# Simple basis for Barnes-Wall lattices in dimension $2^n$

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.

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.
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 $\mathbb{Q}^{2^n}$.