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

Official URL: https://dx.doi.org/10.1109/TIT.2023.3272566

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 )\{0} into 6-subsets of type { a , b , – a – b , – a , – b , a + b } and 3-subsets of type { c , – c , 2 c } 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 .