Constructions of well-rounded algebraic lattices over odd prime degree cyclic number fields

TL;DR

This paper develops new methods for constructing well-rounded algebraic lattices from modules in cyclic number fields of odd prime degree, especially when the prime is ramified, with applications to signal transmission.

Contribution

It generalizes previous results and introduces novel constructions of well-rounded algebraic lattices in ramified cyclic number fields of odd prime degree.

Findings

New lattice constructions for ramified cyclic number fields

Extension of previous results by Tran et al.

Potential applications in signal transmission channels

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.