The pseudo-hyperplanes and homogeneous pseudo-embeddings of AG(n, 4) and PG(n, 4)

We determine all homogeneous pseudo-embeddings of the affine space AG(n, 4) and the projective space PG(n, 4). We give a classification of all pseudo-hyperplanes of AG(n, 4). We also prove that the two homogeneous pseudo-embeddings of the generalized quadrangle Q(4, 3) are induced by the two homogeneous pseudo-embeddings of AG(4, 4) into which Q(4, 3) is fully embeddable

The pseudo-hyperplanes and homogeneous pseudo-embeddings of the generalized quadrangles of order (3, t)

In the paper "as reported by De Bruyn (Adv Geom, to appear)", we introduced the notions of pseudo-hyperplane and pseudo-embedding of a point-line geometry and proved that every generalized quadrangle of order (s, t), 2 a parts per thousand currency sign s < a, has faithful pseudo-embeddings. The present paper focuses on generalized quadrangles of order (3, t). Using the computer algebra system GAP and invoking some theoretical relationships between pseudo-hyperplanes and pseudo-embeddings obtained in "De Bruyn (Adv Geom, to appear)", we are able to give a complete classification of all pseudo-hyperplanes of . We hereby find several new examples of tight sets of generalized quadrangles, as well as a complete classification of all 2-ovoids of . We use the classification of the pseudo-hyperplanes of to obtain a list of all homogeneous pseudo-embeddings of

The homogeneous pseudo-embeddings and hyperovals of the generalized quadrangle H(3,4)

In this paper, we determine all homogeneous pseudo-embeddings of the generalized quadrangle H(3, 4) and give a description of all its even sets. Using this description, we subsequently compute all hyperovals of H(3, 4), up to isomorphism, and give computer free descriptions of them. Several of these hyperovals, but not all of them, have already been described before in the literature. (C) 2020 Elsevier Inc. All rights reserved

Pseudo-embeddings of the (point, k-spaces)-geometry of PG(n, 2) and projective embeddings of DW(2n-1, 2)

We classify all homogeneous pseudo-embeddings of the point-line geometry defined by the points and k-dimensional subspaces of PG(n, 2), and use this to study the local structure of homogeneous full projective embeddings of the dual polar space DW(2n-1, 2). Our investigation allows us to distinguish n possible types for such homogeneous embeddings. For each of these n types, we construct a homogeneous full projective embedding of DW(2n - 1, 2)

On four codes with automorphism group P Sigma L(3,4) and pseudo-embeddings of the large Witt designs

A pseudo-embedding of a point-line geometry is a representation of the geometry into a projective space over the field F-2 such that every line corresponds to a frame of a subspace. Such a representation is called homogeneous if every automorphism of the geometry lifts to an automorphism of the projective space. In this paper, we determine all homogeneous pseudo-embeddings of the three Witt designs that arise by extending the projective plane PG(2, 4). Along our way, we come across some codes with automorphism group P Sigma L(3, 4) and sets of points of PG(2, 4) that have a particular intersection pattern with Baer subplanes or hyperovals

Veronese representation of projective Hjelmslev planes over some quadratic alternative algebras

We geometrically characterise the Veronese representations of ring projective planes over algebras which are analogues of the dual numbers, giving rise to projective Hjelmslev planes of level 2 coordinatised over quadratic alternative algebras. These planes are related to affine buildings of relative type Ã_2 and respective absolute type Ã_2, Ã_5 and Ẽ_6

Characterisations and classifications in the theory of parapolar spaces

This thesis in incidence geometry is divided into two parts, which can both be linked to the geometries of the Freudenthal-Tits magic square. The first and main part consists of an axiomatic characterisation of certain plane geometries, defined via the Veronese mapping using degenerate quadratic alternative algebras (over any field) with a radical that is (as a ring) generated by a single element. This extends and complements earlier results of Schillewaert and Van Maldeghem, who considered such geometries over non-degenerate quadratic alternative algebras. The second and smaller part deals with a classification of parapolar spaces exhibiting the feature that the dimensions of intersections of pairs of symplecta cannot take all possible sensible values, with the only further requirement that, if the parapolar spaces have symplecta of rank 2, then they are strong. This part is based on a joint work with Schillewaert, Van Maldeghem and Victoor