# Barnes–Wall lattice

In mathematics, the Barnes–Wall lattice Λ$_{16}$, discovered by Eric Stephen Barnes and G. E. (Tim) Wall (Barnes & Wall (1959)), is the 16-dimensional positive-definite even integral lattice of discriminant 2$^8$ 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 = 2$^{21}$ 3$^5$ 5$^2$ 7 and has structure 2$^{1+8}$ PSO$_8$$^+$(F$_2$). There are 4320 vectors of norm 4 in the Barnes–Wall lattice (the shortest nonzero vectors in this lattice).
The genus of the Barnes–Wall lattice was described by Scharlau & Venkov (1994) and contains 24 lattices; all the elements other than the Barnes–Wall lattice have root system of maximal rank 16.
The Barnes–Wall lattice is described in detail in (Conway & Sloane 1999, section 4.10).
While Λ$_{16}$ is often referred to as the Barnes-Wall lattice, their original article in fact construct a family of lattices of increasing dimension n=2$^k$ for any integer k, and increasing normalized minimal distance, namely n$^{1/4}$. This is to be compared to the normalized minimal distance of 1 for the trivial lattice $\mathbb {Z} ^{n}$, and an upper bound of $2\cdot \Gamma \left({\frac {n}{2}}+1\right)^{1/n}{\big /}{\sqrt {\pi }}={\sqrt {\frac {2n}{\pi e}}}+o({\sqrt {n}})$ given by Minkowski's theorem applied to Euclidean balls. Interestingly, this family comes with a polynomial time decoding algorithm by Micciancio & Nicolesi (2008).

## Generating matrix

The generator matrix for the Barnes-Wall Lattice $\Lambda _{16}$ is given by the following matrix:
${\frac {1}{2}}\left({\begin{array}{cccccccccccccccc}1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\0&2&0&0&0&0&0&2&0&0&0&2&0&2&0&0\\0&0&2&0&0&0&0&2&0&0&0&2&0&0&2&0\\0&0&0&2&0&0&0&2&0&0&0&2&0&0&0&2\\0&0&0&0&2&0&0&2&0&0&0&0&0&2&2&0\\0&0&0&0&0&2&0&2&0&0&0&0&0&2&0&2\\0&0&0&0&0&0&2&2&0&0&0&0&0&0&2&2\\0&0&0&0&0&0&0&4&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&2&0&0&2&0&2&2&0\\0&0&0&0&0&0&0&0&0&2&0&2&0&2&0&2\\0&0&0&0&0&0&0&0&0&0&2&2&0&0&2&2\\0&0&0&0&0&0&0&0&0&0&0&4&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&2&2&2&2\\0&0&0&0&0&0&0&0&0&0&0&0&0&4&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&4&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&4\end{array}}\right)$
The lattice spanned by the following matrix is isomorphic to the above,
${\frac {1}{2}}\left({\begin{array}{cccccccccccccccc}1&0&0&0&0&1&0&1&0&0&1&1&0&1&1&1\\0&1&0&0&0&1&1&1&1&0&1&0&1&1&0&0\\0&0&1&0&0&0&1&1&1&1&0&1&0&1&1&0\\0&0&0&1&0&1&0&0&1&1&0&1&1&1&0&1\\0&0&0&0&1&0&1&0&0&1&1&0&1&1&1&1\\0&0&0&0&0&2&0&0&0&0&0&0&0&0&0&2\\0&0&0&0&0&0&2&0&0&0&0&0&0&0&0&2\\0&0&0&0&0&0&0&2&0&0&0&0&0&0&0&2\\0&0&0&0&0&0&0&0&2&0&0&0&0&0&0&2\\0&0&0&0&0&0&0&0&0&2&0&0&0&0&0&2\\0&0&0&0&0&0&0&0&0&0&2&0&0&0&0&2\\0&0&0&0&0&0&0&0&0&0&0&2&0&0&0&2\\0&0&0&0&0&0&0&0&0&0&0&0&2&0&0&2\\0&0&0&0&0&0&0&0&0&0&0&0&0&2&0&2\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&2&2\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&4\\\end{array}}\right)$

## Lattice theta function

The lattice theta function for the Barnes Wall lattice $\Lambda _{16}$ is known as
${\begin{aligned}\Theta _{\Lambda _{\text{Barnes-Wall }}}(z)&=1/2\left\{\theta _{2}\left(q\right)^{16}+\theta _{3}\left(q\right)^{16}+\theta _{4}\left(q^{2}\right)^{16}+30\theta _{2}\left(q\right)^{8}\theta _{3}\left(q\right)^{8}\right\}\\&=1+4320q^{2}+61440q^{3}+\cdots \end{aligned}}$
where the thetas are Jacobi theta functions.
${\begin{aligned}&\theta _{2}(q)=\sum _{n=-\infty }^{\infty }q^{(m+1/2)^{2}}\\&\theta _{3}(q)=\sum _{n=-\infty }^{\infty }q^{m^{2}}\\&\theta _{4}(q)=\sum _{n=-\infty }^{\infty }(-q)^{m^{2}}\end{aligned}}$
Note that the lattice theta functions for $D_{4}$, $E_{8}$ are
$\Theta _{D_{4}}(q)=2E_{2}\left(q^{2}\right)-E_{2}(q)=1+24q+24q^{2}+96q^{3}+24q^{4}+144q^{5}+\ldots$
${\begin{aligned}\Theta _{E_{8}}(z)&={\frac {1}{2}}\left(\theta _{2}(q)^{8}+\theta _{3}(q)^{8}+\theta _{4}(q)^{8}\right)\\&=\theta _{2}\left(q^{2}\right)^{8}+14\theta _{2}\left(q^{2}\right)^{4}\theta _{3}\left(q^{2}\right)^{4}+\theta _{3}\left(q^{2}\right)^{8}\\&=1+240q^{2}+2160q^{4}+6720q^{6}+17520q^{8}+\cdots \end{aligned}}$
where $E_{2}(q)=1-24\sum _{n}\sigma _{1}(n)q^{n}$