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Joseph A. Thas

Other affiliations: Miami University
Bio: Joseph A. Thas is an academic researcher from Ghent University. The author has contributed to research in topics: Generalized quadrangle & Projective space. The author has an hindex of 32, co-authored 204 publications receiving 4887 citations. Previous affiliations of Joseph A. Thas include Miami University.


Papers
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MonographDOI
08 Apr 2009

888 citations

BookDOI
01 Jan 2016
TL;DR: In this paper, the authors define Hermitian varieties, Grassmann varieties, Veronese and Segre varieties, and embedded geometries for finite projective spaces of three dimensions.
Abstract: Terminology Quadrics Hermitian varieties Grassmann varieties Veronese and Segre varieties Embedded geometries Arcs and caps Appendix VI. Ovoids and spreads of finite classical polar spaces Appendix VII. Errata for Finite projective spaces of three dimensions and Projective geometries over finite fields Bibliography Index of notation Author index General index.

647 citations

Journal ArticleDOI
Joseph A. Thas1
TL;DR: It is proved that with a set of q upper triangular 2 × 2-matrices over GF(q) of a certain type, there corresponds a generalized quadrangle of order (q2, q) and with each flock of the quadratic cone there corresponds such a setof matrices.
Abstract: A flock of the quadratic cone K of PG(3, q) is a partition of K but its vertex into disjoint conics. If the planes of the q conics of such a flock all contain a common line, then the flock is called linear. With any flock there corresponds a translation plane which is Desarguesian iff the flock is linear. W. M. Kantor showed that with a set of q upper triangular 2 × 2-matrices over GF(q) of a certain type, there corresponds a generalized quadrangle of order (q2, q). We prove that with such a set of q matrices there corresponds a flock of the quadratic cone of PG(3, q), and conversely that with each flock of the quadratic cone there corresponds such a set of matrices. Using this relationship, new flocks, new generalized quadrangles, and probably new translation planes are obtained.

173 citations

Journal ArticleDOI
Joseph A. Thas1
TL;DR: In this article, it was shown that the polar space Q(2n, q) arising from a non-singular elliptic quadric has no ovoid, and the polar spaces H(n,q), q even, and H(q, q even) have no ovo.
Abstract: LetP be a finite classical polar space of rankr, r⩾2. An ovoidO ofP is a pointset ofP, which has exactly one point in common with every totally isotropic subspace of rankr. It is proved that the polar spaceW n (q) arising from a symplectic polarity ofPG(n, q), n odd andn > 3, that the polar spaceQ(2n, q) arising from a non-singular quadric inPG(2n, q), n > 2 andq even, that the polar space Q−(2n + 1,q) arising from a non-singular elliptic quadric inPG(2n + 1,q), n > 1, and that the polar spaceH(n,q 2) arising from a non-singular Hermitian variety inPG(n, q 2)n even andn > 2, have no ovoids. LetS be a generalized hexagon of ordern (⩾1). IfV is a pointset of order n3 + 1 ofS, such that every two points are at distance 6, thenV is called an ovoid ofS. IfH(q) is the classical generalized hexagon arising fromG 2 (q), then it is proved thatH(q) has an ovoid iffQ(6, q) has an ovoid. There follows thatQ(6, q), q=32h+1, has an ovoid, and thatH(q), q even, has no ovoid. A regular system of orderm onH(3,q 2) is a subsetK of the lineset ofH(3,q 2), such that through every point ofH(3,q 2) there arem (> 0) lines ofK. B. Segre shows that, ifK exists, thenm=q + 1 or (q + l)/2.If m=(q + l)/2,K is called a hemisystem. The last part of the paper gives a very short proof of Segre's result. Finally it is shown how to construct the 4-(11, 5, 1) design out of the hemisystem with 56 lines (q=3).

137 citations

Book ChapterDOI
Joseph A. Thas1
01 Jan 1995
TL;DR: In this article, the authors focus on projective geometry over a finite field and define a k-arc in projective plane, PG (n, q) is a set K of k points with k ≥ n + 1 such that no n+ 1 points of K lie in a hyperplane.
Abstract: Publisher Summary This chapter focuses on projective geometry over a finite field. A k-arc in projective plane, PG (n, q) is a set K of k points with k ≥ n + 1 such that no n + 1 points of K lie in a hyperplane. An arc K is complete if it is not properly contained in a larger arc. A normal rational curve of PG(2, q) is an irreducible conic; a normal rational curve of PG(3, q) is a twisted cubic. It is well known that any (n + 3)-arc of PG(n, q) is contained in a unique normal rational curve of this space. For q > n + 1, the osculating hyperplane of the normal rational curve C at the point x ϵ C is the unique hyperplane through x intersecting C at x with multiplicity n. In PG(n, q), n ≥ 3, a set K of k points no three of which are collinear is a k-cap. A k-cap is complete if it is not contained in a (k + 1)-cap. A line of PG(n, q) is a secant, a tangent, or an external line of a k-cap as it meets K in 2, 1, or 0 points.

111 citations


Cited by
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Book ChapterDOI
01 Jan 2007

1,089 citations

MonographDOI
08 Apr 2009

888 citations

Book
01 Jan 2002
TL;DR: In this paper, the value of the variable in each equation is determined by a linear combination of the values of the variables in the equation and the variable's value in the solution.
Abstract: Determine the value of the variable in each equation.

635 citations

Journal ArticleDOI
TL;DR: On etudie les relations entre les codes [n,k] lineaires a deux poids, les ensembles projectifs et certains graphes fortement reguliers as mentioned in this paper.
Abstract: On etudie les relations entre les codes [n,k] lineaires a deux poids, les ensembles (n,k,h 1 h 2 ) projectifs et certains graphes fortement reguliers

609 citations

Journal ArticleDOI
TL;DR: For almost all graphs the answer to the question in the title is still unknown as mentioned in this paper, and the cases for which the answer is known are surveyed in the survey of cases where the Laplacian matrix is known.

605 citations