Every generalized quadrangle of order 5 having a regular point is symplectic

For many years now, one of the most important open problems in the theory of generalized quadrangles has been whether other classes of generalized quadrangles exist besides those that are currently known. This paper reports on an unsuccessful attempt to construct a new generalized quadrangle. As a byproduct of our attempt, however, we obtain the following new characterization result: every generalized quadrangle of order 5 that has at least one regular point is isomorphic to the quadrangle W(5) arising from a symplectic polarity of PG(3, 5). During the classification process, we used the computer algebra system GAP to perform certain computations or to search for an optimal strategy for the proof

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

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

Singer quadrangles

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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

On the Pauli graphs of N-qudits

A comprehensive graph theoretical and finite geometrical study of the commutation relations between the generalized Pauli operators of N-qudits is performed in which vertices/points correspond to the operators and edges/lines join commuting pairs of them. As per two-qubits, all basic properties and partitionings of the corresponding Pauli graph are embodied in the geometry of the generalized quadrangle of order two. Here, one identifies the operators with the points of the quadrangle and groups of maximally commuting subsets of the operators with the lines of the quadrangle. The three basic partitionings are (a) a pencil of lines and a cube, (b) a Mermin's array and a bipartite-part and (c) a maximum independent set and the Petersen graph. These factorizations stem naturally from the existence of three distinct geometric hyperplanes of the quadrangle, namely a set of points collinear with a given point, a grid and an ovoid, which answer to three distinguished subsets of the Pauli graph, namely a set of six operators commuting with a given one, a Mermin's square, and set of five mutually non-commuting operators, respectively. The generalized Pauli graph for multiple qubits is found to follow from symplectic polar spaces of order two, where maximal totally isotropic subspaces stand for maximal subsets of mutually commuting operators. The substructure of the (strongly regular) N-qubit Pauli graph is shown to be pseudo-geometric, i. e., isomorphic to a graph of a partial geometry. Finally, the (not strongly regular) Pauli graph of a two-qutrit system is introduced; here it turns out more convenient to deal with its dual in order to see all the parallels with the two-qubit case and its surmised relation with the generalized quadrangle Q(4, 3), the dual ofW(3).Comment: 17 pages. Expanded section on two-qutrits, Quantum Information and Computation (2007) accept\'

Black Hole Entropy and Finite Geometry

It is shown that the $E_{6(6)}$ symmetric entropy formula describing black holes and black strings in D=5 is intimately tied to the geometry of the generalized quadrangle GQ$(2,4)$ with automorphism group the Weyl group $W(E_6)$. The 27 charges correspond to the points and the 45 terms in the entropy formula to the lines of GQ$(2,4)$. Different truncations with $15, 11$ and 9 charges are represented by three distinguished subconfigurations of GQ$(2,4)$, well-known to finite geometers; these are the "doily" (i. e. GQ$(2,2)$) with 15, the "perp-set" of a point with 11, and the "grid" (i. e. GQ$(2,1)$) with 9 points, respectively. In order to obtain the correct signs for the terms in the entropy formula, we use a non- commutative labelling for the points of GQ$(2,4)$. For the 40 different possible truncations with 9 charges this labelling yields 120 Mermin squares -- objects well-known from studies concerning Bell-Kochen-Specker-like theorems. These results are connected to our previous ones obtained for the $E_{7(7)}$ symmetric entropy formula in D=4 by observing that the structure of GQ$(2,4)$ is linked to a particular kind of geometric hyperplane of the split Cayley hexagon of order two, featuring 27 points located on 9 pairwise disjoint lines (a distance-3-spread). We conjecture that the different possibilities of describing the D=5 entropy formula using Jordan algebras, qubits and/or qutrits correspond to employing different coordinates for an underlying non-commutative geometric structure based on GQ$(2,4)$.Comment: 17 pages, 3 figures, v2 a new paragraph added, typos correcte

Abstract hyperovals, partial geometries, and transitive hyperovals

Includes bibliographical references.2015 Summer.A hyperoval is a (q+2)- arc of a projective plane π, of order q with q even. Let G denote the collineation group of π containing a hyperoval Ω. We say that Ω is transitive if for any pair of points x, y is an element of Ω, there exists a g is an element of G fixing Ω setwise such that xg = y. In1987, Billotti and Korchmaros proved that if 4||G|, then either Ω is the regular hyperoval in PG(2,q) for q=2 or 4 or q = 16 and |G||144. In 2005, Sonnino proved that if |G| = 144, then π is desarguesian and Ω is isomorphic to the Lunelli-Sce hyperoval. For our main result, we show that if G is the collineation group of a projective plane containing a transitivehyperoval with 4 ||G|, then |G| = 144 and Ω is isomorphic to the Lunelli-Sce hyperoval. We also show that if A(X) is an abstract hyperoval of order n ≡ 2(mod 4); then |Aut(A(X))| is odd. If A(X) is an abstract hyperoval of order n such that Aut(A(X)) contains two distinct involutions with |FixX(g)| and |FixX(ƒ)| ≥ 4. Then we show that FixX(g) ≠ FixX(ƒ). We also show that there is no hyperoval of order 12 admitting a group whose order is divisible by 11 or 13, by showing that there is no partial geometry pg(6, 10, 5) admitting a group of order 11 or of order 13. Finally, we were able to show that there is no hyperoval in a projective plane of order 12 with a dihedral subgroup of order 14, by showing that that there is no partial geometry pg(7, 12, 6) admitting a dihedral group of order 14. The latter results are achieved by studying abstract hyperovals and their symmetries