I wrote a program that numerically searches for lattices in $\mathbb{R}^d$ with high sphere packing densities. As I have been running the program, it has been able to find, in addition to well-known lattices such as the laminated lattice $\Lambda_d$ and the Coxeter Todd-related lattices $K_d$, a few interesting looking lattices, which I have been unable to identify. The lattices I have found so far are not as dense as $\Lambda_d$ or $K_d$, but are reasonably dense, and are nice integral lattices. Since I found them through a sort of a local optimization, I suppose they are probably perfect. I looked through the lattices listed on the Sloane-Nebe Catalog of Lattices and did not find any matches, but there do not seem to be many lattices listed there.

Here is an example of one of the lattices I find in $\mathbb{R}^{11}$. The Gram matrix is given by

$$ G = \left(\begin{array}{ccccccccccc} 8&3&3&2&3&4&4&4&4&4&4\\ 3&8&4&4&4&-1&4&-1&4&4&4\\ 3&4&8&0&0&-1&0&-1&0&4&4\\ 2&4&0&8&2&2&2&1&2&4&0\\ 3&4&0&2&8&3&4&-1&4&1&1\\ 4&-1&-1&2&3&8&0&4&0&0&0\\ 4&4&0&2&4&0&8&0&4&2&2\\ 4&-1&-1&1&-1&4&0&8&0&0&0\\ 4&4&0&2&4&0&4&0&8&2&2\\ 4&4&4&4&1&0&2&0&2&8&4\\ 4&4&4&0&1&0&2&0&2&4&8 \end{array}\right)\text. $$

The number of spheres in successive shells (equiv. theta function) are: norm 8, 308; norm 10, 320; norm 12, 680; norm 14, 1472. The packing density is $1/14\sqrt{7}=0.02699\ldots$ (number density for non-overlapping spheres of radius 1, compare to $0.03208\ldots$ for $K_{11}$ and $0.03125$ for $\Lambda_{11}$).

Does anybody know where I might be able to find if these lattices have been studied before?

  • $\begingroup$ You might write Sloane and/or Nebe. Meanwhile Magma says that this lattice has $1536 = 2^9 3$ automorphisms. $\endgroup$ Commented May 14, 2012 at 1:27

1 Answer 1


(Not an answer, but I could not find in the faq how one leaves a comment.)

Your lattice is perfect.

  • $\begingroup$ Thanks for your comment. In the FAQ section on rep, it says you need to have 50 rep points, which you'll soon have, to leave comments. $\endgroup$ Commented May 14, 2012 at 15:08
  • $\begingroup$ @Yoav, Martinet has a book on perfect lattices that Nebe reviewed, maybe I can find a link. Meanwhile, personal math.u-bordeaux.fr/~martinet and then perfect math.u-bordeaux.fr/~martinet/Lattices/index.html book review math.rwth-aachen.de/~Gabriele.Nebe/papers/BookMartinet.pdf $\endgroup$
    – Will Jagy
    Commented May 14, 2012 at 20:12
  • $\begingroup$ @Will, thanks for the reference. I checked the book out of the library, and I am enjoyably perusing it now. $\endgroup$ Commented May 14, 2012 at 21:44
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    $\begingroup$ @Yoav, just watch out for police. Your ordinary driver's license allows you to read, study, review, contemplate, examine, or pore over, but you need a trucker's license to peruse. $\endgroup$
    – Will Jagy
    Commented May 14, 2012 at 22:02

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