Optimal Extensions of Cross-Sections: Sphere Packings in Dimensions 38 to 43
Abstract
We improve the best known sphere packings in every dimension from $38$ to $43$. Our packings in dimensions $38$ to $42$ come from one chain of cross-sections of the extremal even unimodular lattice P48p, √3E6 ⊂ √3E7 ⊂ √3E8 ⊂ K9 ⊂ K10, whose first three members are cut from the fixed lattice √3(E8 ⊥ E8) of an order-three automorphism; their orthogonal complements are lattice packings and set the records in dimensions $42$ down to $38$. Each successive section is a determinant-minimal extension of its predecessor. Our $43$-dimensional packing is an antipode packing: six translates of the complement of a $5$-dimensional section. We also improve some kissing numbers. Conway and Sloane's twelve $1982$ cross-section packings appear never to have had theirs computed; we compute them and find that, in dimensions $42$ to $47$, they exceed the previously tabulated lower bounds. Our chain does better in dimensions $40$, $41$ and $42$, and a further antipode packing beats the record in dimension $45$.
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BibTeX
@misc{dorofeev2026optimal,
title = {{Optimal Extensions of Cross-Sections: Sphere Packings in Dimensions 38 to 43}},
author = {Ivan Dorofeev and Xiaoming Sun and Chengu Wang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2607.20359},
eprint = {2607.20359},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.20359},
}
RIS
TY - UNPB
TI - Optimal Extensions of Cross-Sections: Sphere Packings in Dimensions 38 to 43
AU - Ivan Dorofeev
AU - Xiaoming Sun
AU - Chengu Wang
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2607.20359
UR - https://arxiv.org/abs/2607.20359
AB - We improve the best known sphere packings in every dimension from $38$ to $43$. Our packings in dimensions $38$ to $42$ come from one chain of cross-sections of the extremal even unimodular lattice P_48p, √3E_6 ⊂ √3E_7 ⊂ √3E_8 ⊂ K_9 ⊂ K_10, whose first three members are cut from the fixed lattice √3(E_8 ⊥ E_8) of an order-three automorphism; their orthogonal complements are lattice packings and set the records in dimensions $42$ down to $38$. Each successive section is a determinant-minimal extension of its predecessor. Our $43$-dimensional packing is an antipode packing: six translates of the complement of a $5$-dimensional section. We also improve some kissing numbers. Conway and Sloane's twelve $1982$ cross-section packings appear never to have had theirs computed; we compute them and find that, in dimensions $42$ to $47$, they exceed the previously tabulated lower bounds. Our chain does better in dimensions $40$, $41$ and $42$, and a further antipode packing beats the record in dimension $45$.
ER -