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 $P_{48p}$, $\sqrt{3}{E}_6\subset \sqrt{3}{E}_7\subset \sqrt{3}{E}_8\subset K_9\subset K_{10}$, whose first three members are cut from the fixed lattice $\sqrt{3}(E_8\perp E_8)$ 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 $45$.