11 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Barnes–Wall lattice | 2/2 | https://en.wikipedia.org/wiki/Barnes–Wall_lattice | reference | science, encyclopedia | 2026-05-05T12:04:14.705244+00:00 | kb-cron |
== Simple Construction of a Generating Matrix == According to (Nebe, Rains & Sloane 2002), the generator matrix of
B
W
16
{\displaystyle BW_{16}}
can be constructed in the following way. First, define the matrix
B
=
(
2
0
1
1
)
.
{\displaystyle B={\begin{pmatrix}{\sqrt {2}}&0\\1&1\end{pmatrix}}.}
Next, take its 4th tensor power:
B
⊗
4
=
B
⊗
B
⊗
B
⊗
B
.
{\displaystyle B^{\otimes 4}=B\otimes B\otimes B\otimes B.}
Then, apply the homomorphism of Abelian groups
ϕ
:
Z
[
2
]
→
Z
a
+
b
2
↦
a
+
b
{\displaystyle {\begin{aligned}\phi :\mathbb {Z} [{\sqrt {2}}]&\rightarrow \mathbb {Z} \\a+b{\sqrt {2}}&\mapsto a+b\end{aligned}}}
entrywise to the matrix
B
⊗
4
{\displaystyle B^{\otimes 4}}
. The resulting
16
×
16
{\displaystyle 16\times 16}
integer matrix is a generator matrix for the Barnes–Wall lattice
B
W
16
{\displaystyle BW_{16}}
.
== Lattice theta function == The lattice theta function for the Barnes Wall lattice
B
W
16
{\displaystyle BW_{16}}
is known as
Θ
Λ
Barnes-Wall
(
z
)
=
1
/
2
{
θ
2
(
q
)
16
+
θ
3
(
q
)
16
+
θ
4
(
q
2
)
16
+
30
θ
2
(
q
)
8
θ
3
(
q
)
8
}
=
1
+
4320
q
2
+
61440
q
3
+
⋯
{\displaystyle {\begin{aligned}\Theta _{\Lambda _{\text{Barnes-Wall }}}(z)&=1/2\left\{\theta _{2}\left(q\right)^{16}+\theta _{3}\left(q\right)^{16}+\theta _{4}\left(q^{2}\right)^{16}+30\theta _{2}\left(q\right)^{8}\theta _{3}\left(q\right)^{8}\right\}\\&=1+4320q^{2}+61440q^{3}+\cdots \end{aligned}}}
where the thetas are Jacobi theta functions:
θ
2
(
q
)
=
∑
n
=
−
∞
∞
q
(
n
+
1
/
2
)
2
θ
3
(
q
)
=
∑
n
=
−
∞
∞
q
n
2
θ
4
(
q
)
=
∑
n
=
−
∞
∞
(
−
1
)
n
q
n
2
{\displaystyle {\begin{aligned}&\theta _{2}(q)=\sum _{n=-\infty }^{\infty }q^{(n+1/2)^{2}}\\&\theta _{3}(q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}\\&\theta _{4}(q)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n^{2}}\end{aligned}}}
== The number of vectors of each norm in the ==
B
W
16
{\displaystyle BW_{16}}
The number of vectors
N
(
m
)
{\displaystyle N(m)}
of norm
m
{\displaystyle m}
, as classified by J. H. Conway, is given as follows.
== Notes ==
== 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, MR 0106893 Conway, John Horton; Sloane, Neil J. A. (1999), Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften, vol. 290 (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98585-5, MR 0920369 Scharlau, Rudolf; Venkov, Boris B. (1994), "The genus of the Barnes–Wall lattice.", Comment. Math. Helv., 69 (2): 322–333, CiteSeerX 10.1.1.29.9284, doi:10.1007/BF02564490, MR 1282375 Micciancio, Daniele; Nicolosi, Antonio (2008), "Efficient bounded distance decoders for Barnes-Wall lattices", 2008 IEEE International Symposium on Information Theory, pp. 2484–2488, doi:10.1109/ISIT.2008.4595438, ISBN 978-1-4244-2256-2 Nebe, G.; Rains, E. M.; Sloane, N. J. A. (2002). "A Simple Construction for the Barnes-Wall Lattices". arXiv:math/0207186.
== External links == Barnes–Wall lattice – Sloane's lattice catalog