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Finite geometry
About: Finite geometry is a research topic. Over the lifetime, 909 publications have been published within this topic receiving 17816 citations.
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24 Jan 1980
TL;DR: The first properties of the plane can be found in this article, where the authors define the following properties: 1. Finite fields 2. Projective spaces and algebraic varieties 3. Subspaces 4. Partitions 5. Canonical forms for varieties and polarities 6. The line 7. Ovals 9. Arithmetic of arcs of degree two 10. Cubic curves 12. Arcs of higher degree 13. Blocking sets 14. Small planes 15.
Abstract: 1. Finite fields 2. Projective spaces and algebraic varieties 3. Subspaces 4. Partitions 5. Canonical forms for varieties and polarities 6. The line 7. First properties of the plane 8. Ovals 9. Arithmetic of arcs of degree two 10. Arcs in ovals 11. Cubic curves 12. Arcs of higher degree 13. Blocking sets 14. Small planes Appendix Notation References
1,593 citations
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TL;DR: In this article, it was shown that there is always at least one collineation of period q with respect to any point in the projective plane PG(2, pn) for every prime p and positive integer n.
Abstract: A point in a finite projective plane PG(2, pn), may be denoted by the symbol (Xl, X2, X3), where the coordinates x1, X2, X3 are marks of a Galois field of order pn, GF(pn). The symbol (0, 0, 0) is excluded, and if k is a non-zero mark of the GF(pn), the symbols (X1, X2, X3) and (kxl, kx2, kx3) are to be thought of as the same point. The totality of points whose coordinates satisfy the equation ulxl+u2x2+U3x3 = 0, where u1, U2, u3 are marks of the GF(pn), not all zero, is called a line. The plane then consists of p2n +pn + 1 = q points and q lines; each line contains pn+1 points.t A finite projective plane, PG(2, pn), defined in this way is Pascalian and Desarguesian; it exists for every prime p and positive integer n, and there is only one such PG(2, pn) for a given p and n (VB, p. 247, VY, p. 151). Let Ao be a point of a given PG(2, pn), and let C be a collineation of the points of the plane. (A collineation is a 1-1 transformation carrying points into points and lines into lines.) Suppose C carries Ao into Al, A1 into A2,... , Ak into Ao; or, denoting the product C C by C2, C. C2 by C3, etc., we have C(Ao) =A1, C2(Ao) =A2, . . , Ck(A o) =A o. If k is the smallest positive integer for which C k(A o) =Ao, we call k the period of C with respect to the point A o. If the period of a collineation C with respect to a point Ao is q (=p2n+pn+l), then the period of C with respect to any point in the plane is q, and in this case we will call C simply a collineation of period q. We prove in the first theorem that there is always at least one collineation of period q, and from it we derive some results of interest in finite geometry and number theory. Let
794 citations
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20 Feb 1986
TL;DR: Projective Geometrics Over Finite Fields (OUP, 1979) as mentioned in this paper considers projective spaces of three dimensions over a finite field and examines properties of four and five dimensions, fundamental applications to translation planes, simple groups, and coding theory.
Abstract: This self-contained and highly detailed study considers projective spaces of three dimensions over a finite field It is the second and core volume of a three-volume treatise on finite projective spaces, the first volume being Projective Geometrics Over Finite Fields (OUP, 1979) The present work restricts itself to three dimensions, and considers both topics which are analogous of geometry over the complex numbers and topics that arise out of the modern theory of incidence structures The book also examines properties of four and five dimensions, fundamental applications to translation planes, simple groups, and coding theory
713 citations
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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