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arXiv · 2607.20359

Optimal Extensions of Cross-Sections: Sphere Packings in Dimensions 38 to 43

Ivan Dorofeev·Xiaoming Sun·Chengu Wang
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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 $P_{48p}$, $\sqrt3E_6 \subset \sqrt3E_7 \subset \sqrt3E_8 \subset K_9 \subset K_{10}$, whose first three members are cut from the fixed lattice $\sqrt3(E_8 \perp E_8)$ 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 $45$.

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Ivan Dorofeev, Xiaoming Sun, Chengu Wang. 2026-07-22. Optimal Extensions of Cross-Sections: Sphere Packings in Dimensions 38 to 43. https://arxiv.org/abs/2607.20359

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