View source for Barnes–Wall lattice

[[File:Barns Wall Lattice by MisakiOhta.jpg|thumb|The projection of the 4320 shortest vectors of Barnes Wall lattice]]
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In mathematics, the '''Barnes&ndash;Wall lattice''' Λ<sub>16</sub>, discovered by [[Eric Stephen Barnes]] and G. E. (Tim) Wall ({{harvtxt|Barnes|Wall|1959}}), is the 16-dimensional positive-definite even integral [[lattice (group)|lattice]] of discriminant 2<sup>8</sup> 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&ndash;Todd lattice]].

The automorphism group of the Barnes&ndash;Wall lattice has order 89181388800 = 2<sup>21</sup>&nbsp;3<sup>5</sup>&nbsp;5<sup>2</sup>&nbsp;7 and has structure 2<sup>1+8</sup>&nbsp;PSO<sub>8</sub><sup>+</sup>('''F'''<sub>2</sub>).  There are 4320 vectors of norm 4 in the Barnes&ndash;Wall lattice (the shortest nonzero vectors in this lattice).

The [[Genus of a quadratic form|genus]] of the Barnes&ndash;Wall lattice was described by {{harvtxt|Scharlau|Venkov|1994}} and contains 24 lattices; all the elements other than the Barnes&ndash;Wall lattice have root system of maximal rank 16. 

The Barnes&ndash;Wall lattice is described in detail in {{harv|Conway|Sloane|1999|loc=section 4.10}}.
[[File:Lattice points Barnes Wall.jpg|thumb|right|The projection of the 4320 lattice points without lines]]

While Λ<sub>16</sub> is often referred to as ''the'' Barnes-Wall lattice, their original article in fact construct a family of lattices of increasing dimension n=2<sup>k</sup> for any integer k, and increasing normalized minimal distance, namely n<sup>1/4</sup>. This is to be compared to the normalized minimal distance of 1 for the trivial lattice <math>\mathbb{Z}^n</math>, and an upper bound of <math> 2 \cdot \Gamma\left(\frac n 2 + 1\right)^{1/n} \big/ \sqrt{\pi} = \sqrt{\frac {2n}{\pi e}} + o(\sqrt{n})</math> given by [[Minkowski's theorem]]  applied to Euclidean balls. Interestingly, this family comes with a polynomial time decoding algorithm by {{harvtxt|Micciancio|Nicolesi|2008}}.

==Generating matrix==
The generator matrix for the Barnes-Wall Lattice <math>\Lambda_{16}</math> is given by the following matrix:

<math>
\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)
</math>

The lattice spanned by the following matrix is isomorphic to the above,

<math>
\frac{1}{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)
</math>

==Lattice theta function==
The [[lattice theta function]] for the Barnes Wall lattice <math>\Lambda_{16}</math> is known as

<math>
\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+4320 q^2+61440 q^3+\cdots
\end{aligned}
</math>

where the thetas are [[Jacobi theta function]]s. 

<math>
\begin{aligned}
& \theta_2(q)=\sum_{n=-\infty}^{\infty} q^{(m+1 / 2)^2} \\
& \theta_3(q)=\sum_{n=-\infty}^{\infty} q^{m^2} \\
& \theta_4(q)=\sum_{n=-\infty}^{\infty}(-q)^{m^2}
\end{aligned}
</math>

Note that the lattice theta functions for  <math>D_4</math>, <math>E_8</math> are 

<math>
\Theta_{D_4}(q)=2 E_2\left(q^2\right)-E_2(q)=1+24 q+24 q^2+96 q^3+24 q^4+144 q^5+\ldots
</math>

<math>
\begin{aligned}
\Theta_{E_8}(z) &=\frac{1}{2}\left(\theta_2(q)^8+\theta_3(q)^8+\theta_4(q)^8\right) \\
& =\theta_2\left(q^2\right)^8+14 \theta_2\left(q^2\right)^4 \theta_3\left(q^2\right)^4+\theta_3\left(q^2\right)^8 \\
& =1+240 q^2+2160 q^4+6720 q^6+17520 q^8+\cdots
\end{aligned}
</math>

where <math>E_2(q)=1-24 \sum_n \sigma_1(n) q^n</math>

==References==
*{{citation|mr=0106893 |last1=Barnes|first1= E. S.|last2= Wall|first2= G. E.|title= Some extreme forms defined in terms of Abelian groups|journal=  J. Austral. Math. Soc.|volume=  1 |year= 1959|issue=1|pages= 47–63|doi=10.1017/S1446788700025064|doi-access=free}}
*{{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | last2=Sloane | first2=Neil J. A. | author2-link=Neil Sloane | title=Sphere Packings, Lattices and Groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=3rd | series=Grundlehren der Mathematischen Wissenschaften | isbn=978-0-387-98585-5 | mr=0920369 | year=1999 | volume=290 | url-access=registration | url=https://archive.org/details/spherepackingsla0000conw_b8u0 }}
*{{citation
 |mr=1282375 
 |last1=Scharlau 
 |first1=Rudolf 
 |last2=Venkov 
 |first2=Boris B. 
 |title=The genus of the Barnes&ndash;Wall lattice. 
 |journal=[[Comment. Math. Helv.]] 
 |volume=69 
 |year=1994 
 |issue=2 
 |pages=322–333 
 |doi=10.1007/BF02564490 
|citeseerx=10.1.1.29.9284 
 }}
*{{citation
 |last1=Micciancio
 |first1=Daniele 
 |last2=Nicolesi 
 |first2=Antonio 
 |chapter=Efficient bounded distance decoders for Barnes-Wall lattices 
 |pages=2484–2488 
 |title=2008 IEEE International Symposium on Information Theory
 |year=2008
 |doi=10.1109/ISIT.2008.4595438
 |isbn=978-1-4244-2256-2 
 }}

==External links==
*[http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/LAMBDA16.html Barnes&ndash;Wall lattice] at Sloane's lattice catalogue.

[[Category:Quadratic forms]]
[[Category:Lattice points]]


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