Partial ovoids and partial spreads in symplectic and orthogonal polar spaces

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)]

Some non-existence results for distance-jj ovoids in small generalized polygons

We give a computer-based proof for the non-existence of distance-$2$ ovoids in the dual split Cayley hexagon $\mathsf{H}(4)^D$. Furthermore, we give upper bounds on partial distance-$2$ ovoids of $\mathsf{H}(q)^D$ for $q \in \{2, 4\}$.Comment: 10 page

Ovoids and spreads of finite classical generalized hexagons and applications

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

Dense near octagons with four points on each line, III

This is the third paper dealing with the classification of the dense near octagons of order (3, t). Using the partial classification of the valuations of the possible hexes obtained in [12], we are able to show that almost all such near octagons admit a big hex. Combining this with the results in [11], where we classified the dense near octagons of order (3, t) with a big hex, we get an incomplete classification for the dense near octagons of order (3, t): There are 28 known examples and a few open cases. For each open case, we have a rather detailed description of the structure of the near octagons involved

Hyperplanes of symplectic dual polar spaces : a survey

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