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

05/04/2023

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by   Minjia Shi, et al.

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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,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.