Minimal Blocking Sets in PG(2, 8) and Maximal Partial Spreads in PG(3, 8)

We prove that PG(2, 8) does not contain minimal blocking sets of size 14. Using this result we prove that 58 is the largest size for a maximal partial spread of PG(3, 8). This supports the conjecture that q ²–q+ 2 is the largest size for a maximal partial spread of PG(3, q), q>7.     

Constructing Minimal Blocking Sets Using Field Reduction

We present a construction for minimal blocking sets with respect to ‐spaces in , the ‐dimensional projective space over the finite field of order . The construction relies on the use of blocking cones in the field reduced representation of , extending the well‐known construction of linear blocking sets. This construction is inspired by the construction for minimal blocking sets with respect to the hyperplanes by Mazzocca, Polverino, and Storme (the MPS‐construction); we show that for a suitable choice of the blocking cone over a planar blocking set, we obtain larger blocking sets than the ones obtained from planar blocking sets in F. Mazzocca and O. Polverino, J Algebraic Combin 24(1) (2006), 61–81. Furthermore, we show that every minimal blocking set with respect to the hyperplanes in can be obtained by applying field reduction to a minimal blocking set with respect to ‐spaces in . We end by relating these constructions to the linearity conjecture for small minimal blocking sets. We show that if a small minimal blocking set is constructed from the MPS‐constructionthen it is of Rédei‐type, whereas a small minimal blocking set arises from our cone construction if and only if it is linear.     

Small Blocking Sets in PG(2, p3)

In this paper we show that a small minimalblocking set in PG(2,p³), with p 7,is of Rédei type.     

Two Remarks on Blocking Sets and Nuclei in Planes of Prime Order

In this paper we characterize a sporadic non-Rédei Type blocking set of PG(2,7) having minimum cardinality, and derive an upper bound for the number of nuclei of sets in PG(2,q) having less than q+1 points. Our methods involve polynomials over finite fields, and work mainly for planes of prime order.     

Self-Orthogonal Compact Spreads and Unitals in Topological Translation Planes

A spread of is a set of l-dimensional subspaces L V partitioning V {0}. We construct examples of compact spreads that are identical with their sets of orthogonal spaces L . In the corresponding topological translation planes, every Euclidean sphere is a unital with the additional property that every point at infinity has flat feet.     

Dominating Sets in Projective Planes

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 is smaller than (i.e., twice the size of a Baer subplane), then it contains either all but possibly one points of a line or all but possibly one lines through a point. Furthermore, we completely characterize dominating sets of size at most . In Desarguesian planes, we could rely on strong stability results on blocking sets to show that if a dominating set is sufficiently smaller than 3q, then it consists of the union of a blocking set and a covering set apart from a few points and lines.     

Minimal Blocking Sets in Projective Spaces of Square Order

In this paper minimal m-blocking sets of cardinality at most in projective spaces PG(n,q) of square order q, q 16, are characterized to be (t, 2(m-t-1))-cones for some t with . In particular we will find the smallest m-blocking sets that generate the whole space PG(n,q) for 2m n m.     

Blocking Sets of Type (1, k) in a Finite Projective Plane

Following a previous work of Tallini (J. Geom. (1987), 191–199), we investigate the arithmetical properties of a blocking set in a finite projective plane which is met by any line in either 1 or k points, for a fixed number k. The results are then used to give some characterizations of blocking sets of this type which are preserved by a large collineation group of the plane.     

Minimal Kakeya Sets

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.     

Blocking Sets of Certain Line Sets Related to a Conic

In this paper we classify point sets of minimum size of two types (1) point sets meeting all secants to an irreducible conic of the desarguesian projective plane PG(2,q), q odd; (2) point sets meeting all external lines and tangents to a given irreducible conic of the desarguesian projective plane PG(2,q), q even.     

Blocking Structures of Hermitian Varieties

Given a hermitian variety H(d,q²) and an integer k (d–1)/2, a blocking set with respect to k-subspaces is a set of points of H(d,q²) that meets all k-subspaces of H(d,q²). If H(d,q²) is naturally embedded in PG(d,q²), then linear examples for such a blocking set are the ones that lie in a subspace of codimension k of PG(d,q²). Up to isomorphism there are k+1 non-isomorphic minimal linear blocking sets, and these have different cardinalities. In this paper it is shown for 1 k< (d–1)/2 that all sufficiently small minimal blocking sets of H(d,q²) with respect to k-subspaces are linear. For 1 k< d/2–3, it is even proved that the k+1 minimal linear blocking sets are smaller than all minimal non-linear ones.AMS Classification: 1991 MSC: 51E20, 51E21     

Note on disjoint blocking sets in Galois planes

In this paper, we show that there are at least cq disjoint blocking sets in PG(2,q), where c ≈ 1/3. The result also extends to some non‐Desarguesian planes of order q. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 149–158, 2006     

Blocking structures in finite projective planes

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, 3^2h). Also, we show that these blocking sets of PG(2, q²), regarded as pointsets of the derived André plane , are still minimal blocking sets in . Furthermore, we prove that the new family of blocking semiovals in PG(2, 3^2h) gives rise to a family of blocking semiovals in the André plane as well.     

