Sphere Packing in Dimensions 39 and 43 from the Antipode Construction

We construct non-lattice sphere packings in dimensions $39$ and $43$, improving records that had stood since the Laminated lattices construction by Conway and Sloane (1982).

Both packings come from the antipode construction applied to cross-sections of extremal even unimodular $48$-dimensional lattices.

The $43$-dimensional packing also raises the best known kissing number.

The $39$-dimensional packing is built on the very cross-section that Conway and Sloane used in 1982; only the ten-point antipode cluster placed over it is new.

Machine-checkable certificates for all claims, together with the search and verification code, are publicly available.