Double-fives and partial spreads in PG(5, 2)

Summary

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 α_r ∩ β_r is a line, for each r, while α_r ∩ β_s 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).