Hyperplanes of symplectic dual polar spaces : a survey

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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 $45$ vertices. Finally, several examples of intriguing sets of polar spaces are provided

MUBs inequivalence and affine planes

There are fairly large families of unitarily inequivalent complete sets of N+1 mutually unbiased bases (MUBs) in C^N for various prime powers N. The number of such sets is not bounded above by any polynomial as a function of N. While it is standard that there is a superficial similarity between complete sets of MUBs and finite affine planes, there is an intimate relationship between these large families and affine planes. This note briefly summarizes "old" results that do not appear to be well-known concerning known families of complete sets of MUBs and their associated planes.Comment: This is the version of this paper appearing in J. Mathematical Physics 53, 032204 (2012) except for format changes due to the journal's style policie

Automorphisms and opposition in spherical buildings of classical type

An automorphism of a spherical building is called domestic if it maps no chamber to an opposite chamber. In this paper we classify domestic automorphisms of spherical buildings of classical type

Symplectic polarities of buildings of type E₆

A symplectic polarity of a building Delta of type E (6) is a polarity whose fixed point structure is a building of type F (4) containing residues isomorphic to symplectic polar spaces. In this paper, we present two characterizations of such polarities among all dualities. Firstly, we prove that, if a duality theta of Delta never maps a point to a neighbouring symp, and maps some element to a non-opposite element, then theta is a symplectic duality. Secondly, we show that, if a duality theta never maps a chamber to an opposite chamber, then it is a symplectic polarity. The latter completes the programme for dualities of buildings of type E (6) of determining all domestic automorphisms of spherical buildings, and it also shows that symplectic polarities are the only polarities in buildings of type E (6) for which the Phan geometry is empty

On minimum size blocking sets of the outer tangents to a hyperbolic quadric in PG(3, q)

Let Q(+)(3, q) be a hyperbolic quadric in PG(3, q) and T-1 be the set of all lines of PG(3, q) meeting Q(+)(3, q) in singletons (the so-called outer tangents). If k is the minimum size of a T-1-blocking set in PG(3, q), then we prove that k >= q(2) - 1. It is known that there is no T-1-blocking set of size q(2) - 1 for q > 2 even and that there is a unique (up to isomorphism) T-1-blocking set of size 3 for q = 2. For q = 3, we prove as well that there is a unique T-1-blocking set of size 8. Using a computer, we also classify all T-1-blocking sets of size q(2) - 1 for each prime power q <= 13. On basis of some structural similarities we are subsequently able to recognize three families of blocking sets whose further study shows that they can be constructed from certain objects related to finite fields (like nice subsets or permutations of the latter). This connection with finite fields allows us to obtain some computer free descriptions. (C) 2018 Elsevier Inc. All rights reserved

An infinite family of hyperovals of Q+(5,q)Q^+(5,q), qq even

We construct an infinite family of hyperovals on the Klein quadric $Q^+(5,q)$, $q$ even. The construction makes use of ovoids of the symplectic generalized quadrangle $W(q)$ that is associated with an elliptic quadric which arises as solid intersection with $Q^+(5,q)$. We also solve the isomorphism problem: we determine necessary and sufficient conditions for two hyperovals arising from the construction to be isomorphic