On non-antipodal binary completely regular codes

J. Borges, J. Rifà, V. A. Zinoviev

Research output: Contribution to journalArticleResearchpeer-review

17 Citations (Scopus)

Abstract

Binary non-antipodal completely regular codes are characterized. Using a result on nonexistence of nontrivial binary perfect codes, it is concluded that there are no unknown nontrivial non-antipodal completely regular binary codes with minimum distance d ≥ 3. The only such codes are halves and punctured halves of known binary perfect codes. Thus, new such codes with covering radius ρ = 6 and 7 are obtained. In particular, a half of the binary Golay [23, 12, 7]-code is a new binary completely regular code with minimum distance d = 8 and covering radius ρ = 7. The punctured half of the Golay code is a new completely regular code with minimum distance d = 7 and covering radius ρ = 6. The new code with d = 8 disproves the known conjecture of Neumaier, that the extended binary Golay [24, 12, 8]-code is the only binary completely regular code with d ≥ 8. Halves of binary perfect codes with Hamming parameters also provide an infinite family of binary completely regular codes with d = 4 and ρ = 3. Puncturing of these codes also provide an infinite family of binary completely regular codes with d = 3 and ρ = 2. Both these families of codes are well known, since they are uniformly packed in the narrow sense, or extended such codes. Some of these completely regular codes are new completely transitive codes. © 2007 Elsevier B.V. All rights reserved.
Original languageEnglish
Pages (from-to)3508-3525
JournalDiscrete Mathematics
Volume308
DOIs
Publication statusPublished - 28 Aug 2008

Keywords

  • Antipodality
  • Binary code
  • Completely regular code
  • Golay code
  • Perfect code

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