In mathematics, the Barnes - Wall lattice Λ16, discovered by Eric Stephen Barnes and G. E. (Tim) Wall, is the 16-dimensional positive-definite even integral lattice of discriminant 28 with no norm-2 vectors. It is the sublattice of the Leech lattice fixed by a certain automorphism of order 2, and is analogous to the Coxeter - Todd lattice.
The automorphism group of the Barnes - Wall lattice has order 89181388800 = 221 35 52 7 and has structure 21+8 PSO8+(F2). There are 4320 vectors of norm 4 in the Barnes - Wall lattice (the shortest nonzero vectors in this lattice).
The genus of the Barnes - Wall lattice was described by and contains 24 lattices; all the elements other than the Barnes - Wall lattice have root system of maximal rank 16.
The Barnes - Wall lattice is described in detail in .
While Λ16 is often referred to as the Barnes-Wall lattice, their original article in fact construct a family of lattices of increasing dimension n=2k for any integer k, and increasing normalized minimal distance, namely n1/4. This is to be compared to the normalized minimal distance of 1 for the trivial lattice
Zn
2 ⋅ \Gamma\left(
| n | |
| 2 |
+1\right)1/n/\sqrt{\pi}=\sqrt{
| 2n | |
| \pie |