Characteristics of invariant weights related to code equivalence over rings (original) (raw)

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](/article/10.1007/s10623-012-9671-9#ref-CR5 "Greferath M., Honold T.: Monomial extensions of isometries of linear codes II: invariant weight functions on Z

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              . In: Proceedings of the Tenth International Workshop in Algebraic and Combinatorial Coding Theory (ACCT-10), pp. 106–111. Zvenigorod, Russia (2006).")).

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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

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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
  2. Cathy Mc Fadden
  3. Jens Zumbrägel

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Correspondence toCathy Mc Fadden.

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

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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

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