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The extension theorem for Lee and Euclidean weight codes over integer residue rings

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Abstract

The extension theorem is proved for the Lee and Euclidean weights over the integer residue ring \(\mathbb {Z}_m\), for \(m \ge 2\).

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References

  1. Barra A.: Equivalence theorems and the local-global property. Doctoral dissertation, University of Kentucky (2012).

  2. Barra A.: MacWilliams Equivalence theorem for the Lee weight over \(\mathbb{Z}_{4p + 1}\). Malays. J. Sci. 34(2), 222–226 (2015).

    Article  Google Scholar 

  3. Constantinescu I., Heise W.: A metric for codes over residue class rings. Probl. Inf. Transm. 33, 208–213 (1997). http://mi.mathnet.ru/ppi375

  4. Constantinescu I., Heise W., Honold T.: Monomial extensions of isometries between codes over \(\mathbb{Z}_m\). In: Proceedings of the 5th International Workshop on Algebraic and Combinatorial Coding Theory (ACCT’96), Unicorn Shumen, pp. 98–104 (1996).

  5. Dyshko S., Langevin P., Wood J.A.: Deux analogues au déterminant de Maillet. Comptes Rendus Mathematique 354(7), 649–652 (2016). https://doi.org/10.1016/j.crma.2016.05.004. http://www.sciencedirect.com/science/article/pii/S1631073X16300772.

  6. Greferath M.: Orthogonality matrices for modules over finite Frobenius rings and MacWilliams’ equivalence theorem. Finite Fields Appl. 8(3), 323–331 (2002). https://doi.org/10.1006/ffta.2001.0343. http://www.sciencedirect.com/science/article/pii/S1071579701903430.

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

  8. Greferath M., Nechaev A., Wisbauer R.: Finite quasi-Frobenius modules and linear codes. J. Algebra Appl. 03(03), 247–272 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  9. Greferath M., Honold T., Fadden C.M., Wood J.A., Zumbrägel J.: MacWilliams’ extension theorem for bi-invariant weights over finite principal ideal rings. J. Combin. Theory Ser. A 125, 177–193 (2014). https://doi.org/10.1016/j.jcta.2014.03.005.

    Article  MathSciNet  MATH  Google Scholar 

  10. Hammons A.R., Kumar P.V., Calderbank A.R., Sloane N.J.A., Sole P.: The \(\mathbb{Z}_4\)-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inf. Theory 40(2), 301–319 (1994). https://doi.org/10.1109/18.312154.

    Article  MATH  Google Scholar 

  11. Huppert B.: Character Theory of Finite Groups. No. 25 in De Gruyter Expositions in Mathematics. Walter de Gruyter, Berlin (1998).

  12. Langevin P., Wood J.A.: The extension problem for Lee and Euclidean weights. J. Algebra Combin. Discret. Struct. Appl. 4(2), 207–217 (2017). https://doi.org/10.13069/jacodesmath.284970.

    MathSciNet  Google Scholar 

  13. Langevin P., Wood J.A.: The extension theorem for the Lee and Euclidean weights over \({\mathbb{Z}}/ p^k {\mathbb{Z}}\). J. Pure Appl. Algebra (2018, to appear).

  14. Lee C.: Some properties of nonbinary error-correcting codes. IRE Trans. Inf. Theory 4(2), 77–82 (1958). https://doi.org/10.1109/TIT.1958.1057446.

    Article  MathSciNet  Google Scholar 

  15. MacWilliams J.: Error-correcting codes for multiple-level transmission. Bell Syst. Tech. J. 40(1), 281–308 (1961). https://doi.org/10.1002/j.1538-7305.1961.tb03986.x. http://onlinelibrary.wiley.com/doi/10.1002/j.1538-7305.1961.tb03986.x/abstract.

  16. Mitrinović D.S.: Analytic Inequalities. Springer, Berlin (1970). https://doi.org/10.1007/978-3-642-99970-3.

    Book  MATH  Google Scholar 

  17. Satyanarayana C.: Lee metric codes over integer residue rings (Corresp.). IEEE Trans. Inf. Theory 25(2), 250–254 (1979). https://doi.org/10.1109/TIT.1979.1056017.

    Article  MathSciNet  MATH  Google Scholar 

  18. Terras A.: Fourier Analysis on Finite Groups and Applications. Cambridge University Press, Cambridge (1999).

    Book  MATH  Google Scholar 

  19. Ward H.N., Wood J.A.: Characters and the equivalence of codes. J. Combin. Theory Ser. A 73(2), 348–352 (1996). https://doi.org/10.1016/S0097-3165(96)80011-2. http://www.sciencedirect.com/science/article/pii/S0097316596800112.

  20. Wood J.A.: Duality for modules over finite rings and applications to coding theory. Am. J. Math. 121(3), 555–575 (1999). http://www.jstor.org/stable/25098937.

  21. Wood J.A.: Weight functions and the extension theorem for linear codes over finite rings. Contemp. Math. 225, 231–243 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  22. Wood J.A.: The structure of linear codes of constant weight. Trans. Am. Math. Soc. 354(3), 1007–1026 (2002). https://doi.org/10.1090/S0002-9947-01-02905-1.

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Serhii Dyshko.

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Communicated by I. Landjev.

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Dyshko, S. The extension theorem for Lee and Euclidean weight codes over integer residue rings. Des. Codes Cryptogr. 87, 1253–1269 (2019). https://doi.org/10.1007/s10623-018-0521-2

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