪ 𝕊 and 𝔼 ∪ 𝕊. [ABSTRACT FROM AUTHOR]

Let Q + (3 , q) be a hyperbolic quadric in PG (3 , q) and T be the set of all lines of PG (3 , q) which are tangent to Q + (3 , q). If k is the minimum size of a T -blocking set in PG (3 , q) , then we prove that q 2 + 1 ≤ k ≤ q 2 + q. When q = 3 , we show that: (i) there is no T -blocking set of size 10, and (ii) there are exactly two T -blocking sets of size 11 up to isomorphism. By means of the computer algebra systems GAP (The GAP Group, 2014) and Sage (Sage Mathematics Software (Version 6.3), 2014), we find that there exist no T -blocking sets of size q 2 + 1 for each odd prime power q ≤ 13. [ABSTRACT FROM AUTHOR]

Abstract Let Q + (3 , q) be a hyperbolic quadric in PG (3 , q) and T 1 be the set of all lines of PG (3 , q) meeting Q + (3 , q) in singletons (the so-called outer tangents). If k is the minimum size of a T 1 -blocking set in PG (3 , q) , then we prove that k ≥ q 2 − 1. It is known that there is no T 1 -blocking set of size q 2 − 1 for q > 2 even and that there is a unique (up to isomorphism) T 1 -blocking set of size 3 for q = 2. For q = 3 , we prove as well that there is a unique T 1 -blocking set of size 8. Using a computer, we also classify all T 1 -blocking sets of size q 2 − 1 for each prime power q ≤ 13. On basis of some structural similarities we are subsequently able to recognize three families of blocking sets whose further study shows that they can be constructed from certain objects related to finite fields (like nice subsets or permutations of the latter). This connection with finite fields allows us to obtain some computer free descriptions. [ABSTRACT FROM AUTHOR]

We prove that there is a pencil of hypersurfaces in P n of any given degree over a finite field F q such that every F q -member of the pencil is not blocking with respect to F q -lines. [ABSTRACT FROM AUTHOR]

Using an exhaustive computer search, we prove that the number of inequivalent ( 29 , 5 ) -arcs in PG ( 2 , 7 ) is exactly 22. This generalizes a result of Barlotti (see Barlotti, A. Some Topics in Finite Geometrical Structures, 1965), who constructed the first such arc from a conic. Our classification result is based on the fact that arcs and linear codes are related, which enables us to apply an algorithm for classifying the associated linear codes instead. Related to this result, several infinite families of arcs and multiple blocking sets are constructed. Lastly, the relationship between these arcs and the Barlotti arc is explored using a construction that we call transitioning.

Let H be a hyperbolic quadric in P G ( 3 , q ) , where q is a prime power. Let E (respectively, T ) denote the set of all lines of P G ( 3 , q ) which are external (respectively, tangent) to H . We characterize the minimum size blocking sets in P G ( 3 , q ) , q ≠ 2 , with respect to the line set E ∪ T . We also give an alternate proof characterizing the minimum size blocking sets in P G ( 3 , q ) with respect to the line set E for all odd q . [ABSTRACT FROM AUTHOR]

Paul Erdős and László Lovász established by means of an example that there exists a maximal intersecting family of k -sets with ⌊ ( e − 1 ) k ! ⌋ blocks, where e is the base of natural logarithm. László Lovász conjectured that their example is best known example which has the maximum number of blocks. Later it was disproved. But the quest for such examples remain valid till this date. In this note we compute the transversal size of a certain series of intersecting families of k -sets, which is constructed over the Cyclic graph. It helps to provide an example of maximal intersecting family of k -sets with so many blocks and to present two worthwhile examples which disprove two special cases of one of the conjectures of Frankl et al. [ABSTRACT FROM AUTHOR]

The equivalence problem of F q -linear sets of rank n of PG ( 1 , q n ) is investigated, also in terms of the associated variety, projecting configurations, F q -linear blocking sets of Rédei type and MRD-codes. We call an F q -linear set L U of rank n in PG ( W , F q n ) = PG ( 1 , q n ) simple if for any n -dimensional F q -subspace V of W , L V is P Γ L ( 2 , q n ) -equivalent to L U only when U and V lie on the same orbit of Γ L ( 2 , q n ) . We prove that U = { ( x , Tr q n / q ( x ) ) : x ∈ F q n } defines a simple F q -linear set for each n . We provide examples of non-simple linear sets not of pseudoregulus type for n > 4 and we prove that all F q -linear sets of rank 4 are simple in PG ( 1 , q 4 ) . [ABSTRACT FROM AUTHOR]

