Variations on Five-Dimensional Sphere Packings

We analyze Szöllősi’s recent construction of a conjecturally optimal five-dimensional kissing configuration and produce a new such configuration, the fourth to be discovered. We construct five-dimensional sphere packings from these configurations, which augment Conway and Sloane’s list of conjecturally optimal packings. We also construct a new kissing configuration in nine dimensions. None of these constructions improves on the known records, but they provide geometrically distinct constructions achieving these records.