11 research outputs found
Translation planes of order 23^2
We give a complete classication of translation planes of order 23^2 whose translation complement contains a subgroup G such that the quotient group G modulo scalars is isomorphic to A_6. Up to isomorphisms, there are exactly 23 such planes and six of them have a larger translation complement being modulo scalars isomorphic to S_6
Intriguing sets of strongly regular graphs and their related structures
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 vertices. Finally, several examples of intriguing sets of polar spaces are provided
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)
On 4-general sets in finite projective spaces
A -general set in is a set of points of
spanning the whole and such that no four of them are on a
plane. Such a pointset is said to be complete if it is not contained in a
larger -general set of . In this paper upper and lower
bounds for the size of the largest and the smallest complete -general set in
, respectively, are investigated. Complete -general sets in
, , whose size is close to the theoretical upper
bound are provided. Further results are also presented, including a description
of the complete -general sets in projective spaces of small dimension over
small fields and the construction of a transitive -general set of size in ,
Singer quadrangles
[no abstract available
Characterisations of finite egglike inversive planes
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