Octonions and the two strictly projective tight 5-designs
Algebraic Combinatorics, Volume 5 (2022) no. 3, pp. 401-411.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.215
Classification: 05B99,  17A75,  17C40
Keywords: Projective t-designs, octonions, Leech lattice, generalized hexagon
Nasmith, Benjamin 1

1 Department of Mathematics and Computer Science Royal Military College of Canada Kingston, ON
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Nasmith, Benjamin. Octonions and the two strictly projective tight 5-designs. Algebraic Combinatorics, Volume 5 (2022) no. 3, pp. 401-411. doi : 10.5802/alco.215. https://alco.centre-mersenne.org/articles/10.5802/alco.215/

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