Barnes–Wall lattice

The projection of the 4320 shortest vectors of Barnes Wall lattice

In mathematics, the Barnes–Wall lattice Λ₁₆, discovered by Eric Stephen Barnes and G. E. (Tim) Wall (Template:Harvtxt), 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 Template:Harvtxt 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 Template:Harv.

The projection of the 4320 lattice points without lines

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=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 $2\cdot \mathrm{\Gamma }{(\frac{n}{2}+1)}^{1/n}/\sqrt{\pi }=\sqrt{\frac{2n}{\pi e}}+o(\sqrt{n})$ given by Minkowski's theorem applied to Euclidean balls. Interestingly, this family comes with a polynomial time decoding algorithm by Template:Harvtxt.

Generating matrix

The generator matrix for the Barnes-Wall Lattice ${\mathrm{\Lambda }}_{16}$ 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 ${\mathrm{\Lambda }}_{16}$ is known as

$\begin{array}{cc}{\mathrm{\Theta }}_{{\mathrm{\Lambda }}_{\text{Barnes-Wall }}}(z)& =1/2\{{\theta }_2{\left(q\right)}^{16}+{\theta }_3{\left(q\right)}^{16}+{\theta }_4{\left(q^2\right)}^{16}+30{\theta }_2{\left(q\right)}^8{\theta }_3{\left(q\right)}^8\}\\ & =1+4320q^2+61440q^3+\cdots \end{array}$

where the thetas are Jacobi theta functions.

$\begin{array}{cc}& {\theta }_2(q)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{q}^{(m+1/2{)}^2}\\ & {\theta }_3(q)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{q}^{m^2}\\ & {\theta }_4(q)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}(-q{)}^{m^2}\end{array}$

Note that the lattice theta functions for $D_4$, $E_8$ are

${\mathrm{\Theta }}_{D_4}(q)=2E_2\left(q^2\right)-E_2(q)=1+24q+24q^2+96q^3+24q^4+144q^5+\dots$

$\begin{array}{cc}{\mathrm{\Theta }}_{E_8}(z)& =\frac{1}{2}({\theta }_2(q{)}^8+{\theta }_3(q{)}^8+{\theta }_4(q{)}^8)\\ & ={\theta }_2{\left(q^2\right)}^8+14{\theta }_2{\left(q^2\right)}^4{\theta }_3{\left(q^2\right)}^4+{\theta }_3{\left(q^2\right)}^8\\ & =1+240q^2+2160q^4+6720q^6+17520q^8+\cdots \end{array}$

where $E_2(q)=1-24\sum _n{\sigma }_1(n)q^n$

References

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External links

• Barnes–Wall lattice at Sloane's lattice catalogue.

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