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Barnes-Wall Lattice

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

Special cases are summarized in the following table.

dlattice
2square lattice Z^2
4root lattice D_4
8root lattice E_8
16laminated lattice L_(16,1)

The automorphism group has order

{696729600 for d=3; 2^(d^2+d+1)(2^d-1)product_(i=1)^(d-1)(2^(2i)-1) otherwise,
(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 Lambda_(24).

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

dtheta series generating function
2theta_3^2(q)
41/2[theta_3^4(q)+theta_4^4(q)]
81/2[theta_2^8(q)+theta_3^8(q)+theta_4^8(q)]
161/2[theta_2^(16)(q)+theta_3^(16)(q)+theta_4^(16)(q)+30theta_2^8(q)theta_3^8(q)]

The following table gives the theta series themselves.

dOEIStheta series
2A0040181+4q+4q^2+4q^4+8q^5+4q^8+4q^9+8q^(10)+...
4A0040111+24q^2+24q^4+96q^6+24q^8+144q^(10)+...
8A0040091+240q^2+2160q^4+6720q^6+17520q^8+30240q^(10)+...
16A0084091+4320q^2+61440q^3+522720q^4+2211840q^5+...

See also

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 Lambda_(16)." §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

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