1,604 research outputs found

    Partial ovoids and partial spreads in symplectic and orthogonal polar spaces

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    We present improved lower bounds on the sizes of small maximal partial ovoids and small maximal partial spreads in the classical symplectic and orthogonal polar spaces, and improved upper bounds on the sizes of large maximal partial ovoids and large maximal partial spreads in the classical symplectic and orthogonal polar spaces. An overview of the status regarding these results is given in tables. The similar results for the hermitian classical polar spaces are presented in [J. De Beule, A. Klein, K. Metsch, L. Storme, Partial ovoids and partial spreads in hermitian polar spaces, Des. Codes Cryptogr. (in press)]

    Non-intersecting Ryser hypergraphs

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    A famous conjecture of Ryser states that every rr-partite hypergraph has vertex cover number at most r1r - 1 times the matching number. In recent years, hypergraphs meeting this conjectured bound, known as rr-Ryser hypergraphs, have been studied extensively. It was recently proved by Haxell, Narins and Szab\'{o} that all 33-Ryser hypergraphs with matching number ν>1\nu > 1 are essentially obtained by taking ν\nu disjoint copies of intersecting 33-Ryser hypergraphs. Abu-Khazneh showed that such a characterisation is false for r=4r = 4 by giving a computer generated example of a 44-Ryser hypergraph with ν=2\nu = 2 whose vertex set cannot be partitioned into two sets such that we have an intersecting 44-Ryser hypergraph on each of these parts. Here we construct new infinite families of rr-Ryser hypergraphs, for any given matching number ν>1\nu > 1, that do not contain two vertex disjoint intersecting rr-Ryser subhypergraphs.Comment: 8 pages, some corrections in the proof of Lemma 3.6, added more explanation in the appendix, and other minor change

    The use of blocking sets in Galois geometries and in related research areas

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    Blocking sets play a central role in Galois geometries. Besides their intrinsic geometrical importance, the importance of blocking sets also arises from the use of blocking sets for the solution of many other geometrical problems, and problems in related research areas. This article focusses on these applications to motivate researchers to investigate blocking sets, and to motivate researchers to investigate the problems that can be solved by using blocking sets. By showing the many applications on blocking sets, we also wish to prove that researchers who improve results on blocking sets in fact open the door to improvements on the solution of many other problems

    Direction problems in affine spaces

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    This paper is a survey paper on old and recent results on direction problems in finite dimensional affine spaces over a finite field.Comment: Academy Contact Forum "Galois geometries and applications", October 5, 2012, Brussels, Belgiu

    Large weight code words in projective space codes

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    AbstractRecently, a large number of results have appeared on the small weights of the (dual) linear codes arising from finite projective spaces. We now focus on the large weights of these linear codes. For q even, this study for the code Ck(n,q)⊥ reduces to the theory of minimal blocking sets with respect to the k-spaces of PG(n,q), odd-blocking the k-spaces. For q odd, in a lot of cases, the maximum weight of the code Ck(n,q)⊥ is equal to qn+⋯+q+1, but some unexpected exceptions arise to this result. In particular, the maximum weight of the code C1(n,3)⊥ turns out to be 3n+3n-1. In general, the problem of whether the maximum weight of the code Ck(n,q)⊥, with q=3h (h⩾1), is equal to qn+⋯+q+1, reduces to the problem of the existence of sets of points in PG(n,q) intersecting every k-space in 2(mod3) points

    Dominating sets in projective planes

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    We describe small dominating sets of the incidence graphs of finite projective planes by establishing a stability result which 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 q>81q>81 is smaller than 2q+2[q]+22q+2[\sqrt{q}]+2 (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 2q+q+12q+\sqrt{q}+1. 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.Comment: 19 page

    Minimal symmetric differences of lines in projective planes

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    Let q be an odd prime power and let f(r) be the minimum size of the symmetric difference of r lines in the Desarguesian projective plane PG(2,q). We prove some results about the function f(r), in particular showing that there exists a constant C>0 such that f(r)=O(q) for Cq^{3/2}<r<q^2 - Cq^{3/2}.Comment: 16 pages + 2 pages of tables. This is a slightly revised version of the previous one (Thm 6 has been improved, and a few points explained

    Blocking and double blocking sets in finite planes

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    In this paper, by using properties of Baer subplanes, we describe the construction of a minimal blocking set in the Hall plane of order q(2) of size q(2) + 2q + 2 admitting 1-,2-,3-,4-, (q + 1)- and (q + 2)-secants. As a corollary, we obtain the existence of a minimal blocking set of a non-Desarguesian affine plane of order q(2) of size at most 4q(2)/3 + 5q/3, which is considerably smaller than 2q(2) - 1, the Jamison bound for the size of a minimal blocking set in an affine Desarguesian plane of order q(2). We also consider particular Andre planes of order q, where q is a power of the prime p, and give a construction of a small minimal blocking set which admits a secant line not meeting the blocking set in 1 mod p points. Furthermore, we elaborate on the connection of this problem with the study of value sets of certain polynomials and with the construction of small double blocking sets in Desarguesian projective planes; in both topics we provide some new results
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