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).

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. Interestingly, this family comes with a polynomial time decoding algorithm by Micciancio & Nicolesi (2008).