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The depth distribution of linear cyclic codes over ring Fpk+uFpk

  • School of Mathematics, Hefei University of Technology, Hefei 230009, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2014.12.004
  • Received Date: 09 January 2014
  • Accepted Date: 10 April 2014
  • Rev Recd Date: 10 April 2014
  • Publish Date: 30 December 2014
    Abstract

  • Abstract

    The depth distribution and spectrum of cyclic codes and self-dual codes of an arbitrary length over ring R=Fpk+uFpk were studied. Using the generator polynomials of cyclic codes and the depth distribution of linear codes over ring R, the depth distribution and depth spectrum of cyclic codes and self-dual codes over the ring R were given, along with those of cyclic codes of length pm were also given.

    Abstract

    The depth distribution and spectrum of cyclic codes and self-dual codes of an arbitrary length over ring R=Fpk+uFpk were studied. Using the generator polynomials of cyclic codes and the depth distribution of linear codes over ring R, the depth distribution and depth spectrum of cyclic codes and self-dual codes over the ring R were given, along with those of cyclic codes of length pm were also given.
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    • References(15)
  • [1]
    Etzion T. The depth distribution: A new characterization for linear codes[J]. IEEE Transactions on Information Theory, 1997, 43(4): 1 361-1 363.
    [2]
    Mitchell J C. On integer-valued rational polynomials and depth distributions of binary codes[J]. IEEE Transactions on Information Theory, 1998, 44(7): 3 146-3 150.
    [3]
    Luo Y, Fu F W, Wei V K W. On the depth distribution of linear codes[J]. IEEE Transactions on Information Theory, 2000, 46(6): 2 197-2 203.
    [4]
    Geng Pu, Li Chao. Depth distribution and period distribution of linear codes on finite field[J]. Journal of Applied Sciences, 2007, 25(3): 263-265.
    耿普, 李超. 有限域上线性码的深度分布与周期分布[J]. 应用科学学报, 2007, 25(3): 263-265.
    [5]
    Yang Shanlin, Zhu Shixin, Tong Hongxi. Two recursive algorithms for computing the depth of a codeword on finite ring Z4[J]. Journal of University of Science and Technology of China, 2004, 34(6): 655-660.
    杨善林, 朱士信, 童宏玺. 计算有限环Z4上码字深度的两种递归算法[J]. 中国科学技术大学学报, 2004, 34(6): 655-660.
    [6]
    Zheng X Y, Kong B. The depth spectrums of linear cyclic codes on ring Zpm[C] ∥IEEE Youth Conference on Information, Computing and Telecommunication. IEEE, 2010: 162-165.
    [7]
    Chang Xiaopeng, Zheng Xiying, Kong Bo. Depth spectrums of linear cyclic codes over the ring ZM[J]. Journal of Zhengzhou University(Engineering Science), 2012, 33(3): 110-112.
    常晓鹏, 郑喜英, 孔波. 环ZM上线性循环码的深度谱[J]. 郑州大学学报(工学版), 2012, 33(3): 110-112.
    [8]
    Yu Haifeng, Zhu Shixin. Depth distribution of linear codes over ring F2+uF2[J]. Journal of University of Science and Technology of China, 2008, 38(2): 141-144.
    余海峰, 朱士信. 环F2+uF2上线性码的深度分布[J]. 中国科学技术大学学报, 2008, 38(2): 141-144.
    [9]
    Dougherty S T, Shiromoto K. Maximum distance codes over rings of order 4[J]. IEEE Transactions on Information Theory, 2001, 47(1): 400-404.
    [10]
    Tang Gang. On the depth spectra of linear codes on ring F2+vF2[J]. Journal of Mathematics, 2012, 32(1): 186-190.
    唐刚. 环F2+vF2上线性码的深度谱[J]. 数学杂志, 2012, 32(1): 186-190.
    [11]
    Pu Keli, Liao Qunying. The depth distribution and spectrum of linear codes over the ring R=Fq+uFq(u2=0)[J]. Advances in Mathematics(China), 2014, 43(1): 57-63.
    蒲可莉, 廖群英. 环R=Fq+uFq(u2=0)上线性码的深度分布及深度谱[J].数学进展, 2014, 43(1): 57-63.
    [12]
    Pu Keli, Liao Qunying. A note on the depth spectrum and distribution of linear codes over rings[J]. Journal of Sichuan Normal University (Natural Science), 2013, 36(2): 159-164.
    蒲可莉, 廖群英. 环上线性码的深度谱以及深度分布的一个注记[J]. 四川师范大学学报（自然科学版）, 2013, 36(2): 159-164.
    [13]
    Zheng Xiying, Chang Xiaopeng. The depth distribution of linear cyclic codes over finite chain ring[J]. Journal of Henan University (Natural Science), 2012, 42(4): 347-350.
