The unique extremal threemodular lattice of rank 26, the generalized hexagon (2,8), and the tight Cayley-plane $5$-design
Abstract
We reconstruct the generalized hexagon of order (2,8) from an abstract even rank-$26$ lattice of determinant $3$ and minimum $4$, and conversely reconstruct the lattice from the hexagon. Harmonic theta identities determine the $819$ shortest vectors in a non-zero discriminant class and their association scheme, and show that every non-empty positive-norm shell of the lattice and its dual is a spherical $5$-design. In the converse direction, the rank-$26$ idempotent gives the lattice, with saturation proved by a short dual-coset argument. A positive-definite Niemeier construction proves existence and uniqueness of the lattice and hence of the hexagon. The third moment of a projective $3$-design in the Cayley plane reconstructs the traceless Albert product. This upgrades angle-preserving bijections to elements of F4(ℝ) and proves geometric uniqueness of the tight $819$-point projective $5$-design. Finally, line deletion gives a rootless index-four sublattice of N(A124) with a Golay trio and an oriented gluing. The gluing data form eight root-sign orbits, on which L3(2)≅ L2(7) acts as on P1(F7); two explicit Golay permutations prove transitivity. This proves uniqueness and gives the automorphism-group order. The group is identified afterward as 3D4(2):3, with a central factor C2 for the full lattice group. The reconstruction and deletion arguments parallel the length-$26$ binary code construction.
Related
BibTeX
@misc{hohn2026unique,
title = {{The unique extremal threemodular lattice of rank 26, the generalized hexagon $(2,8)$, and the tight Cayley-plane $5$-design}},
author = {Gerald Höhn},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25086},
eprint = {2609.25086},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25086},
}
RIS
TY - UNPB
TI - The unique extremal threemodular lattice of rank 26, the generalized hexagon (2,8), and the tight Cayley-plane $5$-design
AU - Gerald Höhn
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.25086
UR - https://arxiv.org/abs/2609.25086
AB - We reconstruct the generalized hexagon of order (2,8) from an abstract even rank-$26$ lattice of determinant $3$ and minimum $4$, and conversely reconstruct the lattice from the hexagon. Harmonic theta identities determine the $819$ shortest vectors in a non-zero discriminant class and their association scheme, and show that every non-empty positive-norm shell of the lattice and its dual is a spherical $5$-design. In the converse direction, the rank-$26$ idempotent gives the lattice, with saturation proved by a short dual-coset argument. A positive-definite Niemeier construction proves existence and uniqueness of the lattice and hence of the hexagon. The third moment of a projective $3$-design in the Cayley plane reconstructs the traceless Albert product. This upgrades angle-preserving bijections to elements of F_4(ℝ) and proves geometric uniqueness of the tight $819$-point projective $5$-design. Finally, line deletion gives a rootless index-four sublattice of N(A_1^24) with a Golay trio and an oriented gluing. The gluing data form eight root-sign orbits, on which L_3(2)≅ L_2(7) acts as on P^1(F_7); two explicit Golay permutations prove transitivity. This proves uniqueness and gives the automorphism-group order. The group is identified afterward as ^3D_4(2):3, with a central factor C_2 for the full lattice group. The reconstruction and deletion arguments parallel the length-$26$ binary code construction.
ER -