A coding lattice $\Lambda_c$ and a shaping lattice $\Lambda_s$ forms a nested lattice code $\mathcal{C}$ if $\Lambda_s \subseteq \Lambda_c$. 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.

翻译：编码格 $\Lambda_c$ 与成形格 $\Lambda_s$ 在满足 $\Lambda_s \subseteq \Lambda_c$ 时构成嵌套格码 $\mathcal{C}$。在特定条件下，$\mathcal{C}$ 为通过矩形编码形成的有限循环群。本文给出了此类 $\mathcal{C}$ 存在的条件，并提供了若干设计方案。这些设计对应于线性丢番图方程的解，使得具有任意码本大小 $M$ 的循环格码 $\mathcal C$ 可具备群同构性，这是嵌套格码应用于物理层网络中继技术（如计算转发）的关键性质。