Barnes–Wall lattice


In mathematics, the Barnes-Wall lattice, discovered by Eric Stephen Barnes and G. E. Wall, 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+. There are 4320 vectors of norm 4 in the Barnes-Wall lattice.
The genus of the Barnes-Wall lattice was described by and contains 24 lattices; all the elements other than the Barnes-Wall lattice have root system of maximal rank 16.
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=2k for any integer k, and increasing normalized minimal distance, namely n1/4. This is to be compared to the normalized minimal distance of 1 for the trivial lattice, and an upper bound of given by Minkowski's theorem applied to Euclidean balls. This family comes with a polynomial time decoding algorithm.

Generating matrix

The generator matrix for the Barnes-Wall Lattice is given by the following matrix:
For example, the lattice generated by the above generator matrix has the following vectors as its shortest vectors.
The lattice spanned by the following matrix is isomorphic to the above. Indeed, the following generator matrix can be obtained as the dual lattice of the above generator matrix.

Simple Construction of a Generating Matrix

According to, the generator matrix of can be constructed in the following way.
First, define the matrix
Next, take its 4th tensor power:
Then, apply the homomorphism of Abelian groups
entrywise to the matrix. The resulting integer matrix is a generator matrix for the Barnes–Wall lattice.

Lattice theta function

The lattice theta function for the Barnes Wall lattice is known as
where the thetas are Jacobi theta functions:

The number of vectors of each norm in the BW_{16}

The number of vectors of norm, as classified by J. H. Conway, is given as follows.
mNmN
01328593797600
203411585617920
443203619590534240
6614403825239859200
85227204040979580480
1022118404250877235200
1289606404479783021440
14232243204696134307840
166715440048146902369920
1813516800050172337725440
2031980960052256900127040
2255019520054295487692800
24114764352056431969276160
26177168384058487058227200
28337191552060699846624000
30482660352062776820326400