Octonions and the two strictly projective tight 5-designs

TL;DR

This paper uncovers a new connection between two rare strictly projective tight 5-designs using octonions, linking the Leech lattice and a generalized hexagon in the octonion projective plane.

Contribution

It introduces a novel octonion-based construction that unifies two known strictly projective tight 5-designs and links the Leech lattice with a generalized hexagon structure.

Findings

Octonion involution matrices generate both 5-designs

A new octonion construction relates the Leech lattice and hexagon

Establishes a link between Gh(2,8) and the Leech lattice

Abstract

In addition to the vertices of the regular hexagon and icosahedron, there are precisely two strictly projective tight 5-designs: one constructed from the short vectors of the Leech lattice and the other corresponding to a generalized hexagon structure in the octonion projective plane. This paper describes a new connection between these two strictly projective tight 5-designs -- a common construction using octonions. Certain octonion involutionary matrices act on a three-dimensional octonion vector space to produce the first 5-design and these same matrices act on the octonion projective plane to produce the second 5-design. This result uses the octonion construction of the Leech lattice due to Robert Wilson and provides a new link between the generalized hexagon Gh(2,8) and the Leech lattice.