Octonion Integers and Tight 5-Designs

TL;DR

This paper explores the construction of two tight 5-designs using octonion arithmetic, linking the Leech lattice and octonion projective plane through algebraic and geometric properties.

Contribution

It provides an octonion integer-based construction of the Leech lattice and its automorphism group, unifying previous approaches in a novel algebraic framework.

Findings

Octonion integer construction of the Leech lattice

Description of Leech automorphisms via octonion reflections

Unified construction of tight 5-designs using octonion arithmetic

Abstract

The two strictly projective tight 5-designs are the lines spanned by the short vectors of the Leech lattice and a set of points in the octonion projective plane that define a generalized hexagon of order (2,8). A previous paper introduced a common construction that can generate these two tight 5-designs. This paper describes the same construction in terms of octonion arithmetic. An octonion integer construction of the Leech lattice is described using properties of the octonion integers taken modulo 2. The Leech lattice automorphism group is constructed from octonion reflections. The common construction and the Suzuki subgroup chain of Leech lattice automorphisms are described in terms of octonion integers.