Quasi-cyclic perfect codes in Doob graphs and special partitions of Galois rings

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