---
title: "Barnes–Wall lattice"
chunk: 2/2
source: "https://en.wikipedia.org/wiki/Barnes–Wall_lattice"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T12:04:14.705244+00:00"
instance: "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