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I'm searching for a "simple" description of the basis of the Barnes-Wall lattices in (real) dimension $2^n$, if possible in a basis of minimal vectors, so that I can do some calculations.

Can anyone tell me where to find such a description ?

Note : I'm not looking for examples in fixed dimensions, like the D4, or E8 lattices. I know where to look up those. I'm looking for a formula / systematic description in all dimensions of powers of 2.

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Henry Cohn cited a very nice definition of the Barnes-Wall lattices, but in my opinion, this definition that I just found in a paper by Micciancio and Nicoli is even better. (Although the two definitions are similar.) The Barnes-Wall lattice in $\mathbb{C}^{2^{n-1}}$ is a lattice over the Gaussian integers generated by the rows of the matrix $$\begin{pmatrix} 1 & 1 \\ 0 & 1+i \end{pmatrix}^{\otimes {n-1}}.$$ Then of course to get the generator matrix in $\mathbb{R}^{2^n}$, you use the replacement $$a+ib \mapsto \begin{pmatrix} a & b \\ -b & a \end{pmatrix}.$$ This isn't a basis of minimal vectors. However, if you change the Gaussian generator matrix to $$\begin{pmatrix} 1 & 1 \\ 1 & i \end{pmatrix}^{\otimes {n-1}},$$ then it is a basis of minimal vectors, and the associated real basis is also minimal.

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    $\begingroup$ This is great! I had no idea one could make it even simpler than the Nebe-Rains-Sloane definition. $\endgroup$ – Henry Cohn Oct 24 '11 at 2:32
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Nebe, Rains, and Sloane have a beautiful paper called "A simple construction for the Barnes-Wall lattices" (http://arxiv.org/abs/math/0207186). Their construction is great: consider two vectors of norm $2$ with inner product $\sqrt{2}$, and let $L$ be the $\mathbb{Z}[\sqrt{2}]$ lattice they span. Then up to scaling, the Barnes-Wall lattice of dimension $2^n$ is the sublattice of the $n$-th tensor power $L^{\otimes n}$ fixed by conjugation (i.e., fixed under interchanging $\sqrt{2}$ and $-\sqrt{2}$).

Slightly more explicitly, take $(\sqrt{2},0)$ and $(1,1)$ as a basis of $L$, and then the Barnes-Wall lattice is the intersection of $L^{\otimes n}$ with $\mathbb{Q}^{2^n}$.

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All I can give you is some names and sites. Gabriele Nebe co-owns the Lattices site with Sloane, http://www.math.rwth-aachen.de/~Gabriele.Nebe/ and http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/ and has some work of her own on high-dimensional lattices. She gave a very positive review of a book by Jacques Martinet, Perfect Lattices in Euclidean Spaces. Finally, you might look through every publication by Daniel Allcock, he was the most helpful to me on my final covering radius question, http://www.ma.utexas.edu/users/allcock/

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