Characterization of totally real subfields of $2$-power cyclotomic fields and applications to signal set design

Abstract

A classification of all totally real subfields $\mathbb{K}$ of cyclotomic fields $\mathbb{Q}(\xi_{2^r})$, for any $r\geq 4$, and the fully-diverse related versions of the \linebreak $\mathbb Z^n$-lattice are presented along with closed-form expressions for their minimum \linebreak product distance. Any totally real subfield $\mathbb{K}$ of $\mathbb{Q}(\xi_{2^r})$ must be of the form \linebreak $\mathbb{K}=\mathbb{Q}(\xi_{2^s} + \xi_{2^s}^{-1})$, where $s=r-j$ for some $0 \leq j \leq r-3$. Signal constellations for transmitting information over both Gaussian and Rayleigh fading channels (which can be useful for mobile communications) can be carved out of those lattices.

Received: 31 May 2021 | Accepted: 27 February 2023