10 research outputs found

    Generalized quadrangles of order (p,t) admitting a 2-transitive regulus, p a prime

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    AbstractWe classify generalized quadrangles of order (p,t) admitting a 2-transitive regulus, p a prime

    New and old results on flocks of circle planes

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    Ovoids and spreads of finite classical generalized hexagons and applications

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    One intuitively describes a generalized hexagon as a point-line geometry full of ordinary hexagons, but containing no ordinary n-gons for n<6. A generalized hexagon has order (s,t) if every point is on t+1 lines and every line contains s+1 points. The main result of my PhD Thesis is the construction of three new examples of distance-2 ovoids (a set of non-collinear points that is uniquely intersected by any chosen line) in H(3) and H(4), where H(q) belongs to a special class of order (q,q) generalized hexagons. One of these examples has lead to the construction of a new infinite class of two-character sets. These in turn give rise to new strongly regular graphs and new two-weight codes, which is why I dedicate a whole chapter on codes arising from small generalized hexagons. By considering the (0,1)-vector space of characteristic functions within H(q), one obtains a one-to-one correspondence between such a code and some substructure of the hexagon. A regular substructure can be viewed as the eigenvector of a certain (0,1)-matrix and the fact that eigenvectors of distinct eigenvalues have to be orthogonal often yields exact values for the intersection number of the according substructures. In my thesis I reveal some unexpected results to this particular technique. Furthermore I classify all distance-2 and -3 ovoids (a maximal set of points mutually at maximal distance) within H(3). As such we obtain a geometrical interpretation of all maximal subgroups of G2(3), a geometric construction of a GAB, the first sporadic examples of ovoid-spread pairings and a transitive 1-system of Q(6,3). Research on derivations of this 1-system was followed by an investigation of common point reguli of different hexagons on the same Q(6,q), with nice applications as a result. Of these, the most important is the alternative construction of the Hölz design and a subdesign. Furthermore we theoretically prove that the Hölz design on 28 points only contains Hermitian and Ree unitals (previously shown by Tonchev by computer). As these Hölz designs are one-point extensions of generalized quadrangles, we dedicate a final chapter to the characterization of the affine extension of H(2) using a combinatorial property

    Intriguing sets of strongly regular graphs and their related structures

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    In this paper we outline a technique for constructing directed strongly regular graphs by using strongly regular graphs having a "nice" family of intriguing sets. Further, we investigate such a construction method for rank three strongly regular graphs having at most 4545 vertices. Finally, several examples of intriguing sets of polar spaces are provided

    Hemisystems of small flock generalized quadrangles

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    In this paper, we describe a complete computer classification of the hemisystems in the two known flock generalized quadrangles of order (52,5)(5^2,5) and give numerous further examples of hemisystems in all the known flock generalized quadrangles of order (s2,s)(s^2,s) for s11s \le 11. By analysing the computational data, we identify two possible new infinite families of hemisystems in the classical generalized quadrangle H(3,q2)H(3,q^2).Comment: slight revisions made following referee's reports, and included raw dat

    Characterising substructures of finite projective spaces

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    Part I:

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