Abstract: Exploring the classical Ceva configuration in a Desarguesian projective plane, we construct two families of minimal blocking sets as well as a new family of blocking semiovals in PG(2, 32h). Also, we show that these blocking sets of PG(2, q2), regarded as pointsets of the derived André plane A ( q 2 ), are still minimal blocking sets in A ( q 2 ). Furthermore, we prove that the new family of blocking semiovals in PG(2, 32h) gives rise to a family of blocking semiovals in the André plane A ( 3 2 h ) as well. [ABSTRACT FROM AUTHOR]

Let Π q be an arbitrary finite projective plane of order q . A subset S of its points is called saturating if any point outside S is collinear with a pair of points from S . Applying probabilistic tools we improve the upper bound on the smallest possible size of the saturating set to ⌈ 3 q ln q ⌉ + ⌈ ( q + 1 ) ∕ 2 ⌉ . The same result is presented using an algorithmic approach as well, which points out the connection with the transversal number of uniform multiple intersecting hypergraphs. [ABSTRACT FROM AUTHOR]

We describe small dominating sets of the incidence graphs of finite projective planes by establishing a stability result that shows that dominating sets are strongly related to blocking and covering sets. Our main result states that if a dominating set in a projective plane of order [ABSTRACT FROM AUTHOR]

Abstract A blocking set B in projective plane PG(2,q) is a set of points such that every line in the plane intersect B in at least one point and there exist a line intersect B in only one point, we say that B is minimal if B has no blocking subset. In this research we proved the non_existence of minimal blocking set of size (30) contains 12_secant and not contains 13_secant in PG(2,19).

Minimal blocking sets in PG (2 , q 2) have size at most q 3 + 1. This result is due to Bruen and Thas and the bound is sharp, sets attaining this bound are called unitals. In this paper, we show that the second largest minimal blocking sets have size at most q 3 + 1 − (p − 3) / 2 , if q = p , p ≥ 67 , or q = p h , p > 7 , h > 1. Our proof also works for sets having at least one tangent at each of its points (that is, for tangency sets). [ABSTRACT FROM AUTHOR]

A 1993 result of Alon and Füredi gives a sharp upper bound on the number of zeros of a multivariate polynomial in a finite grid over an integral domain. We give a generalization of the Alon-Füredi Theorem and discuss the relationship between Alon-Füredi, our generalization and the results of DeMillo-Lipton, Schwartz and Zippel. A direct coding theoretic interpretation of our main result in terms of Reed-Muller type affine variety codes is shown, which gives us the minimum Hamming distance of these codes. We also apply the Alon-Füredi Theorem to quickly recover – and sometimes strengthen – some old and new results in finite geometry. [ABSTRACT FROM AUTHOR]

Let E (respectively; T , S ) denote the set of all lines which are external (respectively; tangent, secant) to an irreducible conic in P G ( 2 , q ) . We give a brief survey of the known results regarding the minimum size L -blocking sets of P G ( 2 , q ) for L ∈ { E , S , T ∪ E , T ∪ S } and study the other two cases when L = T , E ∪ S . [ABSTRACT FROM AUTHOR]

Let $$q$$ be a prime power and $$W(q)$$ be the symplectic generalized quadrangle of order $$q$$ . For $$q$$ even, let $$\mathcal {O}$$ (respectively, $$\mathcal {E}$$ , $$\mathcal {T}$$ ) be the binary linear code spanned by the ovoids (respectively, elliptic ovoids, Tits ovoids) of $$W(q)$$ and $$\Gamma $$ be the graph defined on the set of ovoids of $$W(q)$$ in which two ovoids are adjacent if they intersect at one point. For $$\mathcal {A}\in \{\mathcal {E},\mathcal {T},\mathcal {O}\}$$ , we describe the codewords of minimum and maximum weights in $$\mathcal {A}$$ and its dual $$\mathcal {A}^{\perp }$$ , and show that $$\mathcal {A}$$ is a one-step completely orthogonalizable code (Theorem 1.1). We prove that, for $$q>2$$ , any blocking set of $$PG(3,q)$$ with respect to the hyperbolic lines of $$W(q)$$ contains at least $$q^2+q+1$$ points and equality holds if and only if it is a hyperplane of $$PG(3,q)$$ (Theorem 1.3). We deduce that a clique in $$\Gamma $$ has size at most $$q$$ (Theorem 1.4). [ABSTRACT FROM AUTHOR]

In this paper we provide a generalization of the MPS construction of blocking sets of P G ( r , q n ) using subspaces of dimension s ≤ n − 2 . By this construction, we give a new non-planar example in P G ( 3 , q 6 ) . [ABSTRACT FROM AUTHOR]

We present a construction for minimal blocking sets with respect to [ABSTRACT FROM AUTHOR]

In this paper we prove that, in the space 𝒜 of almost continuous functions (with the metric of uniform convergence), the set of functions of the first class of Baire is superporous at each point of this space

Some infinite families of minimal blocking sets on Hermitian generalized quadrangles are constructed. [ABSTRACT FROM AUTHOR]

