On the Existence of Cyclic Lattice Codes

A coding lattice ${\mathrm{\Lambda }}_c$ and a shaping lattice ${\mathrm{\Lambda }}_s$ forms a nested lattice code $\mathcal{C}$ if ${\mathrm{\Lambda }}_s\subseteq {\mathrm{\Lambda }}_c$. Under some conditions, $\mathcal{C}$ is a finite cyclic group formed by rectangular encoding. This paper presents the conditions for the existence of such $\mathcal{C}$ and provides some designs. These designs correspond to solutions to linear Diophantine equations so that a cyclic lattice code $\mathcal{C}$ of arbitrary codebook size $M$ can possess group isomorphism, which is an essential property for a nested lattice code to be applied in physical layer network relaying techniques such as compute and forward.