Abstract

A double-five of planes is a set ψ of 35 points in PG(5, 2) which admits two distinct decompositions ψ = α1 ∪ α2 ∪ α3 ∪ α4 ∪ α5 = β1 ∪ β2 ∪ β3 ∪ β4 ∪ β5 into a set of five mutually skew planes such that αr ∩ βr is a line, for each r, while αr ∩ βs is a point, for r ≠ s. In a recent paper, [Sh96], a construction of a double-five was given, starting out from a (suitably coloured) icosahedron, and some of its main properties were described. The present paper deals first of all with some further properties of double-fives. In particular the existence of an invariant symplectic form is demonstrated and some related duality properties are described.

Secondly the relationship of double-fives to partial spreads of planes in PG(5, 2) is considered. The α-planes, or equally the β-planes, of double-fives provide the only examples of maximal partial spreads. It is shown that one of the planes of a non-maximal partial spread of five planes is always privileged, and this fact is seen to give rise to a nice geometric construction of an overlarge set of nine 3-(8, 4, 1) designs having automorphism group ΓL2(8).

Abstract

This is a survey of all known rank three geometries belonging to a string diagram of type (c*, c)-geometry. There are three types of objects: points, lines, and blocks subject to axioms imposed by the diagram. There are several other formulations described here which are more convenient for presenting certain of the examples. All examples fall into these six classes:

1. Simplicial type, which can easily be characterized.

2. Fischer spaces with no affine planes.

3. Orthogonal types, whose points and lines are exterior points and tangent lines of certain low-dimensional quadrics.

4. Hall type, determined by alternating multilinear forms over the field of two elements.

5. Affine type, whose points are vectors in some d-dimensional space over the integers mod 2. Here, blocks are not subspaces.

6. A few special examples determined by coherent pairs: the construction of odd type of Cameron and Fisher, and two examples of Blokhuis and Brouwer.

Abstract

New algorithms are presented for finding spreads and packings of sets with applications to combinatorial designs and finite geometries. An efficient deterministic method for spread enumeration is used to settle several existence problems for t-designs and partial geometries. Randomized algorithms based on tabu search are employed to construct new Steiner 5-designs and large sets of combinatorial designs. In particular, partitions are found of the 4-subsets of a 16-set into 91 disjoint affine planes of order 4.

Abstract

Buekenhout has given a construction of unitals in PG(2, q2) using the André representation of PG(2, q2) in the space PG(4,q). Metz has shown that this construction produces hermitian and non-hermitian unitals. In this note, we give a geometric criterion in PG(4, q) to decide whether the unital in PG(2, q2) is hermitian or not.