On the existence of rotated $D_n$-lattices constructed via Galois extensions

TL;DR

This paper explores the construction of rotated D_n-lattices using Galois extensions, focusing on their suitability for signal transmission and analyzing the conditions under which such lattices can be constructed.

Contribution

It introduces new methods for constructing rotated D_n-lattices via Galois extensions and identifies cases where such constructions are impossible.

Findings

Constructed families of rotated D_n-lattices with full diversity

Achieved good minimum product distances for signal transmission

Identified Galois extensions where lattice construction is impossible

Abstract

In this paper we construct families of rotated $D_n$-lattices which may be suitable for signal transmission over both Gaussian and Rayleigh fading channels via subfields of cyclotomic fields. These constructions exhibit full diversity and good minimum product distance which are important parameters related to the signal transmission error probability. It is also shown that for some Galois extensions $\matK|\matQ$ it is impossible to construct rotated $D_n$-lattices via fractional ideals of $\mathcal O_{\mattK}$.