Identifying lattices - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T04:51:45Z http://mathoverflow.net/feeds/question/96860 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96860/identifying-lattices Identifying lattices Yoav Kallus 2012-05-14T00:55:15Z 2012-05-14T14:50:16Z <p>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 <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/index.html" rel="nofollow">Catalog of Lattices</a> and did not find any matches, but there do not seem to be many lattices listed there.</p> <p>Here is an example of one of the lattices I find in $\mathbb{R}^{11}$. The Gram matrix is given by</p> <p><code>$$G = \left(\begin{array}{ccccccccccc} 8&amp;3&amp;3&amp;2&amp;3&amp;4&amp;4&amp;4&amp;4&amp;4&amp;4\\ 3&amp;8&amp;4&amp;4&amp;4&amp;-1&amp;4&amp;-1&amp;4&amp;4&amp;4\\ 3&amp;4&amp;8&amp;0&amp;0&amp;-1&amp;0&amp;-1&amp;0&amp;4&amp;4\\ 2&amp;4&amp;0&amp;8&amp;2&amp;2&amp;2&amp;1&amp;2&amp;4&amp;0\\ 3&amp;4&amp;0&amp;2&amp;8&amp;3&amp;4&amp;-1&amp;4&amp;1&amp;1\\ 4&amp;-1&amp;-1&amp;2&amp;3&amp;8&amp;0&amp;4&amp;0&amp;0&amp;0\\ 4&amp;4&amp;0&amp;2&amp;4&amp;0&amp;8&amp;0&amp;4&amp;2&amp;2\\ 4&amp;-1&amp;-1&amp;1&amp;-1&amp;4&amp;0&amp;8&amp;0&amp;0&amp;0\\ 4&amp;4&amp;0&amp;2&amp;4&amp;0&amp;4&amp;0&amp;8&amp;2&amp;2\\ 4&amp;4&amp;4&amp;4&amp;1&amp;0&amp;2&amp;0&amp;2&amp;8&amp;4\\ 4&amp;4&amp;4&amp;0&amp;1&amp;0&amp;2&amp;0&amp;2&amp;4&amp;8 \end{array}\right)\text.$$</code></p> <p>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 $\Lambda_{11}$).</p> <p>Does anybody know where I might be able to find if these lattices have been studied before?</p> http://mathoverflow.net/questions/96860/identifying-lattices/96905#96905 Answer by G.C. for Identifying lattices G.C. 2012-05-14T14:50:16Z 2012-05-14T14:50:16Z <p>(Not an answer, but I could not find in the faq how one leaves a comment.) </p> <p>Your lattice is perfect.</p>