Constructions of full diversity $D_n$-lattices for all $n$

Abstract

We construct some families of full diversity rotated $D_n$-lattices via $\mathbb{Z}$-modules for any $n\ge 3$. We show that $\mathbb{Z}$-modules known in previous works to obtain rotated ${\mathbb{Z}}^n$-lattices with $n$ an odd integer are ideals and we find a sufficient condition for such ideals to be principal ideals. We also present bounds and formulas for the minimum product distance of ${\mathbb{Z}}^n$ and $D_n$ restricted to some conditions.