# 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}). 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).

## 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 - Scharlau, Rudolf; Venkov, Boris B. (1994), "The genus of the Barnes–Wall lattice.",
*Comment. Math. Helv.***69**(2): 322–333, doi:10.1007/BF02564490

## External links

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

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