The pseudo-hyperplanes and homogeneous pseudo-embeddings of \(\mathrm{AG}(n, 4)\) and \(\mathrm{PG}(n, 4)\)
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.
scientific article; zbMATH DE number 6062567