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

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