2026/07/22 by Xiaoming Sun, Chengu Wang
Mathematics · #math.MG #msc:52C17
arxiv created 2026/07/30 · arxiv updated 2026/07/31
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 P48 lattices. Our 43-dimensional packing is built on a new five-dimensional cross-section of the cyclo-quaternionic lattice P48n that supports a six-point antipode cluster; it also raises the best known lower bound for the kissing number. Our 39-dimensional packing is built on the very cross-section of P48p that Conway and Sloane used in 1982, but we place a ten-point antipode cluster over it. Machine-checkable certificates for all claims, together with the search and verification code, are publicly available.