  The Barnes-Wall lattice ${\bf \Lambda}_{16}$ 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 ${\bf \Lambda}_{16}$ is the densest 16 dimensional
lattice. Sometimes it is conjectured that the Barnes-Wall lattice ${\bf
\Lambda}_{16}$ is the only densest lattice and the optimal sphere packing in
${\bf R}^{16}$. 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 ${\bf \Lambda}_n$'s, $n \leq 24, n \neq 11,12,13$ are the only
densest lattices in their dimensions. {abstract}
