The Galois ring GR$(4^\Delta)$ is the residue ring $Z_4[x]/(h(x))$, where $h(x)$ is a basic primitive polynomial of degree $\Delta$ over $Z_4$. For any odd $\Delta$ larger than $1$, we construct a partition of GR$(4^\Delta) \backslash \{0\}$ into $6$-subsets of type $\{a,b,-a-b,-a,-b,a+b\}$ and $3$-subsets of type $\{c,-c,2c\}$ such that the partition is invariant under the multiplication by a nonzero element of the Teichmuller set in GR$(4^\Delta)$ and, if $\Delta$ is not a multiple of $3$, under the action of the automorphism group of GR$(4^\Delta)$. As a corollary, this implies the existence of quasi-cyclic additive $1$-perfect codes of index $(2^\Delta-1)$ in $D((2^\Delta-1)(2^\Delta-2)/{6}, 2^\Delta-1 )$ where $D(m,n)$ is the Doob metric scheme on $Z^{2m+n}$.

翻译：Galois环GR$(4^\Delta)$是剩余环$Z_4[x]/(h(x))$，其中$h(x)$是$Z_4$上次数为$\Delta$的基本本原多项式。对于任意大于1的奇数$\Delta$，我们将GR$(4^\Delta) \backslash \{0\}$划分为$\{a,b,-a-b,-a,-b,a+b\}$型6元子集与$\{c,-c,2c\}$型3元子集，使得该划分在GR$(4^\Delta)$中Teichmüller集非零元的乘法下保持不变，且当$\Delta$不是3的倍数时，在GR$(4^\Delta)$的自同构群作用下亦保持不变。作为推论，这蕴含了在$D((2^\Delta-1)(2^\Delta-2)/{6}, 2^\Delta-1)$（其中$D(m,n)$为$Z^{2m+n}$上的Doob度量方案）中存在指标为$(2^\Delta-1)$的拟循环加法1-完美码。