Dense lattices in low dimensions

The Barnes-Wall lattice ${\mathbf{\Lambda }}_{16}$ with the center density {\{1}{16}} and the kissing number 4320 was found in 1959 and is the only known densest sphere packing in the dimension 16. J. H. Conway and N.J.A. Sloane conjectured that ${\mathbf{\Lambda }}_{16}$ is the densest 16 dimensional lattice. Sometimes it is conjectured that the Barnes-Wall lattice ${\bf \Lambda}_{16}$istheonlydensestlatticeandtheoptimalspherepackingin${\bf R}^{16}$. In this paper two new 16 dimensional lattices with the center density \{1}{16} and the kissing numbers 4224 and 4176 are constructed. This leads to several new 14 and 15 dimensional lattices which have the same center densities but different kissing numbers as the presently known densest lattices in these two dimensions. This gives a negative answer to the long time expectation that ${\mathbf{\Lambda }}_n$’s, $n\le 24,n\mathrm{\ne }11,12,13$ are the only densest lattices in their dimensions. {abstract}