Sphere Packings and Kissing Numbers in Dimensions 39, 43, and 45 from the Antipode Construction

Abstract

We construct non-lattice sphere packings in dimensions $39$ and $43$, improving records that had stood since Conway and Sloane's (1982) Laminated lattices construction. Both packings come from the antipode construction applied to cross-sections of $P_{48}$ lattices. Our $39$-dimensional packing is built on the very cross-section of $P_{48p}$ that Conway and Sloane used in 1982, but we place a ten-point antipode cluster over it. Our $43$-dimensional packing is built on a new five-dimensional cross-section of both $P_{48n}$ and $P_{48m}$ that supports a six-point antipode cluster. We also reconstruct all twelve of the 1982 cross-section packings (dimensions $38$-$47$) in closed form and compute their kissing numbers, which appear never to have been computed: in dimensions $42$-$47$ they exceed the best previously tabulated lower bounds. A further antipode packing in dimension $45$ beats the kissing number record.

Disclosure

“convenience, the search code, the verifier, and machine-checkable certificates for all packings of this paper, including the twelve of Section 4, are available at https://github.com/wcgbg/sphere-packing. Acknowledgments Generative AI tools were used throughout this work: in background research and literature searches, in generating and exploring ideas, in developing the search and verification code, in checking mathematical derivations, in running and analyzing the com”

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2026-08-23T19:01:49.918860Z

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