Bulletin of the Korean Mathematical Society (대한수학회보)

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AN IDENTITY BETWEEN THE m-SPOTTY ROSENBLOOM-TSFASMAN WEIGHT ENUMERATORS OVER FINITE COMMUTATIVE FROBENIUS RINGS

  • Ozen, Mehmet (Department of Mathematics Sakarya University) ;
  • Shi, Minjia (Key Laboratory of Intelligent Computing & Signal Processing Ministry of Education, Anhui University, School of Mathematical Sciences Anhui University) ;
  • Siap, Vedat (Department of Mathematical Engineering Yildiz Technical University)
  • Received : 2014.01.22
  • Published : 2015.05.31
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Abstract

This paper is devoted to presenting a MacWilliams type identity for m-spotty RT weight enumerators of byte error control codes over finite commutative Frobenius rings, which can be used to determine the error-detecting and error-correcting capabilities of a code. This provides the relation between the m-spotty RT weight enumerator of the code and that of the dual code. We conclude the paper by giving three illustrations of the results.

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References

  1. B. C. Chen, L. Lin, and H. Liu, Matrix product codes with Rosenbloom-Tsfasman metric, Acta Math. Sci. Ser. B Engl. Ed. 33 (2013), no. 3, 687-700.
  2. S. T. Dougherty, B. Yildiz, and S. Karadeniz, Codes over Rk, Gray maps and their binary images, Finite Fields Appl. 17 (2011), no. 3, 205-219. https://doi.org/10.1016/j.ffa.2010.11.002
  3. Y. Fan, S. Ling, and H. Liu, Matrix product codes over finite commutative Frobenius rings, Des. Codes Cryptogr. 71 (2014), no. 2, 201-227. https://doi.org/10.1007/s10623-012-9726-y
  4. K. Q. Feng, L. J. Xu, and F. J. Hickernell, Linear error-block codes, Finite Fields Appl. 12 (2006), no. 4, 638-652. https://doi.org/10.1016/j.ffa.2005.03.006
  5. E. Fujiwara, Code Design for Dependable System: Theory and Practical Applications, Wiley & Son, Inc., New Jersey, 2006.
  6. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland Publishing Company, Amsterdam, 1978.
  7. M. Ozen and I. Siap, Codes over Galois rings with respect to the Rosenbloom-Tsfasman metric, J. Franklin Inst. 344 (2007), no. 5, 790-799. https://doi.org/10.1016/j.jfranklin.2006.02.001
  8. M. Ozen and V. Siap, The MacWilliams identity for m-spotty weight enumerators of linear codes over finite fields, Comput. Math. Appl. 61 (2011), no. 4, 1000-1004. https://doi.org/10.1016/j.camwa.2010.12.048
  9. M. Ozen and V. Siap, The MacWilliams identity for m-spotty Rosenbloom-Tsfasman weight enumerator, J. Franklin Inst. 351 (2014), no. 2, 743-750. https://doi.org/10.1016/j.jfranklin.2012.06.002
  10. L. Panek, M. Firer, and M. S. Alves, Classification of Niederreiter-Rosenbloom-Tsfasman block codes, IEEE Trans. Inform. Theory. 56 (2010), no. 10, 5207-5216. https://doi.org/10.1109/TIT.2010.2059590
  11. M. Y. Rosenbloom, M. A. Tsfasman, Codes for the m-metric, Problems Inform. Transmission 33 (1997), no. 1, 45-52.
  12. A. Sharma and A. K. Sharma, On MacWilliams type identities for r-fold joint m-spotty weight enumerators, Discrete Math. 312 (2012), no. 22, 3316-3327. https://doi.org/10.1016/j.disc.2012.07.030
  13. A. Sharma and A. K. Sharma, MacWilliams type identities for some new m-spotty weight enumerators, IEEE Trans. Inform. Theory. 58 (2012), no. 6, 3912-3924. https://doi.org/10.1109/TIT.2012.2189924
  14. A. Sharma and A. K. Sharma, On some new m-spotty Lee weight enumerators, Des. Codes Cryptogr. 71 (2014), no. 1, 119-152. https://doi.org/10.1007/s10623-012-9725-z
  15. M. J. Shi, MacWilliams identity for m-spotty Lee weight enumerators over $\mathbb{F}_2+u\mathbb{F}_2+{\cdot}{\cdot}{\cdot}+u^{m-1}\mathbb{F}_2$, arXiv:1307.2228v1 [cs.IT].
  16. I. Siap, MacWilliams identity for m-spotty Lee weight enumerators, Appl. Math. Lett. 23 (2010), no. 1, 13-16. https://doi.org/10.1016/j.aml.2009.07.019
  17. I. Siap, An identity between the m-spotty weight enumerators of a linear code and its dual, Turkish J. Math. 36 (2012), no. 4, 641-650.
  18. V. Siap and M. Ozen, A MacWilliams Type identity for m-spotty Rosenbloom-Tsfasman weight enumerators over Frobenius rings, CMMSE 2013: Proceedings of the 13th International Conference on Mathematical Methods in Science and Engineering, 1236-1241, Almeria, Spain, 2013.
  19. K. Suzuki and E. Fujiwara, MacWilliams identity for m-spotty weight enumerator, IEICE-Tran. Fund. Elec. Comm & Comp. Sci. E93-A (2010), no. 2, 526-531. https://doi.org/10.1587/transfun.E93.A.526
  20. K. Suzuki, T. Kashiyama, and E. Fujiwara, A general class of m-spotty weight enumerator, IEICE-Tran. Fund. Elec. Comm & Comp. Sci. E90-A (2007), no. 7, 1418-1427. https://doi.org/10.1093/ietfec/e90-a.7.1418
  21. G. Umanesan and E. Fujiwara, A class of random multiple bits in a byte error correcting and single byte error detecting (St/bEC-SbED) codes, IEEE Trans. Comput. 52 (2003), no. 7, 835-847. https://doi.org/10.1109/TC.2003.1214333
  22. J. Wood, Duality for modules over finite rings and application to coding theory, Amer. J. Math. 121 (1999), no. 3, 555-575. https://doi.org/10.1353/ajm.1999.0024