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

Creative Commons License

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

Abstract

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.