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Barnes–Wall lattice

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

Generating matrix

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The generator matrix for the Barnes-Wall Lattice is given by the following matrix:

The lattice spanned by the following matrix is isomorphic to the above,

Lattice theta function

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The lattice theta function for the Barnes Wall lattice is known as

where the thetas are Jacobi theta functions.

Note that the lattice theta functions for , are

where

References

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  • 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
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