Construction of Complex Lattice Codes via Cyclotomic Fields

Abstract

Through algebraic number theory and Construction $A$ we extend an algebraic procedure which generates complex lattice codes from the polynomial ring \mathbb{F}_{2}[x]/(x^{n}-1), where \mathbb{F}_{2}=\{0,1\}, by using ideals from the generalized polynomial ring \frac{\mathbb{F}_{2}[x,\frac{1}{2}\mathbb{Z}_{0}]}{((x^{\frac{1}{2}})^{n}-1)} through the ring of integers $\mathcal{O}_{\mathbb{L}}$ of the cyclotomic field \mathbb{L}=\mathbb{Q}(\zeta_{2^{s}}), where \zeta_{2^{s}} is a 2^{s}-th root of the unit, with s>2.