Well-rounded lattices from odd prime degree number fields in the ramified case

Abstract Recently, algebraic lattices have been widely considered for applications in coding theory and cryptography. Likewise, well-rounded lattices have proven useful for signal transmission in MIMO and SISO wiretap channels. Previous works have conducted extensive studies on well-rounded algebraic lattices. One of them demonstrates the existence of well-rounded lattices obtained through the Minkowski embedding, representing the image of ℤ-modules in the ring of integers of cyclic number fields 𝕂 of odd prime degree p , where p remains unramified in the extension 𝕂/ℚ. In this work, we consider the case where p is ramified in 𝕂/ℚ and introduce a family of ℤ-modules that realize well-rounded algebraic lattices in ℝ p for many (potentially infinitely many) values of p .