Linear codes over F2 + μF2 + νF2 + μνF2

Bahattin Yildiz, Suat Karadeniz

Research output: Contribution to journalArticlepeer-review

81 Scopus citations

Abstract

In this work, we investigate linear codes over the ring F2 + μF2 + νF2 + μνF2. We first analyze the structure of the ring and then define linear codes over this ring which turns out to be a ring that is not finite chain or principal ideal contrary to the rings that have hitherto been studied in coding theory. Lee weights and Gray maps for these codes are defined by extending on those introduced in works such as Betsumiya et al. (Discret Math 275:43-65, 2004) and Dougherty et al. (IEEE Trans Inf 45:32-45, 1999). We then characterize the F2 + μF 2 + νF2 + μνF2 -linearity of binary codes under the Gray map and give a main class of binary codes as an example of F2 + μF2 + νF2 + μνF2 -linear codes. The duals and the complete weight enumerators for Fdbl;2 + μF2 + νF2 + μνF 2 -linear codes are also defined after which MacWilliams-like identities for complete and Lee weight enumerators as well as for the ideal decompositions of linear codes over F2 + μF2 + νF2 + μνF2 are obtained.

Original languageEnglish (US)
Pages (from-to)61-81
Number of pages21
JournalDesigns, Codes, and Cryptography
Volume54
Issue number1
DOIs
StatePublished - Jan 2010
Externally publishedYes

Keywords

  • Complete weight enumerator
  • Gray maps
  • Ideal decompositions
  • Lee weights
  • Mac Williams identities
  • Reed-Muller codes

ASJC Scopus subject areas

  • Computer Science Applications
  • Applied Mathematics

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