Lattices with a circulant generator matrix represent a subclass of cyclic lattices. This subclass can be described by a basis containing a vector and its circular shifts. In this paper, we present certain conditions under which the norm expression of an arbitrary vector of this type of lattice is substantially simplified, and then investigate some of the lattices obtained under these conditions. We exhibit systems of nonlinear equations whose solutions yield lattices as dense as $D_n$ in odd dimensions. As far as even dimensions, we obtain lattices denser than $A_n$ as long as $n \in 2\mathbb{Z} \backslash 4\mathbb{Z}$.

翻译：具有循环生成矩阵的格是循环格的一个子类。该子类可由包含一个向量及其循环移位基组来描述。本文提出了若干条件，在此类条件下此类格中任意向量的范数表达式可大幅简化，并进一步研究了这些条件下获得的某些格。我们构造了非线性方程组系统，其解在奇数维度下可产生与$D_n$格密度相同的格。对于偶数维度，只要$n \in 2\mathbb{Z} \backslash 4\mathbb{Z}$，我们即可获得比$A_n$格更稠密的格。