Blocking sets of Hermitian generalized quadrangles

Some infinite families of minimal blocking sets on Hermitian generalized quadrangles are constructed.     

Constructions of Partial Difference Sets and Relative DifferenceSets Using Galois Rings

We use Galois rings to construct partial difference sets and relative difference sets in non-elementary abelianp-groups. As an example, we also use Galois ringG R(4, 2) to construct a (96,20,4) difference set in Z₄ × Z₄ × Z₆.Dedicated to Hanfried Lenz on the occasion of his 80th birthday     

Largest minimal blocking sets in PG(2,8)

Bruen and Thas proved that the size of a large minimal blocking set is bounded by . Hence, if q = 8, then the maximal possible size is 23. Since 8 is not a square, it was conjectured that a minimal blocking 23‐set does not exist in PG(2,8). We show that this is not the case, and construct such a set. We prove that this is combinatorially unique. We also complete the spectrum problem of minimal blocking sets for PG(2,8) by showing a minimal blocking 22‐set. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 162–169, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10035     

Small proper double blocking sets in Galois planes of prime order

A proper double blocking set in PG(2,p) is a set B of points such that 2?|B∩l|?(p+1)-2 for each line l. The smallest known example of a proper double blocking set in PG(2,p) for large primes p is the disjoint union of two projective triangles of side (p+3)/2; the size of this set is 3p+3. For each prime p?11 such that we construct a proper double blocking set with 3p+1 points, and for each prime p?7 we construct a proper double blocking set with 3p+2 points.     

Spreads admitting net generating regulizations

We say a spread S carries a regulization , if is a collection of reguli contained in S and if each element of S, except at most two lines, is contained either in exactly one regulus of or in all reguli of . Replacement of each regulus of by its complementary regulus (exceptional lines remain unchanged) yields the complementary congruence S^c of S with respect to . If S^c is a hyperbolic or parabolic or elliptic linear congruence of lines, then is called a net generating, in particular, a hyperbolic or parabolic or elliptic regulization, respectively. For hyperbolic and parabolic regulizations we also give other geometric characterizations.     

Normal Spreads

In Dedicata 16 (1984), pp. 291–313, the representation of Desarguesian spreads of the projective space PG(2t – 1, q) into the Grassmannian of the subspaces of rank t of PG(2t – 1, q) has been studied. Using a similar idea, we prove here that a normal spread of PG(rt – 1,q) is represented on the Grassmannian of the subspaces of rank t of PG(rt – 1, q) by a cap V _{r, t} of PG(r ^t – 1, q), which is the GF(q)-scroll of a Segre variety product of t projective spaces of dimension (r – 1), and that the collineation group of PG(r ^t – 1, q) stabilizing V _{r, t} acts 2-transitively on V _{r, t} . In particular, we prove that V _{3, 2} is the union of q ² – q + 1 disjoint Veronese surfaces, and that a Hermitan curve of PG(2, q ²) is represented by a hyperplane section U of V _{3, 2}. For q 0,2 (mod 3) the algebraic variety U is the unitary ovoid of the hyperbolic quadric Q ⁺ (7, q) constructed by W. M. Kantor in Canad. J. Math., 5 (1982), 1195–1207. Finally we study a class of blocking sets, called linear, proving that many of the known examples of blocking sets are of this type, and we construct an example in PG(3, q ²). Moreover, a new example of minimal blocking set of the Desarguesian projective plane, which is linear, has been constructed by P. Polito and O. Polverino.     

Multiple Blocking Sets in PG(n, q), n > 3

This article discusses minimal s-fold blocking sets B in PG (n, q), q = p^h, p prime, q > 661, n > 3, of size |B| > sq + c ^p q ^2/3 - (s - 1) (s - 2)/2 (s > min (c ^p q ^1/6, q ^1/4/2)). It is shown that these s-fold blocking sets contain the disjoint union of a collection of s lines and/or Baer subplanes. To obtain these results, we extend results of Blokhuis–Storme–Szönyi on s-fold blocking sets in PG(2, q) to s-fold blocking sets having points to which a multiplicity is given. Then the results in PG(n, q), n 3, are obtained using projection arguments. The results of this article also improve results of Hamada and Helleseth on codes meeting the Griesmer bound.