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# Barnes-Wall Lattice

The Barnes-Wall lattice is a -dimensional lattice that exists when is a power of 2. It is implemented in the Wolfram Language as LatticeData["BarnesWall", n].

Special cases are summarized in the following table.

 lattice 2 square lattice 4 root lattice 8 root lattice 16 laminated lattice

The automorphism group has order

 (1)

giving the first few terms as 2, 8, 1152, 696729600, 89181388800, 48126558103142400, ... (OEIS A014116).

The 16-dimensional Barnes-Wall lattice can be constructed from the Leech lattice .

The generating function of the theta series for the lattice for various orders are summarized in the following table.

 theta series generating function 2 4 8 16

The following table gives the theta series themselves.

 OEIS theta series 2 A004018 4 A004011 8 A004009 16 A008409

Coxeter-Todd Lattice, Lattice Point, Leech Lattice

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## References

Barnes, E. S. and Wall, G. E. "Some Extreme Forms Defined in Terms of Abelian Groups." J. Austral. Math. Soc. 1, 47-63, 1959.Conway, J. H. and Sloane, N. J. A. "The 16-Dimensional Barnes-Wall Lattice ." §4.10 in Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 127-129, 1993.Sloane, N. J. A. Sequences A004009/M5416,A004011/M5140, A004018, A008409, and A014116 in "The On-Line Encyclopedia of Integer Sequences."Wall, G. E. "On the Clifford Collineation." Nagoya Math. J. 21, 199-222, 1962.

## Referenced on Wolfram|Alpha

Barnes-Wall Lattice

## Cite this as:

Weisstein, Eric W. "Barnes-Wall Lattice." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Barnes-WallLattice.html