# Barnes–Wall lattice

In mathematics, the **Barnes–Wall lattice** Λ_{16}, discovered by Eric Stephen Barnes and G. E. (Tim) Wall ((Barnes Wall)), 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}).

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).

## References

- Barnes, E. S.; Wall, G. E. (1959), "Some extreme forms defined in terms of Abelian groups",
*J. Austral. Math. Soc.***1**(1): 47–63, doi:10.1017/S1446788700025064 - Conway, John Horton; Sloane, Neil J. A. (1999),
*Sphere Packings, Lattices and Groups*, Grundlehren der Mathematischen Wissenschaften,**290**(3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98585-5, https://archive.org/details/spherepackingsla0000conw_b8u0 - {{citation

|mr=1282375 |last=Scharlau |first=Rudolf |last2=Venkov |first2=Boris B. |title=The genus of the Barnes–Wall lattice. |journal=Comment. Math. Helv. |volume=69 |year=1994 |issue=2 |pages=322–333 |url=http://retro.seals.ch/digbib/view?did=c1:421661&sdid=c1:422358 |doi=10.1007/BF02564490

|citeseerx=10.1.1.29.9284

## External links

- Barnes–Wall lattice at Sloane's lattice catalogue.

Original source: https://en.wikipedia.org/wiki/ Barnes–Wall lattice.
Read more |