Editorial: Finite geometries
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Editorial: Finite geometries D. Ghinelli · J. W. P. Hirschfeld · D. Jungnickel · J. A. Thas
Received: 24 November 2012 / Accepted: 28 November 2012 / Published online: 18 December 2012 © Springer Science+Business Media New York 2012
Finite geometries are of modern interest for both their pure and applied aspects. Much of the investigation concerns combinatorial structures in finite projective and affine spaces: these include arcs, blocking sets, caps, curves, hypersurfaces, ovoids, spreads and partial spreads. The connections to diagram geometries, graphs, groups, and linear codes provide many interlinked problems, many of which remain to be solved. This volume contains 26 papers covering a significant range of material. They may be divided into seven categories with their numbers of papers as follows, although many of the papers cover more than one of these categories. (1) Finite Projective Spaces of Three or More Dimensions, 5: De Beule examines partial ovoids of the non-singular quadric in PG(4, q). Beukemann and Metsch look at tight sets on a hyperbolic quadric in any odd dimension. Sziklai and Van de Voorde characterise a class of blocking sets. Rodgers finds new Cameron– Liebler line classes in PG(3, q). Glynn constructs independent sets on a Veronese variety.
D. Ghinelli Dipartimento di Matematica, Università di Roma “La Sapienza”, 2, Piazzale Aldo Moro, 00185 Roma, Italy e-mail: [email protected] J. W. P. Hirschfeld (B) Department of Mathematics, University of Sussex, Pevensey Building, Brighton BN1 9QH, UK e-mail: [email protected] D. Jungnickel Mathematical Institute, University of Augsburg, 86135 Augsburg, Germany e-mail: [email protected] J. A. Thas Department of Mathematics, Ghent University, Krijgslaan 281 - S22, 9000 Gent, Belgium e-mail: [email protected]
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(2) Desarguesian Planes, 3: Betten, Cheon, Kim and Maruta construct blocking sets from hyperovals. Giulietti and Korchmáros obtain properties of (q − 1)-arcs in PG(2, q). Giulietti, Korchmáros, Marcugini and Pambianco investigate a certain class of k-arcs and their completeness. (3) Non-Desarguesian Planes, 2: Dempwolff constructs translation planes using particular polynomials. Honold and Kiermaier find a class of arcs in certain Hjelmslev planes. (4) Designs and Graphs, 4: Amarra, Giudici and Praeger classify a class of graphs of diameter two. Chen and Feng construct association schemes from certain finite groups. Polhill, Davis and Smith find a new class of partial difference sets. Jungnickel and Tonchev present new invariants for a wide variety of incidence structures. (5) Semifields and Hypercubes, 3: Coolsaet classifies a class of hypercubes. Glynn constructs semifields from a class of hypercubes. Lavrauw describes the classification of semifields using nonsingular tensors. (6) Diagram Geometries, 5: Cardinali and Pasini consider symplectic dual polar spaces. De Bruyn looks at the structure of generalised quadrangles with four points on a line. Kasikova and Van Maldeghem examine spherical buildi
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