In mathematics, the Barnes–Wall lattice , discovered by Eric Stephen Barnes and G. E. (Tim) 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+(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 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 (up to a suitable scaling factor) 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 The number of vectors of norm , as classified by J. H. Conway, is given as follows. m N(m) m N(m) 0 1 32 8593797600 2 0 34 11585617920 4 4320 36 19590534240 6 61440 38 25239859200 8 522720 40 40979580480 10 2211840 42 50877235200 12 8960640 44 79783021440 14 23224320 46 96134307840 16 67154400 48 146902369920 18 135168000 50 172337725440 20 319809600 52 256900127040 22 550195200 54 295487692800 24 1147643520 56 431969276160 26 1771683840 58 487058227200 28 3371915520 60 699846624000 30 4826603520 62 776820326400
Notes
References
External links Barnes–Wall lattice – Sloane's lattice catalog