38 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Barnes–Wall lattice | 1/2 | https://en.wikipedia.org/wiki/Barnes–Wall_lattice | reference | science, encyclopedia | 2026-05-05T12:04:14.705244+00:00 | kb-cron |
In mathematics, the Barnes–Wall lattice
B
W
16
{\displaystyle BW_{16}}
, discovered by Eric Stephen Barnes and G. E. (Tim) 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 (1994) and contains 24 lattices; all the elements other than the Barnes–Wall lattice have root system of maximal rank 16. While Λ16 is often referred to as the Barnes-Wall lattice, their original article in fact construct a family of lattices of increasing dimension n=2k for any integer k, and increasing normalized minimal distance, namely n1/4. This is to be compared to the normalized minimal distance of 1 for the trivial lattice
Z
n
{\displaystyle \mathbb {Z} ^{n}}
, and an upper bound of
2
⋅
Γ
(
n
2
+
1
)
1
/
n
/
π
=
2
n
π
e
+
o
(
n
)
{\displaystyle 2\cdot \Gamma \left({\frac {n}{2}}+1\right)^{1/n}{\big /}{\sqrt {\pi }}={\sqrt {\frac {2n}{\pi e}}}+o({\sqrt {n}})}
given by Minkowski's theorem applied to Euclidean balls. This family comes with a polynomial time decoding algorithm.
== Generating matrix == The generator matrix for the Barnes-Wall Lattice
B
W
16
{\displaystyle BW_{16}}
is given by the following matrix:
M
B
W
16
=
1
2
(
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
2
0
0
0
0
0
2
0
0
0
2
0
2
0
0
0
0
2
0
0
0
0
2
0
0
0
2
0
0
2
0
0
0
0
2
0
0
0
2
0
0
0
2
0
0
0
2
0
0
0
0
2
0
0
2
0
0
0
0
0
2
2
0
0
0
0
0
0
2
0
2
0
0
0
0
0
2
0
2
0
0
0
0
0
0
2
2
0
0
0
0
0
0
2
2
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
2
0
2
2
0
0
0
0
0
0
0
0
0
0
2
0
2
0
2
0
2
0
0
0
0
0
0
0
0
0
0
2
2
0
0
2
2
0
0
0
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
)
{\displaystyle M_{BW_{16}}={\frac {1}{2}}\left({\begin{array}{cccccccccccccccc}1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\0&2&0&0&0&0&0&2&0&0&0&2&0&2&0&0\\0&0&2&0&0&0&0&2&0&0&0&2&0&0&2&0\\0&0&0&2&0&0&0&2&0&0&0&2&0&0&0&2\\0&0&0&0&2&0&0&2&0&0&0&0&0&2&2&0\\0&0&0&0&0&2&0&2&0&0&0&0&0&2&0&2\\0&0&0&0&0&0&2&2&0&0&0&0&0&0&2&2\\0&0&0&0&0&0&0&4&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&2&0&0&2&0&2&2&0\\0&0&0&0&0&0&0&0&0&2&0&2&0&2&0&2\\0&0&0&0&0&0&0&0&0&0&2&2&0&0&2&2\\0&0&0&0&0&0&0&0&0&0&0&4&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&2&2&2&2\\0&0&0&0&0&0&0&0&0&0&0&0&0&4&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&4&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&4\end{array}}\right)}
For example, the lattice
B
W
16
{\displaystyle BW_{16}}
generated by the above generator matrix has the following vectors as its shortest vectors.
v
1
=
(
1
2
,
1
2
,
1
2
,
1
2
,
1
2
,
1
2
,
1
2
,
1
2
,
1
2
,
1
2
,
1
2
,
1
2
,
−
1
2
,
−
1
2
,
−
1
2
,
−
1
2
)
v
2
=
(
0
,
1
,
1
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
1
,
1
,
0
)
{\displaystyle {\begin{aligned}v_{1}=&\left({\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}},-{\frac {1}{2}},-{\frac {1}{2}},-{\frac {1}{2}},-{\frac {1}{2}}\right)\\v_{2}=&(0,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0)\end{aligned}}}
The lattice spanned by the following matrix is isomorphic to the above. Indeed, the following generator matrix can be obtained as the dual lattice (up to a suitable scaling factor) of the above generator matrix.
M
~
B
W
16
1
2
=
(
1
0
0
0
0
1
0
1
0
0
1
1
0
1
1
1
0
1
0
0
0
1
1
1
1
0
1
0
1
1
0
0
0
0
1
0
0
0
1
1
1
1
0
1
0
1
1
0
0
0
0
1
0
1
0
0
1
1
0
1
1
1
0
1
0
0
0
0
1
0
1
0
0
1
1
0
1
1
1
1
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
)
{\displaystyle {\widetilde {M}}_{BW_{16}}{\frac {1}{\sqrt {2}}}=\left({\begin{array}{cccccccccccccccc}1&0&0&0&0&1&0&1&0&0&1&1&0&1&1&1\\0&1&0&0&0&1&1&1&1&0&1&0&1&1&0&0\\0&0&1&0&0&0&1&1&1&1&0&1&0&1&1&0\\0&0&0&1&0&1&0&0&1&1&0&1&1&1&0&1\\0&0&0&0&1&0&1&0&0&1&1&0&1&1&1&1\\0&0&0&0&0&2&0&0&0&0&0&0&0&0&0&2\\0&0&0&0&0&0&2&0&0&0&0&0&0&0&0&2\\0&0&0&0&0&0&0&2&0&0&0&0&0&0&0&2\\0&0&0&0&0&0&0&0&2&0&0&0&0&0&0&2\\0&0&0&0&0&0&0&0&0&2&0&0&0&0&0&2\\0&0&0&0&0&0&0&0&0&0&2&0&0&0&0&2\\0&0&0&0&0&0&0&0&0&0&0&2&0&0&0&2\\0&0&0&0&0&0&0&0&0&0&0&0&2&0&0&2\\0&0&0&0&0&0&0&0&0&0&0&0&0&2&0&2\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&2&2\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&4\\\end{array}}\right)}