For each odd integer k≥5, we prove that, if M is a simple rank-r binary matroid with no odd circuit of length less than k and with |M|>k2r-k+1, then M is isomorphic to a restriction of the rank-r binary affine geometry; this bound is tight for all r≥k-1. We use this to give a simpler proof of the following result of Govaerts and Storme: for each integer n≥2, if M is a simple rank-r binary matroid with no PG(n-1,2)-restriction and with |M|>(1-112n+2)2r, then M has critical number at most n-1. That result is a geometric analogue of a theorem of Andrásfai, Erd's and Sós in extremal graph theory. [ABSTRACT FROM AUTHOR]

A blocking semioval in a projective plane is a set of points which is both a semioval and a blocking set. In this paper, blocking semiovals in the Desaguesian projective plane $$\mathrm{PG}(2,s^2)$$ admitting an order $$s+1$$ homology group are considered. The geometry of the point-orbits of such a group is studied. Using this geometry two new blocking semiovals are constructed in $$\mathrm{PG}(2,5^2)$$ . Their automorphism group is also discussed. [ABSTRACT FROM AUTHOR]

Abstract: A weak c-colouring of a balanced incomplete block design (BIBD) is a colouring of the points of the design with c colours in such a way that no block of the design has all of its vertices receive the same colour. A BIBD is said to be weakly c-chromatic if c is the smallest number of colours with which the design can be weakly coloured. In this paper we show that for all and with , the obvious necessary conditions for the existence of a -BIBD are asymptotically sufficient for the existence of a weakly c-chromatic -BIBD. [Copyright &y& Elsevier]

Blokhuis and Mazzocca (A. Blokhuis and F. Mazzocca, The finite field Kakeya problem (English summary). Building bridges. Bolyai Soc Math Stud 19 (2008) 205-218) provide a strong answer to the finite field analog of the classical Kakeya problem, which asks for the minimum size of a point set in an affine plane π that contains a line in every direction. In this article, we consider the related problem of minimal Kakeya sets, namely Kakeya sets containing no smaller Kakeya sets, and provide an interesting infinite family of minimal Kakeya sets that are not of extremal size. [ABSTRACT FROM AUTHOR]

We present three families of multiple blocking sets in Desarguesian projective planes of even characteristic. The first and the third construction apply to any translation hyperoval. The second construction applies to arbitrary hyperovals. [ABSTRACT FROM AUTHOR]

In this paper, we show that a small minimal blocking set with exponent e in PG( n, p), p prime, spanning a ( t/ e − 1)-dimensional space, is an $${\mathbb{F}_{p^e}}$$ -linear set, provided that p > 5( t/ e)−11. As a corollary, we get that all small minimal blocking sets in PG( n, p), p prime, p > 5 t − 11, spanning a ( t − 1)-dimensional space, are $${\mathbb{F}_p}$$ -linear, hence confirming the linearity conjecture for blocking sets in this particular case. [ABSTRACT FROM AUTHOR]

Abstract: Let be a -covering with excess E and a blocking set T. Let be all connected components of E with at least two vertices (note that if ). The blocking set T is called tight if further and for . In this paper we give a complete solution for the existence of a minimum -covering admitting a blocking set (BS), or a tight blocking set (TBS) for any λ and when and . [Copyright &y& Elsevier]

Abstract: A blocking semioval in a projective plane is a set of points which is both a semioval and a blocking set. In this paper, blocking semiovals in the Desaguesian projective plane admitting an order homology group are considered. By looking at the geometry of the orbits on points of such a group we find two new blocking semiovals of large size in . [Copyright &y& Elsevier]

Abstract: A t-semiarc of a finite projective plane is a pointset with the property that the number of tangent lines to at each of its points is t. Some characterization theorems for t-semiarcs with large collinear subsets are given. [Copyright &y& Elsevier]

The generalized quadrangle $Q(4,q)$ arising from the parabolic quadric in $PG(4,q)$ always has an ovoid. It is not known whether a minimal blocking set of size smaller than $q^2 + q$ (which is not an ovoid) exists in $Q(4,q)$, $q$ odd. We present results on smallest blocking sets in $Q(4,q)$, $q$ odd, obtained by a computer search. For $q = 5,7,9,11$ we found minimal blocking sets of size $q^2 + q - 2$ and we discuss their structure. By an exhaustive search we excluded the existence of a minimal blocking set of size $q^2 + 3$ in $Q(4,7)$.

Abstract: Let be a -covering with excess and a blocking set . Let , , , be all connected components of with at least two vertices (note that if ). The blocking set is called tight if further and for . In this paper, we give a complete solution for the existence of a minimum -covering admitting a blocking set (BS), or a tight blocking set (TBS) for any and when and . [Copyright &y& Elsevier]