In this paper by Viazovska, she said that:

- "The E8-lattice sphere packing π«E8 is the union of open Euclidean balls with centers at the lattice points and radius $1/\sqrt{2}$." So I think the distance between centers of the two balls is $$D=2 \cdot 1/\sqrt{2}=\sqrt{2}.$$

She then said:

- The Leech lattice Ξ24 was constructed by J. Leech in 1967. This lattice is an even unimodular lattice of rank 24. There exist 24 isomorphism classes of such lattices. Among these 24, the Leech lattice is the unique one having the shortest non-zero vector of length 2. In the other 23 classes, the shortest vector has minimal possible for even lattices length $$D=β2.$$

As the minimal distance between two points in Ξ24 is $$D=2,$$ it is a good candidate for a dense sphere packing

## My question is this:

Leech lattice $Ξ24$ has larger distance between centers of the two balls (namely $D=2$), in contrast with other 23 classes of 24-dimensional lattice which as $D=β2$, also in contrast with the E8 lattice which as $D=β2$. Am I correct to summarize this way?

What are the sizes of unit balls? Of radius $R=1/\sqrt{2}$ for the case of E8 and of 23 other 24-dimensional lattices? Of radius also $R=1/\sqrt{2}$ or $R=1$ for the Leech lattice? How come the unit ball sizes are different with different radius $R$ for these cases??

Is the relation: shortest vector $$D=2R?$$ as twice of the radius of unit ball always hold?

- If so, how can larger $D$ give rise to denser sphere packing? It is very
**counter intuitive**! Can we explain in some intuitive ways then on the larger $D$ denser sphere packing?

Thanks for replying and answering!