Showing papers by "Joseph A. Thas published in 2007"

The uniqueness of 1-systems of W5(q) satisfying the BLT-property, with q odd

Abstract: In the symplectic polar space W 5(q) every 1-system which satisfies the BLT-property (and then q is odd) defines a generalized quadrangle (GQ) of order (q 2,q 3). In this paper, we show that this 1-system is unique, so that the only GQ arising in this way is isomorphic to the classical GQ H(4,q 2), q odd.

Finite geometries: classical problems and recent developments

Abstract: In recent years there has been an increasing interest in finite projective spaces, and important applications to practical topics such as coding theory, cryptography and design of experiments have made the field even more attractive Pioneering work has been done by B Segre and each of the four topics of this paper is related to his work; two classical problems and two recent developments will be discussed First I will mention a purely combinatorial characterization of Hermitian curves in PG(2, q); here, from the beginning, the considered pointset is contained in PG(2, q) A second approach is where the object is described as an incidence structure satisfying certain properties; here the geometry is not a priori embedded in a projective space This will be illustrated by a characterization of the classical inversive plane in the odd case A recent beautiful result in Galois geometry is the discovery of an infinite class of hemisystems of the Hermitian variety in PG(3, q), leading to new interesting classes of incidence structures, graphs and codes; before this result, just one example for GF(9), due to Segre, was known An exemplary example of research combining combinatorics, incidence geometry, Galois geometry and group theory is the determination of embeddings of generalized polygons in finite projective spaces As an illustration I will discuss the embedding of the generalized quadrangle of order (4,2), that is, the Hermitian variety H(3, 4), in PG(3,K) with K any commutative field