Well-rounded algebraic lattices coming from the maximal real subfield of the 4q-th cyclotomic field, where q is an odd number

Lattices are discrete additive subgroups of $$\mathbb {R}^n$$ known to be useful in finding solutions to important mathematical problems, such as the Sphere Packing Problem, and in applications for coding theory and post-quantum cryptography. In particular, well-rounded lattices have been considered for data transmission on Wiretap channels. A lattice obtained as an image of a free $$\mathbb {Z}$$ -submodule of the ring of algebraic integers of a number field through the canonical embedding, or a twisted version of it, is called an algebraic lattice. In 2012, Fukshansky and Petersen showed that an algebraic lattice obtained as image of the whole ring of algebraic integers of a number field $$\mathbb {K}$$ through the canonical embedding is well-rounded if and only if $$\mathbb {K}$$ is a cyclotomic field. In this work, we prove that the lattice obtained as image of the ring of algebraic integers of the maximal real subfield of the 4q-th cyclotomic field via some twisted embedding is well-rounded for any odd integer number q.