Quasi-Cyclic Perfect Codes in Doob Graphs and Special Partitions of Galois Rings

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Shi M., Li X., Krotov D. S., ÖZBUDAK F.

IEEE Transactions on Information Theory, vol.69, no.9, pp.5597-5603, 2023 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 69 Issue: 9
  • Publication Date: 2023
  • Doi Number: 10.1109/tit.2023.3272566
  • Journal Name: IEEE Transactions on Information Theory
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, PASCAL, Aerospace Database, Applied Science & Technology Source, Business Source Elite, Business Source Premier, Communication Abstracts, Compendex, Computer & Applied Sciences, INSPEC, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
  • Page Numbers: pp.5597-5603
  • Keywords: 1â perfect code, Doob graph, galois ring, quasicyclic code
  • Middle East Technical University Affiliated: Yes


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,b -ab-b,b-a,b-b, a+b} and 3-subsets of type {c,b-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 b- 1) in D((2Δb-1)(2Δb-2)/6, 2Δb-1) where D(m, n) is the Doob metric scheme on Z2m+n.