    郑喜英, 常晓鹏. 有限链环上循环码的深度分布[J]. 河南大学学报(自然科学版), 2012, 42(4): 47-350.
    [14]
    Li Ping, Zhu Shixin. Cyclic codes of arbitrary lengths over the ring Fq+uFq[J]. Journal of University of Science and Technology of China, 2008, 38(12): 1 392-1 395.
    李平, 朱士信. 环Fq+uFq上任意长度的循环码[J]. 中国科学技术大学学报, 2008, 38(12): 1392-1395.
    [15]
    Zhang Zhentao, Yang Yixian, Hu Zhengming, et al. Research on the depth distribution of linear code[J]. Journal of Beijing University of Posts and Telecommunications, 2002, 25(3): 68-72.
    张振涛, 杨义先, 胡正名, 等. 关于线性码深度分布的研究[J]. 北京邮电大学学报, 2002, 25(3): 68-72.
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    [1]
    Etzion T. The depth distribution: A new characterization for linear codes[J]. IEEE Transactions on Information Theory, 1997, 43(4): 1 361-1 363.
    [2]
    Mitchell J C. On integer-valued rational polynomials and depth distributions of binary codes[J]. IEEE Transactions on Information Theory, 1998, 44(7): 3 146-3 150.
    [3]
    Luo Y, Fu F W, Wei V K W. On the depth distribution of linear codes[J]. IEEE Transactions on Information Theory, 2000, 46(6): 2 197-2 203.
    [4]
    Geng Pu, Li Chao. Depth distribution and period distribution of linear codes on finite field[J]. Journal of Applied Sciences, 2007, 25(3): 263-265.
    耿普, 李超. 有限域上线性码的深度分布与周期分布[J]. 应用科学学报, 2007, 25(3): 263-265.
    [5]
    Yang Shanlin, Zhu Shixin, Tong Hongxi. Two recursive algorithms for computing the depth of a codeword on finite ring Z4[J]. Journal of University of Science and Technology of China, 2004, 34(6): 655-660.
    杨善林, 朱士信, 童宏玺. 计算有限环Z4上码字深度的两种递归算法[J]. 中国科学技术大学学报, 2004, 34(6): 655-660.
    [6]
    Zheng X Y, Kong B. The depth spectrums of linear cyclic codes on ring Zpm[C] ∥IEEE Youth Conference on Information, Computing and Telecommunication. IEEE, 2010: 162-165.
    [7]
    Chang Xiaopeng, Zheng Xiying, Kong Bo. Depth spectrums of linear cyclic codes over the ring ZM[J]. Journal of Zhengzhou University(Engineering Science), 2012, 33(3): 110-112.
    常晓鹏, 郑喜英, 孔波. 环ZM上线性循环码的深度谱[J]. 郑州大学学报(工学版), 2012, 33(3): 110-112.
    [8]
    Yu Haifeng, Zhu Shixin. Depth distribution of linear codes over ring F2+uF2[J]. Journal of University of Science and Technology of China, 2008, 38(2): 141-144.
    余海峰, 朱士信. 环F2+uF2上线性码的深度分布[J]. 中国科学技术大学学报, 2008, 38(2): 141-144.
    [9]
    Dougherty S T, Shiromoto K. Maximum distance codes over rings of order 4[J]. IEEE Transactions on Information Theory, 2001, 47(1): 400-404.
    [10]
    Tang Gang. On the depth spectra of linear codes on ring F2+vF2[J]. Journal of Mathematics, 2012, 32(1): 186-190.
    唐刚. 环F2+vF2上线性码的深度谱[J]. 数学杂志, 2012, 32(1): 186-190.
    [11]
    Pu Keli, Liao Qunying. The depth distribution and spectrum of linear codes over the ring R=Fq+uFq(u2=0)[J]. Advances in Mathematics(China), 2014, 43(1): 57-63.
    蒲可莉, 廖群英. 环R=Fq+uFq(u2=0)上线性码的深度分布及深度谱[J].数学进展, 2014, 43(1): 57-63.
    [12]
    Pu Keli, Liao Qunying. A note on the depth spectrum and distribution of linear codes over rings[J]. Journal of Sichuan Normal University (Natural Science), 2013, 36(2): 159-164.
    蒲可莉, 廖群英. 环上线性码的深度谱以及深度分布的一个注记[J]. 四川师范大学学报（自然科学版）, 2013, 36(2): 159-164.
    [13]
    Zheng Xiying, Chang Xiaopeng. The depth distribution of linear cyclic codes over finite chain ring[J]. Journal of Henan University (Natural Science), 2012, 42(4): 347-350.
    郑喜英, 常晓鹏. 有限链环上循环码的深度分布[J]. 河南大学学报(自然科学版), 2012, 42(4): 47-350.
    [14]
    Li Ping, Zhu Shixin. Cyclic codes of arbitrary lengths over the ring Fq+uFq[J]. Journal of University of Science and Technology of China, 2008, 38(12): 1 392-1 395.
    李平, 朱士信. 环Fq+uFq上任意长度的循环码[J]. 中国科学技术大学学报, 2008, 38(12): 1392-1395.
    [15]
    Zhang Zhentao, Yang Yixian, Hu Zhengming, et al. Research on the depth distribution of linear code[J]. Journal of Beijing University of Posts and Telecommunications, 2002, 25(3): 68-72.
    张振涛, 杨义先, 胡正名, 等. 关于线性码深度分布的研究[J]. 北京邮电大学学报, 2002, 25(3): 68-72.
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