In mathematics, the Barnes–Wall lattice Λ
16, discovered by Eric Stephen Barnes and G. E. (Tim) Wall (), is the 16-dimensional positive-definite even integral
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
of discriminant 2
8 with no norm-2 vectors. It is the sublattice of the
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by ...
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
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
of the Barnes–Wall lattice was described by 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 .
References
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*
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External links
Barnes–Wall latticeat Sloane's lattice catalogue.
Quadratic forms
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