Barnes–Wall lattice
# Barnes–Wall lattice

In mathematics, the Barnes–Wall lattice $Λ_{16},$ discovered by Eric Stephen Barnes and G. E. (Tim) Wall (Barnes & Wall (1959)), is the 16-dimensional positive-definite even integral lattice of discriminant $2^8$ 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 = $2^{21}$ $3^5$ $5^2$ 7 and has structure $2^{1+8}$ $PSO_8^+(F_2).$ 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 Scharlau & Venkov (1994) 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 (Conway & Sloane 1999, section 4.10).

