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Linear codes over \({\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}\)

  • Bahattin Yildiz
  • Suat Karadeniz
Article

Abstract

In this work, we investigate linear codes over the ring \({\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}\) . 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 \({\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}\) -linearity of binary codes under the Gray map and give a main class of binary codes as an example of \({\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}\) -linear codes. The duals and the complete weight enumerators for \({\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}\) -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 \({\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}\) are obtained.

Keywords

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

Mathematics Subject Classifications (2000)

94B05 94B99 11T71 13M99 

References

  1. 1.
    Betsumiya K., Ling S., Nemenzo F.R.: Type II codes over \({\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}}\) . Discrete Math. 275, 43–65 (2004)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Carlet C.: \({\mathbb{Z}_{2^k}}\) -linear codes. IEEE Trans. Inform. Theory 44, 1543–1547 (1998)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Dougherty S.T., Gaborit P., Harada M., Munemasa A., Solé P.: Type IV self-dual codes over rings. IEEE Trans. Inform. Theory 45, 2345–2360 (1999)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Dougherty S.T., Gaborit P., Harada M., Solé P.: Type II codes over \({\mathbb{F}_2+u\mathbb{F}_2}\) . IEEE Trans. Inform. Theory 45, 32–45 (1999)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Hammons A.R., Kumar V., Calderbank A.R., Sloane N.J.A., Solé P.: The \({\mathbb{Z}_4}\) -linearity of Kerdock, Preparata, Goethals and related codes. IEEE Trans. Inform. Theory 40, 301–319 (1994)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Huffman W.C.: Decompositions and extremal Type II codes over \({\mathbb{Z}_4}\) . IEEE Trans. Inform. Theory 44, 800–809 (1998)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    MacWilliams F.J., Sloane N.J.A.: The theory of error-correcting codes. North-Holland, Amsterdam (1977)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsFatih UniversityIstanbulTurkey

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