  In this paper we prove that given any two point lattices $\Lambda_1 \subset
\mathbb{R}^n$ and $ \Lambda_2 \subset \nobreak \mathbb{R}^{n-k}$, there is a
set of $k$ vectors $\bm{v}_i \in \Lambda_1$ such that $\Lambda_2$ is, up to
similarity, arbitrarily close to the projection of $\Lambda_1$ onto the
orthogonal complement of the subspace spanned by $\bm{v}_1, \ldots, \bm{v}_k$.
This result extends the main theorem of \cite{Sloane2} and has applications in
communication theory.
