TITLE: Best non-lattice sphere packings
QUESTION [9 upvotes]: Consider a dense sphere packing in $\mathbb{R}^n$, i.e. an arrangement of mutually disjoint solid open spheres, all of the same radius. 
In dimensions $2, 3, 8,$ and $24$, it is known that lattice packings (packings where center of the spheres form a discrete subgroup of $\mathbb{R}^n$) are optimal.
It is widely believed that, in high enough dimensions, the best packings will be non-lattice packings. Is anything known about the best non-lattice packings? Do the methods of Viazovska et al. and Hales give values for the density of the best non-lattice packing? Is anything known (or conjectured) about the the ratio of the density of the best non-lattice packing to the density of the best lattice packing?

REPLY [5 votes]: I recently posted a preprint together with Alexei Andreanov in which we enumerate all the locally optimal 2-periodic sphere packings (also known as double-lattice packings) in dimensions up to $d=5$. This enumeration includes lattices, since they can be represented as 2-periodic arrangements by using a sublattice of index 2 as the unit cell. For the optimal packing density (among 2-periodic packings), we found that in $d=3$, there is one non-lattice that achieves it (the hexagonal close packing), in $d=4$ there are no non-lattices that achieve it, and for $d=5$ there are two. The highest locally optimal non-lattice double-lattice density in $d=4$ is the intriguing $\delta = 1/(2 \sqrt{9+4\sqrt{5}}) \approx 0.118$ (compared to $\delta = 1/8 = 0.125$ for the $D_4$ lattice).