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Λ_c和成形格Λ_s构成嵌套格码C，当满足Λ_s⊆Λ_c。在某些条件下，C是由矩形编码形成的有限循环群。本文给出此类C存在的条件，并提供若干设计方案。这些设计方案对应于线性丢番图方程的解，使得任意码本大小M的循环格码C能够具备群同构性质——该性质是嵌套格码应用于物理层网络中继技术（如计算转发）的必要特性。