On projections of arbitrary lattices

TL;DR

This paper proves that any two point lattices in different dimensions can be closely related through projections involving a set of vectors from the higher-dimensional lattice, extending previous theorems with applications in communication theory.

Contribution

It extends the main theorem of previous work by showing arbitrary lattice projections can be approximated using vectors from the original lattice.

Findings

Any two lattices in different dimensions can be related via projections.

The result generalizes previous theorems on lattice projections.

Applications in communication theory are demonstrated.

Abstract

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.