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

Primarily AI-generated textHuman understanding: no partsmath.MG — Metric Geometrymath.CO — Combinatoricsmath.RA — Rings and Algebrasmath.AG — Algebraic Geometrycs.LO — Logic in Computer Science

Contributed by Robert R. MacGregor ↗

Version 1 / Nov 12, 2026 / CC BY-SA 4.0

Abstract

Cohn, Kumar and Minton constructed a four-parameter family of tight 3939-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 F4F_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 2828-dimensional space modulo F4F_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 3939-point codes modulo F4F_4 contains a nonempty open subset homeomorphic to an open subset of R10\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.

Formalizations

Tools used

Anthropic
Claude FableVersion 5.2
OpenAI
Astra GPTVersion 6
Lean FRO
Lean 4Version v4.35.0-rc2
Mathlib community
MathlibVersion commit 0653561

References

  1. H. Cohn, A. Kumar and G. Minton, Optimal simplices and codes in projective spaces , Geom. Topol. 20 (2016), no. 3, 1289--1357, doi:10.2140/gt.2016.20.1289 (https://doi.org/10.2140/gt.2016.20.1289); arXiv:1308.3188v4 (https://arxiv.org/abs/1308.3188v4). Section, theorem and equation numbers agree in both versions; page numbers cited here are those of arXiv v4.arXivDOI
  2. S. G. Hoggar, Parameters of tt-designs in FPd−1\mathbb{F}P^{d-1} , European J. Combin. 5 (1984), no. 1, 29--36, doi:10.1016/S0195-6698(84)80015-3 (https://doi.org/10.1016/S0195-6698(84)80015-3).DOI
  3. H. Cohn and A. Kumar, Universally optimal distribution of points on spheres , J. Amer. Math. Soc. 20 (2007), no. 1, 99--148, doi:10.1090/S0894-0347-06-00546-7 (https://doi.org/10.1090/S0894-0347-06-00546-7).DOI
  4. J. C. Baez, The octonions , Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 2, 145--205, doi:10.1090/S0273-0979-01-00934-X (https://doi.org/10.1090/S0273-0979-01-00934-X); arXiv:math/0105155v4 (https://arxiv.org/abs/math/0105155v4).arXivDOI
  5. H. Cohn, Y. Jiao, A. Kumar and S. Torquato, Rigidity of spherical codes , Geom. Topol. 15 (2011), no. 4, 2235--2273, doi:10.2140/gt.2011.15.2235 (https://doi.org/10.2140/gt.2011.15.2235); arXiv:1102.5060v2 (https://arxiv.org/abs/1102.5060v2).arXivDOI
  6. W. Tadej and K. Życzkowski, Defect of a unitary matrix , Linear Algebra Appl. 429 (2008), no. 2--3, 447--481, doi:10.1016/j.laa.2008.02.036 (https://doi.org/10.1016/j.laa.2008.02.036); arXiv:math/0702510v2 (https://arxiv.org/abs/math/0702510v2).arXivDOI
  7. N. Barros e Sá and I. Bengtsson, Families of complex Hadamard matrices , Linear Algebra Appl. 438 (2013), no. 7, 2929--2957, doi:10.1016/j.laa.2012.10.029 (https://doi.org/10.1016/j.laa.2012.10.029); arXiv:1202.1181v2 (https://arxiv.org/abs/1202.1181v2). Page numbers cited refer to the arXiv version.arXivDOI
  8. R. Connelly and W. Whiteley, Second-order rigidity and prestress stability for tensegrity frameworks , SIAM J. Discrete Math. 9 (1996), no. 3, 453--491, doi:10.1137/S0895480192229236 (https://doi.org/10.1137/S0895480192229236).DOI
  9. S. Łojasiewicz, Sur le problème de la division , Studia Math. 18 (1959), no. 1, 87--136, doi:10.4064/sm-18-1-87-136 (https://doi.org/10.4064/sm-18-1-87-136).DOI
