# Densest sphere packing for a given lattice

There is a great deal of fascinating work on the sphere packing problem. I was wondering if there exist methods to find the densest possible packing of a sublattice of a given lattice. In particular, I am thinking of the Niemeier lattices (not including the Leech).

Thanks!

Unfortunately, this is equivalent to the question of finding the densest lattice sphere packing in general. In particular, for any lattices $L$ and $L'$, there exists a sequence of sublattices $L_n$ of $L$ such that the density of $L_n$ approaches the density of $L'$ as $n \to \infty$.
To see this, simply note that if $L' \subset \mathbb{Q}^n$ is a rational lattice and $L = \mathbb{Z}^n$ is the integer lattice, then by clearing denominators, we find a single fixed sublattice of $L$ that is a scaling of $L'$. By applying a rational linear transformation, we see that the same holds for any two rational lattices (with full rank) $L, L' \subset \mathbb{Q}^n$. The full result for (full-rank) lattices $L,L' \subset \mathbb{Q}^n$ follows by taking appropriate sequences of rational lattices that converge to $L$ and $L'$.
• I'm not sure. One way to think about this is to consider a random sublattice of a given index. (It's mildly convenient to take the index to be a prime $p$, in which case these are sublattices are given by lattice vectors whose coordinates in some lattice basis satisfy a random linear equation mod $p$.) As the index gets large, this distribution looks more and more similar to the Haar measure over all lattices of the appropriate determinant. Morally, as this distribution gets closer to the Haar measure, the question becomes less interesting. – Noah Stephens-Davidowitz Aug 2 '17 at 0:59
• Right, so if $p$ is sufficiently large (say $\sqrt{n}$) then we can approximate the density of good ''random'' lattices. What happens in between might depend on $L$. My guess is that the question asked usually arises in ''algebraic'' contexts where $p$ is fairly small, say $2$ or $3$ for the Leech lattice. This is just a guess - it would be nice to have the motivation from the OP. – Campello Aug 2 '17 at 10:12