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 (Barnes & Wall (1959)), 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 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).

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 $2\cdot \Gamma \left({\frac {n}{2}}+1\right)^{1/n}{\big /}{\sqrt {\pi }}={\sqrt {\frac {2n}{\pi e}}}+o({\sqrt {n}})$ 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 $E_{2}(q)=1-24\sum _{n}\sigma _{1}(n)q^{n}$

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; Nicolesi, 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

External links

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