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. 



Simple basis for Barnes-Wall lattices in dimension `$2^n$`
Henry Cohn cited a very nice definition of the Barnes-Wall lattices, but in my opinion, [this definition](http://www.cs.stevens.edu/~nicolosi/papers/isit08.pdf) 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.


