Finite Geometries: Proceedings Of The Fourth Isle Of Thorns Conference (developments In Mathematics)

When? These are the proceedings of Finite Geometries, the Fourth Isle of Thorns Conference, which took place from Sunday 16 to Friday 21 July, 2000. It was organised by the editors of this volume. The Third Conference in 1990 was published as Advances in Finite Geometries and Designs by Oxford University Press and the Second Conference in 1980 was published as Finite Geometries and Designs by Cambridge University Press. The main speakers were A. R. Calderbank, P. J. Cameron, C. E. Praeger, B. Schmidt, H. Van Maldeghem. There were 64 participants and 42 contributions, all listed at the end of the volume. Conference web site http://www. maths. susx. ac. uk/Staff/JWPH/ Why? This collection of 21 articles describes the latest research and current state of the art in the following inter-linked areas: • combinatorial structures in finite projective and affine spaces, also known as Galois geometries, in which combinatorial objects such as blocking sets, spreads and partial spreads, ovoids, arcs and caps, as well as curves and hypersurfaces, are all of interest; • geometric and algebraic coding theory; • finite groups and incidence geometries, as in polar spaces, gener­ alized polygons and diagram geometries; • algebraic and geometric design theory, in particular designs which have interesting symmetric properties and difference sets, which play an important role, because of their close connections to both Galois geometry and coding theory. Front Matter....Pages i-x Point-Line Geometries with a Generating Set that Depends on the Underlying Field....Pages 1-25 On a Class of Symmetric Divisible Designs which are Almost Projective Planes....Pages 27-34 Generalized Elliptic Cubic Curves, Part 1....Pages 35-48 Fixed Points and Cycles....Pages 49-60 RWPRI Geometries for the Alternating Group A 8 ....Pages 61-97 A Characterization of Truncated D n -Buildings as Flag-Transitive PG.PG*-Geometries....Pages 99-119 Group-Theoretic Characterizations of Classical Ovoids....Pages 121-131 A General Framework for Subexponential Discrete Logarithm Algorithms in Groups of Unknown Order....Pages 133-146 Generalized Quadrangles and Pencils of Quadrics....Pages 147-158 The Desarguesian Plane of Order Thirteen....Pages 159-170 Two Characterizations of the Hermitian Spread in the Split Cayley Hexagon....Pages 171-176 Epimorphisms of Generalized Polygons, Part 2: Some Existence and Non-Existence Results....Pages 177-200 The Packing Problem in Statistics, Coding Theory and Finite Projective Spaces: Update 2001....Pages 201-246 The Geometric Approach to Linear Codes....Pages 247-256 Flocks and Locally Hermitian 1-Systems of Q (6, q )....Pages 257-275 Diagrams for Embeddings of Polygons....Pages 277-293 The Law—Penttila q -Clan Geometries....Pages 295-303 Implications of Line-Transitivity for Designs....Pages 305-317 Exponent Bounds....Pages 319-331 On Symmetries and Translation Generalized Quadrangles....Pages 333-345 Complete Caps Projective Space Which are Disjoint from a Subspace of Codimension Two....Pages 347-361 Back Matter....Pages 363-367 When? These are the proceedings of Finite Geometries, the Fourth Isle of Thorns Conference, which took place from Sunday 16 to Friday 21 July, 2000. It was organised by the editors of this volume. The Third Conference in 1990 was published as Advances in Finite Geometries and Designs by Oxford University Press and the Second Conference in 1980 was published as Finite Geometries and Designs by Cambridge University Press. The main speakers were A. R. Calderbank, P. J. Cameron, C. E. Praeger, B. Schmidt, H. Van Maldeghem. There were 64 participants and 42 contributions, all listed at the end of the volume. Conference web site http://www. maths. susx. ac. uk/Staff/JWPH/ Why? This collection of 21 articles describes the latest research and current state of the art in the following inter-linked areas: " combinatorial structures in finite projective and affine spaces, also known as Galois geometries, in which combinatorial objects such as blocking sets, spreads and partial spreads, ovoids, arcs and caps, as well as curves and hypersurfaces, are all of interest; " geometric and algebraic coding theory; " finite groups and incidence geometries, as in polar spaces, generƯ alized polygons and diagram geometries; " algebraic and geometric design theory, in particular designs which have interesting symmetric properties and difference sets, which play an important role, because of their close connections to both Galois geometry and coding theory This is a graduate-level book aimed at experienced researchers as well as beginning graduate students in finite geometry, incidence geometry, design theory, coding theory and combinatorics. The 21 articles arise from the Fourth Isle of Thorns Conference on Finite Geometries, held in July 2000. The book contains new high-level results at the forefront of the field as well as authoritative surveys. The articles are written so that beginning graduate students can appreciate them. Among the surveys by the invited speakers are P.J. Cameron `Fixed Points and cycles', C.E. Praeger `Implications of line-transitivity for designs', B. Schmidt, `Exponent bounds', which explains recent results on difference sets, and several papers with H. Van Maideghem as co-author on generalized polygons. The editors also contribute: A. Blokhuis, D. Jungnickel, and B. Schmidt `On a class of symmetric divisible designs which are almost projective planes', and D. Luyckx and J.A. Thas discuss the geometry of the quadric in six dimensions. There is also a lengthy and complete survey by J.W.P. Hirschfeld and L. Storme `The packing problem in statistics, coding theory and finite projective spaces: update 2001'. Audience: Researchers as well as beginning graduate students in the areas of finite geometry, incidence geometry, design theory, coding theory and combinatorics will appreciate the surveys and original papers by leaders in these fields