Tiling of Constellations

TL;DR

This paper investigates the problem of partitioning finite constellations in integer vector spaces into subsets, especially when the constellation lacks an abelian group structure, with applications in secure communication.

Contribution

It provides sufficient and necessary conditions for tiling constellations generated by linear codes, extending tiling theory to non-abelian settings.

Findings

Established a sufficient condition for tiling existence.

Derived a necessary condition for tiling.

Discussed tiling results when the constellation is an abelian group.

Abstract

Motivated by applications in reliable and secure communication, we address the problem of tiling (or partitioning) a finite constellation in $\mathbb{Z}_{2^L}^n$ by subsets, in the case that the constellation does not possess an abelian group structure. The property that we do require is that the constellation is generated by a linear code through an injective mapping. The intrinsic relation between the code and the constellation provides a sufficient condition for a tiling to exist. We also present a necessary condition. Inspired by a result in group theory, we discuss results on tiling for the particular case when the finer constellation is an abelian group as well.