  We give a simple description of rectangular matrices that can be implemented
by a post-selected stabilizer circuit. Given a matrix with entries in dyadic
cyclotomic number fields $\mathbb{Q}(\exp(i\frac{2\pi}{2^m}))$, we show that it
can be implemented by a post-selected stabilizer circuit if it has entries in
$\mathbb{Z}[\exp(i\frac{2\pi}{2^m})]$ when expressed in a certain
non-orthogonal basis. This basis is related to Barnes-Wall lattices. Our result
is a generalization to a well-known connection between Clifford groups and
Barnes-Wall lattices. We also show that minimal vectors of Barnes-Wall lattices
are stabilizer states, which may be of independent interest. Finally, we
provide a few examples of generalizations beyond standard Clifford groups.
