Construction of Complex Lattice Codes via Cyclotomic Fields

ABSTRACT

Through algebraic number theory and Construction A we extend an algebraic procedure which generates nested complex lattice codes from the polynomial ring ${\mathbb{F}}_2\left[x\right]/\left(x^n-1\right)$, where ${\mathbb{F}}_2=\left\{0,1\right\}$, by using ideals from the generalized polynomial ring $\frac{{\mathbb{F}}_2\left[x,{\displaystyle \frac{1}{2}}{\mathrm{\mathbb{Z}}}_0\right]}{\left({\left(x^{{\displaystyle \frac{1}{2}}}\right)}^{2n}-1\right)}$ through the ring of integers 𝒪_𝕃 of the cyclotomic field $\mathbb{L}=\mathrm{\mathbb{Q}}\left({\zeta }_{2^s}\right)$, where ${\zeta }_{2^s}$ is a 2^s -th root of the unit, with s > 2.

Keywords:
complex lattice codes; binary cyclotomic fields; monoid Rings