Constructions of well-rounded algebraic lattices over odd prime degree cyclic number fields
Robson Ricardo de Araujo, Antonio Aparecido de Andrade, Trajano Pires da Nóbrega Neto, Jéfferson Luiz Rocha Bastos
TL;DR
This work extends the construction of well-rounded algebraic lattices to cyclic number fields of odd prime degree $p$, focusing on ramified cases. By leveraging the canonical embedding and carefully chosen $\mathbb{Z}$-modules in $\mathcal{O}_{\mathbb{K}}$, the authors generalize prior cubic-field results and derive explicit trace forms, minimum norms, and center densities for resulting lattices. They develop two main families of modules in the ramified setting, including $\mathcal{M}_m$ and $\mathcal{M}_{m,c}$, and identify conditions under which the associated lattices are well-rounded, particularly distinguishing cases when $p$ divides $m$ versus when it does not. The paper also connects these constructions to prime-ideal decompositions and provides explicit basis and density formulas, offering new, practical lattice designs for signaling applications in channels. Overall, the results yield concrete, provably well-rounded algebraic lattices in ramified odd prime degree cyclic number fields and extend the toolkit for algebraic lattice constructions via $\mathbb{Z}$-modules.
Abstract
Algebraic lattices are those obtained from modules in the ring of integers of algebraic number fields through the canonical or twisted embeddings. In turn, well-rounded lattices are those with maximal cardinality of linearly independent vectors in its set of minimal vectors. Both classes of lattices have been applied for signal transmission in some channels, such as wiretap channels. Recently, some advances have been made in the search for well-rounded lattices that can be realized as algebraic lattices. Moreover, some works have been published studying algebraic lattices obtained from modules in cyclic number fields of odd prime degree $p$. In this work, we generalize some results of a recent work of Tran et al. and we provide new constructions of well-rounded algebraic lattices from a certain family of modules in the ring of integers of each of these fields when $p$ is ramified in its extension over the field of rational numbers.