Barnes–Wall lattice

In mathematics, the Barnes–Wall lattice ${\displaystyle BW_{16}}$, discovered by Eric Stephen Barnes and G. E. (Tim) Wall,^[1] 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.^[2]

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.^[3]

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 ${\displaystyle \mathbb {Z} ^{n}}$, and an upper bound of ${\displaystyle 2\cdot \Gamma \left({\frac {n}{2}}+1\right)^{1/n}{\big /}{\sqrt {\pi }}={\sqrt {\frac {2n}{\pi e}}}+o({\sqrt {n}})}$ given by Minkowski's theorem applied to Euclidean balls. This family comes with a polynomial time decoding algorithm.^[4]

The generator matrix for the Barnes-Wall Lattice ${\displaystyle BW_{16}}$ is given by the following matrix: $${\displaystyle M_{BW_{16}}={\frac {1}{2}}\left({\begin{array}{cccccccccccccccc}1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\0&2&0&0&0&0&0&2&0&0&0&2&0&2&0&0\\0&0&2&0&0&0&0&2&0&0&0&2&0&0&2&0\\0&0&0&2&0&0&0&2&0&0&0&2&0&0&0&2\\0&0&0&0&2&0&0&2&0&0&0&0&0&2&2&0\\0&0&0&0&0&2&0&2&0&0&0&0&0&2&0&2\\0&0&0&0&0&0&2&2&0&0&0&0&0&0&2&2\\0&0&0&0&0&0&0&4&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&2&0&0&2&0&2&2&0\\0&0&0&0&0&0&0&0&0&2&0&2&0&2&0&2\\0&0&0&0&0&0&0&0&0&0&2&2&0&0&2&2\\0&0&0&0&0&0&0&0&0&0&0&4&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&2&2&2&2\\0&0&0&0&0&0&0&0&0&0&0&0&0&4&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&4&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&4\end{array}}\right)}$$

For example, the lattice ${\displaystyle BW_{16}}$ generated by the above generator matrix has the following vectors as its shortest vectors.

$${\displaystyle {\begin{aligned}v_{1}=&\left({\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}},-{\frac {1}{2}},-{\frac {1}{2}},-{\frac {1}{2}},-{\frac {1}{2}}\right)\\v_{2}=&(0,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0)\end{aligned}}}$$

The lattice spanned by the following matrix is isomorphic to the above. Indeed, the following generator matrix can be obtained as the dual lattice (up to a suitable scaling factor) of the above generator matrix.

$${\displaystyle {\widetilde {M}}_{BW_{16}}{\frac {1}{\sqrt {2}}}=\left({\begin{array}{cccccccccccccccc}1&0&0&0&0&1&0&1&0&0&1&1&0&1&1&1\\0&1&0&0&0&1&1&1&1&0&1&0&1&1&0&0\\0&0&1&0&0&0&1&1&1&1&0&1&0&1&1&0\\0&0&0&1&0&1&0&0&1&1&0&1&1&1&0&1\\0&0&0&0&1&0&1&0&0&1&1&0&1&1&1&1\\0&0&0&0&0&2&0&0&0&0&0&0&0&0&0&2\\0&0&0&0&0&0&2&0&0&0&0&0&0&0&0&2\\0&0&0&0&0&0&0&2&0&0&0&0&0&0&0&2\\0&0&0&0&0&0&0&0&2&0&0&0&0&0&0&2\\0&0&0&0&0&0&0&0&0&2&0&0&0&0&0&2\\0&0&0&0&0&0&0&0&0&0&2&0&0&0&0&2\\0&0&0&0&0&0&0&0&0&0&0&2&0&0&0&2\\0&0&0&0&0&0&0&0&0&0&0&0&2&0&0&2\\0&0&0&0&0&0&0&0&0&0&0&0&0&2&0&2\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&2&2\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&4\\\end{array}}\right)}$$

According to (Nebe, Rains & Sloane 2002), the generator matrix of ${\displaystyle BW_{16}}$ can be constructed in the following way.^[5]

First, define the matrix $${\displaystyle B={\begin{pmatrix}{\sqrt {2}}&0\\1&1\end{pmatrix}}.}$$ Next, take its 4th tensor power: $${\displaystyle B^{\otimes 4}=B\otimes B\otimes B\otimes B.}$$ Then, apply the ring homomorphism $${\displaystyle {\begin{aligned}\phi :\mathbb {Z} [{\sqrt {2}}]&\rightarrow \mathbb {Z} \\a+b{\sqrt {2}}&\mapsto a+b\end{aligned}}}$$ entrywise to the matrix ${\displaystyle B^{\otimes 4}}$. The resulting ${\displaystyle 16\times 16}$ integer matrix is a generator matrix for the Barnes–Wall lattice ${\displaystyle BW_{16}}$.^[5]

The lattice theta function for the Barnes Wall lattice ${\displaystyle BW_{16}}$ is known as $${\displaystyle {\begin{aligned}\Theta _{\Lambda _{\text{Barnes-Wall }}}(z)&=1/2\left\{\theta _{2}\left(q\right)^{16}+\theta _{3}\left(q\right)^{16}+\theta _{4}\left(q^{2}\right)^{16}+30\theta _{2}\left(q\right)^{8}\theta _{3}\left(q\right)^{8}\right\}\\&=1+4320q^{2}+61440q^{3}+\cdots \end{aligned}}}$$ where the thetas are Jacobi theta functions: $${\displaystyle {\begin{aligned}&\theta _{2}(q)=\sum _{n=-\infty }^{\infty }q^{(m+1/2)^{2}}\\&\theta _{3}(q)=\sum _{n=-\infty }^{\infty }q^{m^{2}}\\&\theta _{4}(q)=\sum _{n=-\infty }^{\infty }(-q)^{m^{2}}.\end{aligned}}}$$

The number of vectors ${\displaystyle N(m)}$ of norm ${\displaystyle m}$, as classified by J. H. Conway,^[6] is given as follows.

• 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
• Nebe, G.; Rains, E. M.; Sloane, N. J. A. (2002). "A Simple Construction for the Barnes-Wall Lattices". arXiv. arXiv:math/0207186.

• Barnes–Wall lattice – Sloane's lattice catalog