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.