A New Family of Extended Generalized Quadrangles

A Construction of Extended Generalized Quadrangles Using the Veronesean

Given a finite generalized quadrangle Q, an extended generalized quadrangle is called an extension of Q if each point-residue is isomorphic to Q. There are several known examples of extended generalized quadrangles [4, §1.1], among which those admitting flag-transitive automorphism groups have been extensively studied in their close relations with group theory. Those flag-transitive geometries ...

Characterizations of Generalized Quadrangles by Generalized Homologies

Let S= (P, B, I) be a generalized quadrangle of order (s, t). For X, y E P, X+ y. let X(x, y) be the group of all collineations of S fixing x and y linewise. If ZE (4 YJL> then the set of all points incident with the line xz (resp. y;) is denoted by z(resp. 3. The generalized quadrangle S = (P, B, I) is said to be (x, y)-transitive, x+ y, if X(x, y) is transitive on each set Z{x, z} and jZ’-{ y...

Finite Generalized Quadrangles

Foundations of elation generalized quadrangles

In these notes we are interested in the following fundamental question: “Given a thick generalized quadrangle S(x) with elation point (respectively center of transitivity) x , when does the set of all elations about x form a group?”. It was the general belief for a long time that this is usually the case. However, the goal of these notes is to show that this belief is wrong. To that end, we wil...

Authentication codes from generalized quadrangles

Authentication codes were introduced by Simmons in [3]. Many combinatorial structures can be used to construct authentication codes, and interesting combinatorial bounds can be obtained, see e.g. [7], [8] and [9]. We investigate authentication codes arising from generalized quadrangles (GQs), which was first done in [1]. The use of intricated techniques and constructions from the theory of GQs ...