  10. E. R. Avakov, A. A. Agrachev and A. V. Arutyunov, The level set of a smooth mapping in a neighborhood of a singular point, and zeros of a quadratic mapping , Mat. Sb. 182 (1991), no. 8, 1091--1104 (Russian); English transl., Math. USSR-Sb. 73 (1992), no. 2, 455--466, doi:10.1070/SM1992v073n02ABEH002555 (https://doi.org/10.1070/SM1992v073n02ABEH002555).DOI
Show all 22 referencesShow fewer references
  1. A. F. Izmailov and M. V. Solodov, Error bounds for 22-regular mappings with Lipschitzian derivatives and their applications , Math. Program. 89 (2001), no. 3, 413--435, doi:10.1007/PL00011406 (https://doi.org/10.1007/PL00011406).DOI
  2. K. G. H. Vollbrecht and R. F. Werner, Why two qubits are special , J. Math. Phys. 41 (2000), no. 10, 6772--6782, doi:10.1063/1.1286032 (https://doi.org/10.1063/1.1286032); arXiv:quant-ph/9910064v1 (https://arxiv.org/abs/quant-ph/9910064v1).arXivDOI
  3. S. G. Hoggar, Parameters of association schemes , Ars Combin. 16-B (1983), 325--339.
  4. S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury and F. Vatan, A new proof for the existence of mutually unbiased bases , Algorithmica 34 (2002), no. 4, 512--528, doi:10.1007/s00453-002-0980-7 (https://doi.org/10.1007/s00453-002-0980-7); arXiv:quant-ph/0103162v3 (https://arxiv.org/abs/quant-ph/0103162v3).arXivDOI
  5. I. Yokota, Exceptional Lie groups , arXiv:0902.0431v1 (https://arxiv.org/abs/0902.0431v1), 2009.arXiv
  6. E. Knill, Group representations, error bases and quantum codes , Los Alamos National Laboratory report, 1996, doi:10.2172/378680 (https://doi.org/10.2172/378680); arXiv:quant-ph/9608049v1 (https://arxiv.org/abs/quant-ph/9608049v1).arXivDOI
  7. E. Svegborn, J. Pauwels, N. Gisin and A. Pozas-Kerstjens, Uncountably many inequivalent maximally entangled measurements for two qutrits , arXiv:2607.16396v1 (https://arxiv.org/abs/2607.16396v1), 2026.arXiv
  8. L. de Moura and S. Ullrich, The Lean 4 theorem prover and programming language , in: Automated Deduction -- CADE 28, Lecture Notes in Comput. Sci. 12699 , Springer, Cham, 2021, 625--635, doi:10.1007/978-3-030-79876-5_37 (https://doi.org/10.1007/978-3-030-79876-5_37).DOI
  9. The mathlib Community, The Lean mathematical library , in: Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (CPP '20), ACM, New York, 2020, 367--381, doi:10.1145/3372885.3373824 (https://doi.org/10.1145/3372885.3373824).DOI
  10. Palomar registry of machine-checked Lean formalizations, https://palomar-registry.org/about, entry PALOMAR-2026-10-29-000034 , version 1 (https://palomar-registry.org/entry?id=PALOMAR-2026-10-29-000034&version=1).link
  11. K. McCrimmon, A Taste of Jordan Algebras , Universitext, Springer, New York, 2004, doi:10.1007/b97489 (https://doi.org/10.1007/b97489).DOI
  12. W. Tadej and K. Życzkowski, A concise guide to complex Hadamard matrices , Open Syst. Inf. Dyn. 13 (2006), no. 2, 133--177, doi:10.1007/s11080-006-8220-2 (https://doi.org/10.1007/s11080-006-8220-2); arXiv:quant-ph/0512154v2 (https://arxiv.org/abs/quant-ph/0512154v2).arXivDOI

Version history

  1. v1 · 55 pagesSubmitted by Robert R. MacGregor55 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.CurrentNov 11, 2026