Skip to main content
SpringerLink
Log in

Characteristics of invariant weights related to code equivalence over rings

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

The Equivalence Theorem states that, for a given weight on an alphabet, every isometry between linear codes extends to a monomial transformation of the entire space. This theorem has been proved for several weights and alphabets, including the original MacWilliams’ Equivalence Theorem for the Hamming weight on codes over finite fields. The question remains: What conditions must a weight satisfy so that the Extension Theorem will hold? In this paper we provide an algebraic framework for determining such conditions, generalising the approach taken in Greferath and Honold (Proceedings of the Tenth International Workshop in Algebraic and Combinatorial Coding Theory (ACCT-10), pp. 106–111. Zvenigorod, Russia, 2006).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Clark W.E., Drake D.A.: Finite chain rings. Abh. Math. Sem. Univ. Hamburg, 39, 147–153 (1973)

    Article  MathSciNet  Google Scholar 

  2. Constantinescu I., Heise W.: On the concept of code-isomorphy. J. Geom. 57(1–2), 63–69 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Constantinescu I., Heise W., Honold T.: Monomial extensions of isometries between codes over Z m . In: Proceedings of the Fifth International Workshop in Algebraic and Combinatorial Coding Theory (ACCT-5), pp. 98–104. Sozopol, Bulgaria (1996).

  4. Greferath M., Honold T.: On weights allowing for MacWilliams equivalence theorem. In: Proceedings of the Fourth International Workshop in Optimal Codes and Related Topics, pp. 182–192. Pamporovo, Bulgaria (2005).

  5. Greferath M., Honold T.: Monomial extensions of isometries of linear codes II: invariant weight functions on Z m . In: Proceedings of the Tenth International Workshop in Algebraic and Combinatorial Coding Theory (ACCT-10), pp. 106–111. Zvenigorod, Russia (2006).

  6. Greferath M., Schmidt S.E.: Finite-ring combinatorics and MacWilliams’ equivalence theorem. J. Comb. Theory Ser. A 92(1), 17–28 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Greferath M., Nechaev A., Wisbauer R.: Finite quasi-Frobenius modules and linear codes. J. Algebra Appl. 3(3), 247–272 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hammons A.R. Jr., 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. Inf. Theory 40(2), 301–319 (1994)

    Article  MATH  Google Scholar 

  9. MacWilliams J.: A theorem on the distribution of weights in a systematic code. Bell Syst. Tech. J. 42, 79–94 (1963)

    MathSciNet  Google Scholar 

  10. McDonald B.R.: Finite Rings with Identity. Marcel Dekker Inc., New York (1974)

    MATH  Google Scholar 

  11. Nechaev A.A.: Finite rings of principal ideals. Mat. Sb. (N.S.) 91(133), 350–366, 471 (1973)

    MathSciNet  Google Scholar 

  12. Nechaev A.A., Honold T.: Fully weighted modules and representations of codes. Problemy Peredachi Informatsii 35(3), 18–39 (1999)

    MathSciNet  Google Scholar 

  13. Roman S.: Coding and Information Theory Volume 134 of Graduate Texts in Mathematics. Springer, New York (1992)

    Google Scholar 

  14. Stanley R.P.: Enumerative Combinatorics. Volume 1, Volume 9 of Cambridge Studies in Advanced Mathematics, 2nd edn. Cambridge University Press, Cambridge (2012)

    Google Scholar 

  15. Ward H.N., Wood J.A.: Characters and the equivalence of codes. J. Comb. Theory Ser. A 73(2), 348–352 (1996)

    MathSciNet  MATH  Google Scholar 

  16. Wood J.A.: Extension theorems for linear codes over finite rings. In: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (Toulouse, 1997), Volume 1255 of Lecture Notes in Computer Science, pp. 329–340. Springer, Berlin (1997).

  17. Wood J.A.: Duality for modules over finite rings and applications to coding theory. Am. J. Math. 121(3), 555–575 (1999)

    Article  MATH  Google Scholar 

  18. Wood J.A.: Factoring the semigroup determinant of a finite commutative chain ring. In: Coding Theory, Cryptography and Related Areas (Guanajuato, 1998), pp 249–259. Springer, Berlin (2000).

  19. Wood J.A.: Code equivalence characterizes finite Frobenius rings. Proc. Am. Math. Soc. 136(2), 699–706 (2008) (electronic).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University College Dublin, Dublin, Republic of Ireland

    Marcus Greferath, Cathy Mc Fadden & Jens Zumbrägel

  2. Claude Shannon Institute for Discrete Mathematics, Coding, Cryptography, and Information Security, Dublin, Republic of Ireland

    Marcus Greferath, Cathy Mc Fadden & Jens Zumbrägel

Authors
  1. Marcus Greferath
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cathy Mc Fadden
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jens Zumbrägel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cathy Mc Fadden.

Additional information

This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Greferath, M., Mc Fadden, C. & Zumbrägel, J. Characteristics of invariant weights related to code equivalence over rings. Des. Codes Cryptogr. 66, 145–156 (2013). https://doi.org/10.1007/s10623-012-9671-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-012-9671-9

Keywords

Mathematics Subject Classification

Search

Navigation