The Galois ring GR(4Δ) is the residue ring Z4[x]/(h(x)), where
h(x) is a basic primitive polynomial of degree Δ over Z4. For any
odd Δ larger than 1, we construct a partition of GR(4Δ)\{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(4Δ) and, if Δ is not a multiple of 3, under the action of
the automorphism group of GR(4Δ).
As a corollary, this implies the existence of quasi-cyclic additive
1-perfect codes of index (2Δ−1) in D((2Δ−1)(2Δ−2)/6,2Δ−1) where D(m,n) is the Doob metric scheme on Z2m+n.Comment: Accepted version; 7 IEEE TIT page