Designs, Codes and Cryptography

, Volume 72, Issue 2, pp 331–344 | Cite as

Relative one-weight linear codes

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Abstract

Relative one-weight linear codes were introduced by Liu and Chen over finite fields. These codes can be defined just as simply for egalitarian and homogeneous weights over Frobenius bimodule alphabets. A key lemma helps describe the structure of relative one-weight codes, and certain known types of two-weight linear codes can then be constructed easily. The key lemma also provides another approach to the MacWilliams extension theorem.

Keywords

Relative one-weight codes Two-weight codes Extension theorem Homogeneous weight Egalitarian weight 

Mathematics Subject Classification (2000)

94B05 
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Notes

Acknowledgments

I thank the Department of Mathematics, Huazhong Normal University, Wuhan, China, and especially Professors Yun Fan and Hongwei Liu, for their hospitality during the summer of 2011, when much of the research for this paper was conducted. I thank the referees for their helpful comments, especially those related to the history of homogeneous weights, and for saving me from some embarassing typos. I also thank my wife Elizabeth S. Moore for her continuing support and encouragement. This work was partially supported by a sabbatical leave from Western Michigan University.

References

  1. 1.
    Assmus Jr, E.F., Mattson Jr, H.F.: Error-correcting codes: an axiomatic approach. Inform. Control 6, 315–330 (1963).Google Scholar
  2. 2.
    Bogart K., Goldberg D., Gordon J.: An elementary proof of the MacWilliams theorem on equivalence of codes. Inform. Control 37(1), 19–22 (1978).Google Scholar
  3. 3.
    Calderbank A.R., Kantor W.M.: The geometry of two-weight codes. Bull. London Math. Soc. 18, 97–122 (1986).Google Scholar
  4. 4.
    Camion P.: Codes and association schemes: basic properties of association schemes relevant to coding. In: Handbook of Coding Theory, vols. I, II, pp. 1441–1566. North-Holland, Amsterdam (1998).Google Scholar
  5. 5.
    Chen W., Kløve T.: The weight hierarchies of \(q\)-ary codes of dimension \(4\). IEEE Trans. Inform. Theory 42 (6, part 2), 2265–2272 (1996).Google Scholar
  6. 6.
    Constantinescu I., Heise W.: On the concept of code-isomorphy. J. Geom. 57(1–2), 63–69 (1996).Google Scholar
  7. 7.
    Greferath M.: Orthogonality matrices for modules over finite Frobenius rings and MacWilliams’ equivalence theorem. Finite Fields Appl. 8(3), 323–331 (2002).Google Scholar
  8. 8.
    Greferath M., Nechaev A., Wisbauer R.: Finite quasi-Frobenius modules and linear codes. J. Algebra Appl. 3(3), 247–272 (2004).Google Scholar
  9. 9.
    Greferath M., Schmidt S.E.: Finite-ring combinatorics and MacWilliams’s equivalence theorem. J. Combin. Theory Ser. A 92(1), 17–28 (2000).Google Scholar
  10. 10.
    Heise W., Honold T.: Homogeneous and egalitarian weights on finite rings. In: Proceedings of the Seventh International Workshop on Algebraic and Combinatorial Coding Theory (ACCT-2000), pp. 183–188. Bansko, Bulgaria (2000).Google Scholar
  11. 11.
    Hirano Y.: On admissible rings. Indag. Math. (N.S.) 8(1), 55–59 (1997).Google Scholar
  12. 12.
    Honold T.: Characterization of finite Frobenius rings. Arch. Math. (Basel) 76(6), 406–415 (2001).Google Scholar
  13. 13.
    Honold T.: Further results on homogeneous two-weight codes. Proceedings of Optimal Codes and Related Topics. Bulgaria (2007).Google Scholar
  14. 14.
    Honold T., Landjev I.: Linear codes over finite chain rings. Electron. J. Combin. 7, Research Paper 11, 22 (2000). http://www.combinatorics.org/Volume_7/Abstracts/v7i1r11.html.
  15. 15.
    Honold T., Nechaev A.A.: Weighted modules and representations of codes. Problems Inform. Transm. 35(3), 205–223 (1999).Google Scholar
  16. 16.
    Lam T.Y.: Lectures on modules and rings, Graduate Texts in Math., vol. 189. Springer, New York (1999).Google Scholar
  17. 17.
    Liu Z., Chen W.: Notes on the value function. Des. Codes Cryptogr. 54(1), 11–19 (2010).Google Scholar
  18. 18.
    Liu Z., Chen W., Sun Z., Zeng X.: Further results on support weights of certain subcodes. Des. Codes Cryptogr. 61(2), 119–129 (2011).Google Scholar
  19. 19.
    MacWilliams F.J.: Error-correcting codes for multiple-level transmission. Bell System Tech. J. 40, 281–308 (1961).Google Scholar
  20. 20.
    Peterson W.W.: Error-Correcting Codes. The MIT Press, Cambridge (1961).Google Scholar
  21. 21.
    Stanley R.P.: Enumerative combinatorics, The Wadsworth& Brooks/Cole Mathematics Series, vol. I. Wadsworth& Brooks/Cole Advanced Books& Software, Monterey, CA (1986).Google Scholar
  22. 22.
    Terras A.: Fourier analysis on finite groups and applications, London Mathematical Society Student Texts, vol. 43. Cambridge University Press, Cambridge (1999).Google Scholar
  23. 23.
    Tsfasman M.A., Vlăduţ S.G.: Algebraic-geometric codes. In: Mathematics and its Applications (Soviet Series), vol. 58. Kluwer Academic Publishers Group, Dordrecht (1991).Google Scholar
  24. 24.
    Wood J.A.: Duality for modules over finite rings and applications to coding theory. Am. J. Math. 121(3), 555–575 (1999).Google Scholar
  25. 25.
    Wood J.A.: The structure of linear codes of constant weight. Trans. Am. Math. Soc. 354(3), 1007–1026 (2002).Google Scholar
  26. 26.
    Wood J.A.: Foundations of linear codes defined over finite modules: the extension theorem and the MacWilliams identities. In: Codes Over Rings (Ankara, 2008), Ser. Coding Theory Cryptol., vol. 6, pp. 124–190. World Sci. Publ., Hackensack (2009).Google Scholar
  27. 27.
    Wood J.A.: Applications of finite Frobenius rings to the foundations of algebraic coding theory. In: Iyama, O. (ed.) Proceedings of the 44th Symposium on Rings and Representation Theory (Okayama University, Japan, September 25–27, 2011), pp. 223–245. Nagoya (2012).Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of MathematicsWestern Michigan UniversityKalamazooUSA

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