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)]

Hyperplanes of symplectic dual polar spaces : a survey

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Constant rank-distance sets of hermitian matrices and partial spreads in hermitian polar spaces

In this paper we investigate partial spreads of $H(2n-1,q^2)$ through the related notion of partial spread sets of hermitian matrices, and the more general notion of constant rank-distance sets. We prove a tight upper bound on the maximum size of a linear constant rank-distance set of hermitian matrices over finite fields, and as a consequence prove the maximality of extensions of symplectic semifield spreads as partial spreads of $H(2n-1,q^2)$. We prove upper bounds for constant rank-distance sets for even rank, construct large examples of these, and construct maximal partial spreads of $H(3,q^2)$ for a range of sizes

Tight sets in finite classical polar spaces

We show that every i-tight set in the Hermitian variety H(2r + 1, q) is a union of pairwise disjoint (2r + 1)-dimensional Baer subgeometries PG(2r + 1, root q) and generators of H(2r + 1, q), if q >= 81 is an odd square and i < (q(2/3) - 1)/2. We also show that an i-tight set in the symplectic polar space W(2r + 1, q) is a union of pairwise disjoint generators of W(2r + 1, q), pairs of disjoint r-spaces {Delta,Delta(perpendicular to)}, and (2r + 1)-dimensional Baer subgeometries. For W(2r + 1, q) with r even, pairs of disjoint r-spaces {Delta,Delta(perpendicular to)} cannot occur. The (2r + 1)-dimensional Baer subgeometries in the i-tight set of W(2r + 1, q) are invariant under the symplectic polarity perpendicular to of W(2r + 1, q) or they arise in pairs of disjoint Baer subgeometries corresponding to each other under perpendicular to. This improves previous results where i < q(5/8)/root 2+ 1 was assumed. Generalizing known techniques and using recent results on blocking sets and minihypers, we present an alternative proof of this result and consequently improve the upper bound on i to (q(2/3) - 1)/2. We also apply our results on tight sets to improve a known result on maximal partial spreads in W(2r + 1, q)

Points and hyperplanes of the universal embedding space of the dual polar space DW(5,q), q odd

It was proved earlier that there are 6 isomorphism classes of hyperplanes in the dual polar space (5,q)$,$ even, which arise from its Grassmann-embedding. In the present paper, we extend these results to the case that $is odd. Specifically, we determine the orbits of the full automorphism group of (5,q)$, $odd, on the projective points (or equivalently, the hyperplanes) of the projective space (13,q)$ which affords the universal embedding of (5,q)$

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

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

The uniqueness of the SDPS-set of the symplectic dual polar space DW(4n−1,q)DW(4n-1,q), n≥2n \geq 2

SDPS-sets are very nice sets of points in dual polar spaces which themselves carry the structure of dual polar spaces. They were introduced in \cite{DB-V:2} because they gave rise to new valuations and hyperplanes of dual polar spaces. In the present paper, we show that the symplectic dual polar space (4n-1,q)$, \geq 2$, has up to isomorphisms a unique SDPS-set

A combinatorial characterisation of embedded polar spaces

Some classical polar spaces admit polar spaces of the same rank as embedded polar spaces (often arisen as the intersection of the polar space with a non-tangent hyperplane). In this article we look at sets of generators that behave combinatorially as the set of generators of such an embedded polar space, and we prove that they are the set of generators of an embedded polar space