Designs, Codes and Cryptography

, Volume 57, Issue 2, pp 169–179 | Cite as

New bounds for codes over finite Frobenius rings

  • Eimear Byrne
  • Marcus Greferath
  • Axel Kohnert
  • Vitaly Skachek
Article

Abstract

We give further results on the question of code optimality for linear codes over finite Frobenius rings for the homogeneous weight. This article improves on the existing Plotkin bound derived in an earlier paper (Greferath and O’Sullivan, Discr Math 289:11–24, 2004) and suggests a version of a Singleton bound. We also present some families of codes meeting these new bounds.

Keywords

Codes over rings Finite Frobenius rings Homogeneous weights Plotkin and Singleton bounds 

Mathematics Subject Classification (2000)

94B65 16L60 05E99 

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References

  1. 1.
    Byrne E., Greferath M., O’Sullivan M.E.: The linear programming bound for codes over finite Frobenius rings. Des. Codes Cryptogr. 42(3), 289–301 (2007).MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bachoc C.: Applications of coding theory to the construction of modular lattices. J. Comb. Theory Ser. A 78-1, 92–119 (1997).CrossRefMathSciNetGoogle Scholar
  3. 3.
    Constantinescu I.: Lineare Codes über Restklassenringen ganzer Zahlen und ihre Automorphismen bezüglich einer verallgemeinerten Hamming-Metrik. Ph.D. thesis, Technische Universität München (1995).Google Scholar
  4. 4.
    Constantinescu I., Heise W.: A metric for codes over residue class rings of integers. Prob. Peredachi Inform. 33(3), 22–28 (1997).MathSciNetGoogle Scholar
  5. 5.
    Greferath M., Schmidt S.E.: Finite-ring combinatorics and MacWilliams equivalence theorem. J. Comb. Theory A 92, 17–28 (2000).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Greferath M., O’Sullivan M.E.: On bounds for codes over Frobenius rings under homogeneous weights. Discr. Math. 289, 11–24 (2004).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Greferath M., McGuire G., O’Sullivan M.E.: On Plotkin optimal codes over finite Frobenius rings. J. Algebra Appl. 5(6), 799–815 (2006).MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hammons A.R., Kumar P.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
  9. 9.
    Heise W., Honold T., Nechaev A.A.: Weighted modules and representations of codes. Proceedings of the ACCT 6 Pskov, Russia, pp. 123–129 (1998).Google Scholar
  10. 10.
    Honold T.: A characterization of finite Frobenius rings. Arch. Math. (Basel) 76, 406–415 (2001).MATHMathSciNetGoogle Scholar
  11. 11.
    Kaya R., Plaumann P., Strambach K.: Rings and geometry, NATO ASI Series, Reidel (1984).Google Scholar
  12. 12.
    Nakayama T., Ikeda M.: Supplementary remarks on Frobeniusean algebras II. Osaka Math J. 2(1), 7–12 (1950).MATHMathSciNetGoogle Scholar
  13. 13.
    Nechaev A.A.: Kerdock codes in a cyclic form. Discr. Math. Appl. 1, 365–384 (1991).MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Wood J.A.: Duality for modules over finite rings and applications to coding theory. Am. J. Math. 121, 555–575 (1999).MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Eimear Byrne
    • 1
  • Marcus Greferath
    • 1
  • Axel Kohnert
    • 2
  • Vitaly Skachek
    • 1
    • 3
  1. 1.School of Mathematical SciencesUniversity College DublinBelfield, Dublin 4Ireland
  2. 2.Department of MathematicsUniversity of BayreuthBayreuthGermany
  3. 3.Nanyang Technological UniversitySingaporeSingapore

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