Algebraic lattices coming from $${\mathbb {Z}}$$ -modules generalizing ramified prime ideals in odd prime degree cyclic number fields

Abstract

Lattice theory has shown to be useful in information theory, and rotated lattices with high modulation diversity have been extensively studied as an alternative approach for transmission over a Rayleigh fading channel, where the performance depends on the minimum product distance to achieve coding gains. The maximum diversity of a rotated lattice is guaranteed when we use totally real number fields and the minimum product distance is optimized by considering fields with minimum discriminants. With the construction of full-diversity algebraic lattices as our goal, in this work we present and study constructions of full-diversity algebraic lattices in odd prime dimensional Euclidean spaces from families of modules in cyclic number fields. These families include all the ramified prime ideals in each of these number fields. As immediate applications of our results, we present algebraic constructions from the densest lattices in dimensions 3 and 5.