Robert R. MacGregor

math.MG — Metric Geometry

On the local geometry of the tight 39-point code in the octonionic projective plane

Contributed by Robert R. MacGregor

Cohn, Kumar and Minton constructed a four-parameter family of tight $39$-point codes in the octonionic projective plane. We show that near each member outside a closed nowhere-dense set of resonant parameters, the labeled tight codes modulo $F_4$ form exactly four ten-dimensional families, any two meeting in the four-parameter family, although the first-order deformations of the member form a $28$-dimensional space modulo $F_4$. The originally constructed code is resonant, and we do not determine the tight codes near it; but every neighborhood of it in the space of unlabeled tight $39$-point codes modulo $F_4$ contains a nonempty open subset homeomorphic to an open subset of $\mathbb R^{10}$, so any neighborhood of it that is a manifold has dimension ten.

Author comments: 55 pages, 5 figures. Ancillary files: anc/native-data.json holds the exact coefficients of Appendix C, and anc/finite-identities.py replays the finite identities of Proposition 6.5 from it in exact arithmetic. Lean source: https://github.com/macgregor-sudo/tight-39-code at commit 46f01c1.