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MacWilliams identities of linear codes over Mn×s(Rk) with respect to RT metric

  • School of Mathematical Sciences of Anhui University. Hefei 230601, China

Funds:  Supported by NNSF of China (61202068), Talents youth Fund of Anhui Province Universities (2012SQRL020ZD).
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2015.01.006
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  • Author Bio:

    YAO Ting, female, born in 1989, master candidate. Research field: Algebraic codingtheory. E-mail: yaoting_1649@163.com

  • Corresponding author: SHI Minjia
  • Received Date: 05 June 2014
  • Accepted Date: 29 July 2014
  • Rev Recd Date: 29 July 2014
  • Publish Date: 30 January 2015
    Abstract

  • Abstract

    The definitions of the Lee complete ρ weight enumerator and the exact complete ρ weight enumerator over Mn×s(Rk)(u2i=0,uiuj=ujui) were given, and the MacWilliams identities with respect to the RT metric for these two weight enumerators of linear codes over Mn×s(Rk) were obtained, respectively. Finally, two examples were presented to illustrate our obtained results.

    Abstract

    The definitions of the Lee complete ρ weight enumerator and the exact complete ρ weight enumerator over Mn×s(Rk)(u2i=0,uiuj=ujui) were given, and the MacWilliams identities with respect to the RT metric for these two weight enumerators of linear codes over Mn×s(Rk) were obtained, respectively. Finally, two examples were presented to illustrate our obtained results.
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    • References(10)
  • [1]
    Xu H Q, Zhu S X. MacWilliams identities of linear codes over ring Mn×s(F2+uF2) with respect to RT metric[J]. Journal of University of Science and Technology of China, 2008, 38(9):1 075-1 080.
    许和乾, 朱士信. 环Mn×s(F2+uF2)上线性码关于RT距离的MacWilliams恒等式[J]. 中国科学技术大学学报, 2008, 38(9): 1 075-1 080.
    [2]
    Xu H Q, Du W. A MacWilliams identity respecting to ρ metric[J]. Computer Engineering, 2012, 38(19): 122-125.
    许和乾, 杜炜. 关于ρ度量的一个MacWilliam恒等式[J]. 计算机工程, 2012, 38(19): 122-125.
    [3]
    Zhu S X, Xu H Q, Shi M J. MacWilliams identity with respect to RT metric over ring Z4[J]. Acta Electronica Sinica, 2009, 37(5): 1 115-1 118.
    朱士信, 许和乾, 施敏加. 环Z4上线性码关于RT距离的MacWilliams恒等式[J]. 电子学报, 2009, 37(5): 1 115-1 118.
    [4]
    Rosenbloom M Y, Tsfasman M A. Codes for the m-metric[J]. Journal of Problems Information Transmission, 1997, 33(1): 45-52.
    [5]
    Shi M J, Solé P, Wu B. Cyclic codes and the weight enumerator of linear codes over F2+υF2+υ2F2[J]. Applied and Computational Mathematics, 2013, 12(2):247-255.
    [6]
    Siap I. The complete weight enumerator for codes over Mn×s(Fq)[J]. Lecture Notes on Computer Sciences, 2001, 2260(8): 20-26.
    [7]
    Siap I. A MacWilliams type identity[J]. Turkey Journal of Mathematics, 2002, 26(4): 465-473.
    [8]
    Siap I, Ozen M. The complete weight enumerator for codes over Mn×s(R)[J]. Applied Mathematics Letters, 2004, 17(1): 65-69.
    [9]
    Liu Y, Shi M J. The MacWilliams identity of linear codes with respect to RT metric over Mn×s(Fp+uFp+υFp+uυFp) with respect to RT metric[C]// The International Conference on Computers and Information Processing Technologies. Shanghai, China: IEEE Press, 2014.
    [10]
    Dougherty S T, Yildiz B, Karadeniz S. Codes over Rk, gray maps and their binary images[J]. Finite Fields and Their Applications, 2011, 17(3): 205-219.
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    [1]
    Xu H Q, Zhu S X. MacWilliams identities of linear codes over ring Mn×s(F2+uF2) with respect to RT metric[J]. Journal of University of Science and Technology of China, 2008, 38(9):1 075-1 080.
    许和乾, 朱士信. 环Mn×s(F2+uF2)上线性码关于RT距离的MacWilliams恒等式[J]. 中国科学技术大学学报, 2008, 38(9): 1 075-1 080.
    [2]
    Xu H Q, Du W. A MacWilliams identity respecting to ρ metric[J]. Computer Engineering, 2012, 38(19): 122-125.
    许和乾, 杜炜. 关于ρ度量的一个MacWilliam恒等式[J]. 计算机工程, 2012, 38(19): 122-125.
    [3]
    Zhu S X, Xu H Q, Shi M J. MacWilliams identity with respect to RT metric over ring Z4[J]. Acta Electronica Sinica, 2009, 37(5): 1 115-1 118.
    朱士信, 许和乾, 施敏加. 环Z4上线性码关于RT距离的MacWilliams恒等式[J]. 电子学报, 2009, 37(5): 1 115-1 118.
    [4]
    Rosenbloom M Y, Tsfasman M A. Codes for the m-metric[J]. Journal of Problems Information Transmission, 1997, 33(1): 45-52.
    [5]
    Shi M J, Solé P, Wu B. Cyclic codes and the weight enumerator of linear codes over F2+υF2+υ2F2[J]. Applied and Computational Mathematics, 2013, 12(2):247-255.
    [6]
    Siap I. The complete weight enumerator for codes over Mn×s(Fq)[J]. Lecture Notes on Computer Sciences, 2001, 2260(8): 20-26.
    [7]
    Siap I. A MacWilliams type identity[J]. Turkey Journal of Mathematics, 2002, 26(4): 465-473.
    [8]
    Siap I, Ozen M. The complete weight enumerator for codes over Mn×s(R)[J]. Applied Mathematics Letters, 2004, 17(1): 65-69.
    [9]
    Liu Y, Shi M J. The MacWilliams identity of linear codes with respect to RT metric over Mn×s(Fp+uFp+υFp+uυFp) with respect to RT metric[C]// The International Conference on Computers and Information Processing Technologies. Shanghai, China: IEEE Press, 2014.
    [10]
    Dougherty S T, Yildiz B, Karadeniz S. Codes over Rk, gray maps and their binary images[J]. Finite Fields and Their Applications, 2011, 17(3): 205-219.
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