Barnes–Wall lattice

From HandWiki

In mathematics, the Barnes–Wall lattice Λ16, discovered by Eric Stephen Barnes and G. E. (Tim) Wall ((Barnes 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 (Scharlau Venkov) 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).


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