## Combinatorics '86Recent developments in all aspects of combinatorial and incidence geometry are covered in this volume, including their links with the foundations of geometry, graph theory and algebraic structures, and the applications to coding theory and computer science. Topics covered include Galois geometries, blocking sets, affine and projective planes, incidence structures and their automorphism groups. Matroids, graph theory and designs are also treated, along with weak algebraic structures such as near-rings, near-fields, quasi-groups, loops, hypergroups etc., and permutation sets and groups. The vitality of combinatorics today lies in its important interactions with computer science. The problems which arise are of a varied nature and suitable techniques to deal with them have to be devised for each situation; one of the special features of combinatorics is the often sporadic nature of solutions, stemming from its links with number theory. The branches of combinatorics are many and various, and all of them are represented in the 56 papers in this volume. |

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### Contents

1 | |

7 | |

21 | |

31 | |

Chapter 5 Blocking Sets in the Projective Plane of Order Four | 43 |

Chapter 6 Kalahari and the Sequence Sloane No 377 | 51 |

Chapter 7 Enciphered Geometry Some Applications of Geometry to Cryptography | 59 |

Chapter 8 On Finite Grassmann Spaces | 69 |

Chapter 30 Irreducible Representations of Hecke Algebras of Rank 2 Geometries | 277 |

Chapter 31 A Characterization of Pappian Affine Hjelmslev Planes | 281 |

Chapter 32 Embedding Locally Projective Planar Spaces in to Projective Spaces | 293 |

Chapter 33 On Topological Incidence Groupoids | 297 |

Chapter 34 Isomorphisms of Finite Hypergroupoids | 301 |

Chapter 35 Seminversive Planes | 311 |

Chapter 36 Geometric and Algebraic Methods in the Classification of Geometries Belonging to Lie Diagrams | 315 |

Chapter 37 The ThasFisher Generalized Quadrangles | 357 |

Chapter 9 The Regular Subgroups of the Sharply 3Transitive Finite Permutation Groups | 75 |

Chapter 10 Hyperovals in Desarguesian Planes of Even Order | 87 |

Chapter 11 Circular Block Designs from Planar NearRings | 95 |

Chapter 12 Extending the Concept of Decomposability for Triple Systems | 107 |

Chapter 13 Translation Partial Geometries | 117 |

Chapter 14 On Admissible Sets with Two Intersection Numbers in a Projective PLane | 137 |

Chapter 15 Commutative Finite AHypergroups of Length Two | 147 |

Chapter 16 On Sets of Fixed Parity in Steiner Systems | 157 |

Chapter 17 Blocking Sets of Index Two | 169 |

Chapter 18 A Short Proof that Ordered Linear Spaces a r e Locally Projective | 177 |

Chapter 19 Midpoints and Midlines in a Finite Hyperbolic Plane | 181 |

Chapter 20 HallRyser Type Theorems for Relative Difference Sets | 189 |

Chapter 21 Coordination of Generalized Quadrangles | 195 |

Chapter 22 Construction of Some Planar Translation Spaces | 209 |

Chapter 23 Regular Sets in Geometries | 217 |

Chapter 24 Group Preserving Extensions of Skew Parabola Planes | 225 |

Chapter 25 Products of Involutions in Orthogonal Groups | 231 |

Chapter 26 Examples of Ovoidal MBbius Planes of Hering Class II1 | 249 |

Chapter 27 A Construction of Pairs and Triples of kIncomplete Orthogonal Arrays | 251 |

Chapter 28 Relative Infinity in Projective De Sitter Spacetime and Its Relation to Proper Time | 257 |

Chapter 29 Affine Hjelmslev Rings and Planes | 265 |

Chapter 38 On Group Spaces Defined by Semidirect Products of Groups | 367 |

Chapter 39 On Permutation Properties for Finitely Generated Semigroups | 375 |

Chapter 40 On kSets of Type Omn in Srq with Three Exterior Hyperplanes | 377 |

Chapter 41 An Algorithm for LS colourations | 385 |

Chapter 42 A Blocking Set in PG 3 q q 5 | 391 |

Chapter 43 A Characterization of all Abelian Groups whose Lattice of Precompact Group Topologies Represents a Projective Geometry | 395 |

Chapter 44 Groups of Homologies in 4Dimensional Stable Planes are Classical | 399 |

Chapter 45 Polynomial Species and Connections among Bases of the Symmetric Polynomials | 405 |

Chapter 46 Set and Sequence Closure for Finite Permutation Groups | 413 |

Chapter 47 PCyclic Hypergroups with Three Characteristic Elements | 421 |

Chapter 48 Order and Uniform Structure in Projective Geometry | 427 |

Chapter 49 On Blocking Sets in Finite Projective and Affine Spaces | 433 |

Chapter 50 Symmetric Designs without Ovals and Extremal SelfDual Codes | 451 |

Chapter 51 Groups in Hypergroups | 459 |

Chapter 52 The PerronFrobenius Projection in the Theory of Graphs Digraphs Designs and Stochastic Processes | 469 |

Chapter 53 On the NonExistence of Certain Difference Sets | 479 |

Chapter 54 On Complete 12Arcs i n Projective Planes of Order 12 | 485 |

Chapter 55 Block Designs Admitting Flag Transitive Groups of Automorphisms | 493 |

Chapter 56 An Independence Theorem on the Conditions for Incidence Loops | 497 |