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Latin cubes of order $4$ and $4320$-point quaternionic constructions in ℝ16

  • Rustem Takhanov
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Abstract

Possibly the most enduring lower-bound record for kissing numbers in dimensions greater than $8$ (excluding $24$) is κ16≥ 4320. This lower bound is attained by a shell of the famous Barnes--Wall lattice, which was discovered in 1959. Besides this configuration, at least one alternative $4320$-point kissing arrangement is currently known. Recently, it was noticed that all these configurations can be described in terms of the so-called quaternionic construction, a general type of construction that was instrumental in explaining the geometry of the $841$-point kissing arrangement in ℝ12. We give a simple generalization of quaternionic constructions in ℝ16 obtained from record configurations. They are parameterized by three Latin cubes of order $4$, six permutations of $3$-element set, and are all infinitesimally jammed. Computer-aided counting shows that there are 4 072 728 621 isometry classes in this family of $4320$-point kissing arrangements.

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Cite

BibTeXRIS
BibTeX
@misc{takhanov2026latin,
  title = {{Latin cubes of order $4$ and $4320$-point quaternionic constructions in $\mathbb R^16$}},
  author = {Rustem Takhanov},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00927},
  eprint = {2611.00927},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00927},
}
RIS
TY  - UNPB
TI  - Latin cubes of order $4$ and $4320$-point quaternionic constructions in ℝ^16
AU  - Rustem Takhanov
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00927
UR  - https://arxiv.org/abs/2611.00927
AB  - Possibly the most enduring lower-bound record for kissing numbers in dimensions greater than $8$ (excluding $24$) is κ_16≥ 4320. This lower bound is attained by a shell of the famous Barnes--Wall lattice, which was discovered in 1959. Besides this configuration, at least one alternative $4320$-point kissing arrangement is currently known. Recently, it was noticed that all these configurations can be described in terms of the so-called quaternionic construction, a general type of construction that was instrumental in explaining the geometry of the $841$-point kissing arrangement in ℝ^12. We give a simple generalization of quaternionic constructions in ℝ^16 obtained from record configurations. They are parameterized by three Latin cubes of order $4$, six permutations of $3$-element set, and are all infinitesimally jammed. Computer-aided counting shows that there are 4 072 728 621 isometry classes in this family of $4320$-point kissing arrangements.
ER  -