On projections of arbitrary lattices

In this paper we prove that given any two point lattices $Λ_1 \subset \mathbb{R}^n$ and $Λ_2 \subset \nobreak \mathbb{R}^{n-k}$, there is a set of $k$ vectors $\bm{v}_i \in Λ_1$ such that $Λ_2$ is, up to similarity, arbitrarily close to the projection of $Λ_1$ 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.

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