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).
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).
- 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
|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
- Barnes–Wall lattice at Sloane's lattice catalogue.
Original source: https://en.wikipedia.org/wiki/ Barnes–Wall lattice. Read more