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A plane curve C ⊂ P 2 of degree d is called blocking if every F q -line in the plane meets C at some F q -point. We prove that the proportion of blocking curves among those of degree d is o (1) when d ≥ 2 q − 1 and q → ∞. We also show that the same conclusion holds for smooth curves under the somewhat weaker condition d ≥ 3 p and d , q → ∞. Moreover, the two events in which a random plane curve is smooth and respectively blocking are shown to be asymptotically independent. Extending a classical result on the number of F q -roots of random polynomials, we find that the limiting distribution of the number of F q -points in the intersection of a random plane curve and a fixed F q -line is Poisson with mean 1. We also present an explicit formula for the proportion of blocking curves involving statistics on the number of F q -points contained in a union of k lines for k = 1 , 2 , ... , q 2 + q + 1. [ABSTRACT FROM AUTHOR]

In this paper we completely characterize the words with second minimum weight in the p -ary linear code generated by the rows of the incidence matrix of points and hyperplanes of PG ( n , q ) , with q = p h and p prime, proving that they are the scalar multiples of the difference of the incidence vectors of two distinct hyperplanes of PG ( n , 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: We introduce generator blocking sets of finite classical polar spaces. These sets are a generalisation of maximal partial spreads. We prove a characterization of these minimal sets of the polar spaces , and , in terms of cones with vertex a subspace contained in the polar space and with base a generator blocking set in a polar space of rank 2. [Copyright &y& Elsevier]

Abstract: In this paper we prove a conjecture of Metsch about the maximum number of lines intersecting a pointset in , presented at the conference “Combinatorics 2002”. As a consequence, we give a short proof of the famous Jamison, Brouwer and Schrijver bound on the size of the smallest affine blocking set in . [Copyright &y& Elsevier]

Abstract: We construct minimal blocking sets with respect to generators on the Hermitian surfaces when n and q are both odd. A blocking set arises from quadrics in whose polarities commute with a unitary polarity, constructed from the union of Baer sub-quadrics with the common intersections deleted. [Copyright &y& Elsevier]

Abstract: Methods for constructing large families of permutation polynomials of finite fields are introduced. For some of these permutations the cycle structure and the inverse mapping are determined. The results are applied to lift minimal blocking sets of to those of . [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we show that a small minimal k-blocking set in , , , p prime, , intersecting every -space in points, is linear. As a corollary, this result shows that all small minimal k-blocking sets in , p prime, , are -linear, proving the linearity conjecture (see Sziklai, 2008 ) in the case , p prime, . [ABSTRACT FROM AUTHOR]

Abstract: Let be the p-ary linear code defined by the incidence matrix of points and k-spaces in , , p prime, . In this paper, we show that there are no codewords of weight in the open interval in which implies that there are no codewords with this weight in if . In particular, for the code of points and hyperplanes of , we exclude all codewords in with weight in the open interval . This latter result implies a sharp bound on the weight of small weight codewords of , a result which was previously only known for general dimension for q prime and , with p prime, , and in the case , for , [K. Chouinard, On weight distributions of codes of planes of order 9, Ars Combin. 63 (2002) 3–13; V. Fack, Sz.L. Fancsali, L. Storme, G. Van de Voorde, J. Winne, Small weight codewords in the codes arising from Desarguesian projective planes, Des. Codes Cryptogr. 46 (2008) 25–43; M. Lavrauw, L. Storme, G. Van de Voorde, On the code generated by the incidence matrix of points and hyperplanes in and its dual, Des. Codes Cryptogr. 48 (2008) 231–245; M. Lavrauw, L. Storme, G. Van de Voorde, On the code generated by the incidence matrix of points and k-spaces in and its dual, Finite Fields Appl. 14 (2008) 1020–1038]. [Copyright &y& Elsevier]

Abstract: We prove that a small minimal blocking set of is “very close” to be a linear blocking set over some subfield . This implies that (i) a similar result holds in for small minimal blocking sets with respect to k-dimensional subspaces () and (ii) most of the intervals in the interval-theorems of Szőnyi and Szőnyi–Weiner are empty. [Copyright &y& Elsevier]

