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

Bruyn, Bart De

doi:10.1007/s10623-012-9705-3

Cited by 8 publications

(26 citation statements)

References 14 publications

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“…In De Bruyn [2], we used the computer algebra system GAP [3] to show that Q(4, 3) has, up to isomorphism, two homogeneous pseudo-embeddings, the universal pseudo-embedding in PG(14, 2) and a certain homogeneous pseudo-embedding in PG (8,2). No direct constructions for these two homogeneous embeddings were however given in [2].…”

Section: Basic Definitions and Main Resultsmentioning

confidence: 99%

“…Let S be a point-line geometry with the property that the number of points on each line is finite and at least three, and let G be a group of automorphisms of S. In this section, we give a criterion, proved in De Bruyn [2], to decide whether a given pseudo-embedding of S is G-homogeneous. This criterion was used in [2] to determine all homogeneous pseudo-embeddings of all generalized quadrangles of order (3, t).…”

Section: The Recognition Of G-homogeneous Pseudo-embeddingsmentioning

confidence: 99%

“…This criterion was used in [2] to determine all homogeneous pseudo-embeddings of all generalized quadrangles of order (3, t). In the present paper, we will use this criterion to determine all homogeneous pseudo-embeddings of PG(n, 4) and AG(n, 4).…”

Section: The Recognition Of G-homogeneous Pseudo-embeddingsmentioning

confidence: 99%

“…In the present paper, we will use this criterion to determine all homogeneous pseudo-embeddings of PG(n, 4) and AG(n, 4). While the classification of the homogeneous pseudo-embeddings in [2] needed the use of a computer (GAP), the classification of the homogeneous pseudo-embeddings in the present paper will be computer free.…”

Section: The Recognition Of G-homogeneous Pseudo-embeddingsmentioning

confidence: 99%

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

Bruyn

2011

Des. Codes Cryptogr.

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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.Keywords: homogeneous pseudo-embedding, pseudo-hyperplane MSC2000: 51E20, 05B25 Basic definitions and main resultsThe aim of this section is to state the main results of this paper and to define the basic notions which are necessary to understand these results. Throughout this section, S = (P, L, I) is a point-line geometry with the property that the number of points on each line is finite and at least three.Suppose V is a vector space over the field F 2 of order 2. A pseudo-embedding of S into the projective space Σ = PG(V ) is a mapping e from P to the point set of Σ satisfying: (1) < e(P) > Σ = Σ; (2) if L is a line of S with points x 1 , x 2 , . . . , x k , then the points e(x 1 ), e(x 2 ), . . . , e(x k−1 ) of Σ are linearly independent and e(x k ) = wherev i , i ∈ {1, 2, . . . , k − 1}, is the unique vector of V for which e(x i ) = Σ . Two pseudo-embeddings e 1 : S → Σ 1 and e 2 : S → Σ 2 of S are called isomorphic (e 1 ∼ = e 2 ) if there exists an isomorphism φ : Σ 1 → Σ 2 such that e 2 = φ • e 1 . The notion pseudoembedding was introduced in De Bruyn [1].Suppose e : S → PG(V ) is a pseudo-embedding of S and G is a group of automorphisms of S. We say that e is G-homogeneous if for every θ ∈ G, there exists a (necessarily unique) projectivity η θ of PG(V ) such that e(x θ ) = e(x) η θ for every point x of S. If G is the full automorphism group of S, then e is also called a homogeneous pseudo-embedding.Suppose e : S → Σ is a pseudo-embedding of S and α is a subspace of Σ satisfying the following two properties: 1

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Section: Basic Definitions and Main Resultsmentioning

confidence: 99%

Section: The Recognition Of G-homogeneous Pseudo-embeddingsmentioning

confidence: 99%

Section: The Recognition Of G-homogeneous Pseudo-embeddingsmentioning

confidence: 99%

Section: The Recognition Of G-homogeneous Pseudo-embeddingsmentioning

confidence: 99%

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

Bruyn

2011

Des. Codes Cryptogr.

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“…We hope that the theory of pseudo-embeddings and pseudo-hyperplanes will find further applications in related areas. Some of these applications will occur in the forthcoming paper [11], where we will study the pseudo-hyperplanes and pseudo-embeddings of generalized quadrangles of order (3, t), together with some of their applications to so-called m-ovoids and tight sets.…”

Section: Introductionmentioning

confidence: 99%

Pseudo-embeddings and pseudo-hyperplanes

Bruyn¹

2013

Advances in Geometry

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We generalize some known results regarding hyperplanes and projective embeddings of point-line geometries with three points per line to geometries with an arbitrary but finite number of points on each line. In order to generalize these results, we need to introduce the new notions of pseudo-hyperplane, (universal) pseudo-embedding, pseudo-embedding rank and pseudo-generating rank. We prove several connections between these notions and address the problem of the existence of (certain) pseudo-embeddings. We apply this to several classes of point-line geometries. We also determine the pseudo-embedding rank and the pseudo-generating rank of the projective space PG(n, 4) and the affine space AG(n, 4).

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