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.

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.