Abstract: Given any linear code C over a finite field we show how C can be described in a transparent and geometrical way by using the associated Bruen–Silverman code. Then, specializing to the case of MDS codes we use our new approach to offer improvements to the main results currently available concerning MDS extensions of linear MDS codes. We also sharply limit the possibilities for constructing long non-linear MDS codes. Our proofs make use of the connection between the work of Rédei [L. Rédei, Lacunary Polynomials over Finite Fields, North-Holland, Amsterdam, 1973. Translated from the German by I. Földes. ] and the Rédei blocking sets that was first pointed out over thirty years ago in [A.A. Bruen, B. Levinger, A theorem on permutations of a finite field, Canad. J. Math. 25 (1973) 1060–1065]. The main results of this paper significantly strengthen those in [A. Blokhuis, A.A. Bruen, J.A. Thas, Arcs in , MDS-codes and three fundamental problems of B. Segre—Some extensions, Geom. Dedicata 35 (1–3) (1990) 1–11; A.A. Bruen, J.A. Thas, A.Blokhuis, On M.D.S. codes, arcs in with q even, and a solution of three fundamental problems of B. Segre, Invent. Math. 92 (3) (1988) 441–459]. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate subquadrangles of order s of flock generalized quadrangles of order , s odd, with base point , where the subquadrangle does not contain the point . We prove that if the flock generalized quadrangle has such a subquadrangle, then is classical. This solves “Remaining case (a)” in Brown and Thas [M.R. Brown, J.A. Thas, Subquadrangles of order s of generalized quadrangles of order , II, J. Combin. Theory Ser. A 106 (2004) 33–48] (“Remaining case (b)” was already handled in K. Thas [K. Thas, Symmetrieën in eindige veralgemeende vierhoeken (Symmetries in finite generalized quadrangles), Master thesis, Ghent University, Ghent, 1999, 186 p.]). As a corollary we have: if is an egg in for which the translation dual is good at the tangent space of at its element π and if there is a pseudo-oval on not containing π, then is classical. As a byproduct we prove that if the flock GQ of order , s odd, has a regular point x collinear with, but distinct from, the base point , then is a translation generalized quadrangle with base line . [Copyright &y& Elsevier]

Abstract: We determine the smallest nontrivial blocking sets with respect to t-spaces in , . For , they are skeletons of solids in ; for , they are cones with vertex an -space and base the set of points on the edges of a tetrahedron in a solid skew to . [Copyright &y& Elsevier]

Abstract: We find a relationship between semifield spreads of , small Rédei minimal blocking sets of , disjoint from a Baer subline of a Rédei line, and translation ovoids of the hermitian surface . [Copyright &y& Elsevier]

It was shown recently that Q(6,q), q>3, q a prime has no ovoids. We improve this result by showing that the smallest cardinality of a set of points of Q(6,q), q>3 prime, meeting all generators of Q(6,q) is q3+q. Up to isomorphism there is only one example of this size. At last, we generalize this result to Q(2n,q). [Copyright &y& Elsevier]

A proof is presented that shows that the number of directions determined by a function over a finite field GF(q) is either 1, at least (q+3)/2, or between q/s+1 and (q−1)/(s−1) for some s where GF(s) is a subfield of GF(q). Moreover, the graph of those functions that determine less than half the directions is GF(s)-linear. This completes the unresolved cases s=2 and 3 of the main theorem in Blokhuis et al. (J. Combin. Theory Ser. A 86 (1999) 187). [Copyright &y& Elsevier]

Intersection sets and blocking sets play an important role in contemporary finite geometry. There are cryptographic applications depending on their construction and combinatorial properties. This paper contributes to this topic by answering the question: how many circles of an inversive plane will be blocked by a d-element set of points that has successively been constructed using a greedy type algorithm? We derive a lower bound for this number and thus obtain an upper bound for the cardinality of an intersection set of smallest size. Defining a coefficient called greedy index, we finally give an asymptotic analysis for the blocking capabilities of circles and subplanes of inversive planes. [Copyright &y& Elsevier]

The characterisation by Blokhuis, Ball, Brouwer, Storme, and Szo¨nyi of certain kinds of blocking sets of Re´dei type is extended to specify the type of polynomial which defines the blocking set. A graphic characterisation called the profile of the set is also given, and the correspondence between the two is demonstrated with relatively small polynomial sizes. [Copyright &y& Elsevier]

A particular class of minihypers was studied previously by the authors (in press, Des. Codes Cryptogr.). For q square, this paper improves the results of that work, under the assumption that no weights occur in the minihyper. Using the link between these minihypers and maximal partial s-spreads of PG(N, q), (s+1) ∣ (N+1), the findings on minihypers translate immediately into results on the extendability of partial s-spreads with small positive deficiency. Other applications of this characterisation of minihypers are given by P. Govaerts et al. (in press, European J. Combin.). [Copyright &y& Elsevier]