Variations on five-dimensional sphere packings

TL;DR

This paper explores new five- and nine-dimensional sphere packings and kissing configurations, adding to the set of known conjecturally optimal arrangements without surpassing existing records.

Contribution

It introduces new five- and nine-dimensional sphere packings and kissing configurations, expanding the variety of known geometrically distinct optimal arrangements.

Findings

Produced a new five-dimensional kissing configuration

Constructed a new nine-dimensional kissing configuration

Augmented existing lists of conjecturally optimal packings

Abstract

We analyze Sz\"oll\H{o}si'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.