The pseudo-hyperplanes and homogeneous pseudo-embeddings of \(\mathrm{AG}(n, 4)\) and \(\mathrm{PG}(n, 4)\)
type: Item
instance of: scholarly article
title: The pseudo-hyperplanes and homogeneous pseudo-embeddings of \(\mathrm{AG}(n, 4)\) and \(\mathrm{PG}(n, 4)\)
zbMATH Open document ID: 1248.51001
author: Bart De Bruyn
published in: Designs, Codes and Cryptography
publication date: 2012-07-31T00:00:00Z
review text: A pseudo-embedding from a geometry \(S\) into the projective space \(\Sigma\) is a mapping \(e\) from the point set \(P\) of \(S\) to the point set of the projective space, satisfying \(\langle e(P) \rangle _{\Sigma} = \Sigma\), and if \(L\) is a line of \(S\) with points \(x_1,\dots,x_k\), then their images are linearly independent with \(e(x_k) = \langle \overline{v_1} + \overline{v_2} + \dots + \overline {v_{k-1}} \rangle\), where \(\overline{v_i}\) is the unique vector for which \(e(x_i) = \langle \overline{v_i} \rangle_{\Sigma}.\) The author determines all homogeneous pseudo-embeddings of the affine space \(\mathrm{AG}(n, 4)\) and the projective space \(\mathrm{PG}(n, 4)\). While previous work was computer dependent, the classification of the homogeneous pseudo-embeddings here are computer free. A pseudo-hyperplane is a proper subset \(H\) of the point set such that every line contains an even number of points in the compliment of the subset. The author gives a classification of all pseudo-hyperplanes of \(\mathrm{AG}(n, 4)\) and proves that the homogeneous pseudo-embeddings of the generalized quadrangle \(Q(4,3)\) are induced by the two homogeneous pseudo-embeddings of \(\mathrm{AG}(4, 4)\) into which \(Q(4,3)\) is fully embeddable.
reviewed by: Steven T. Dougherty
Mathematics Subject Classification ID: 51A45
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 51E12
zbMATH DE Number: 6062567
zbMATH Keywords: homogeneous pseudo-embedding
zbMATH Keywords: pseudo-hyperplane
zbMATH Keywords: projective space
zbMATH Keywords: affine space
zbMATH Keywords: generalized quadrangle
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: s10623-011-9577-y
OpenAlex ID: W1974548643
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DOI: 10.1007/S10623-011-9577-Y
description: scientific article; zbMATH DE number 6062567