Barnes-Wall lattice

E. S. Barnes and G. E. Wall, Some extreme forms defined in terms of Abelian groups, J. Austral. Math. Soc. 1 (1959), 47–63, doi:10.1017/S1446788700025064.

This packing consists of a lattice in 16 dimensions. A Gram matrix for the lattice is given below as comma-separated values. See the table of densities for how this packing compares with other dimensions.

16 × 16 Gram matrix

4,2,-2,2,2,2,2,2,2,2,2,2,2,1,1,-1
2,4,-2,2,2,2,2,2,2,2,2,2,1,2,1,-1
-2,-2,4,0,0,-2,-2,-2,0,-2,-2,-2,-1,-1,-2,0
2,2,0,4,2,2,0,0,2,2,0,0,1,1,0,-2
2,2,0,2,4,2,0,0,2,1,1,1,2,2,-1,-2
2,2,-2,2,2,4,0,0,1,2,1,1,2,2,1,-2
2,2,-2,0,0,0,4,2,1,1,2,2,1,0,1,1
2,2,-2,0,0,0,2,4,1,1,2,2,0,1,1,1
2,2,0,2,2,1,1,1,4,2,0,0,2,2,-1,-2
2,2,-2,2,1,2,1,1,2,4,0,0,2,2,1,-2
2,2,-2,0,1,1,2,2,0,0,4,2,1,0,1,1
2,2,-2,0,1,1,2,2,0,0,2,4,0,1,1,1
2,1,-1,1,2,2,1,0,2,2,1,0,4,2,-1,-1
1,2,-1,1,2,2,0,1,2,2,0,1,2,4,-1,-1
1,1,-2,0,-1,1,1,1,-1,1,1,1,-1,-1,4,0
-1,-1,0,-2,-2,-2,1,1,-2,-2,1,1,-1,-1,0,4