Take the tour ×  Historically, the lattices with high density
were studied intensively, e.g. E_8 lattice or Leech Lattice.
However, there are situations that lattices with low kissing
number are required. Specifically, I am wondering if there is
any infinite family of n-dimension lattices which the kissing
number is bounded by poly(n), and still relatively dense?
share|improve this question add comment  3 Answers  If the
question is about the asymptotic behavior in high dimensions
then I do not think it is known or even conjectured that the
kissing number for the densest packing is not polynomial in n.
To the best of my memory (but check the book by Conway and
Sloane) there are no known examples of lattices where the
kissing number is exponential, and the best known bound are only
quasi-polynomial ($n^{\log n}$).  So "dense lattice packing" and
"huge kissing numbers" are not the same in very high dimensions.
(There are the same in dimension 24!). If you are interested in
interesting notions of low density pa