# Best non-lattice configurations for the Kissing number problem

I am studying the Kissing Number problem. We know the optimal kissing number for dimensions 1, 2, 3, 4, 8 and 24. For all of these, the optimal configurations stem from highly symmetrical lattices. The best known arrangements for dimensions 5, 6 and 7 also stem from lattices.

I am wondering, what are the best known non-lattice configurations for each dimension?

• In dimension 3 at least it is possible to get the same center density and kissing number with a non-lattice construction. The idea is described in Conway & Sloane. You get the lattice by repeating a chosen 2D-layer with constant displacement from one layer to the next. But you get the same amount of kissing by using non-constant displacement (hence non-lattice). May 6, 2021 at 2:38
• Thank you @JyrkiLahtonen! Do you know if the same can be done in higher dimensions? I suppose one could always start with a lattice and then shift some of the spheres by a small amount. Although not sure if there is any room to do that in the $E_8$ lattice? May 6, 2021 at 2:42
• By the way, are you interested in configurations around a single sphere, or do you still want a pattern that can continue in the entire space? May 6, 2021 at 2:43
• I just want a configuration around a single central sphere. May 6, 2021 at 2:44
• My guess is that $E_8$ and Leech are so unique that there is no wiggle room, but I may be wrong about this. May 6, 2021 at 2:44