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Negacyclic codes of arbitrary lengths over ring Fq+uFq+u2Fq

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

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2014.12.005
  • Received Date: 25 December 2013
  • Accepted Date: 10 April 2014
  • Rev Recd Date: 10 April 2014
  • Publish Date: 30 December 2014
    Abstract

  • Abstract

    According to the structure of negacyclic codes over ring Fq+uFq, the negacyclic codes of arbitrary lengths over ring Fq+uFq+u2Fq were studied. Bying ring homomorphism, the structure of negacyclic codes over ring Fq+uFq+u2Fq of length n were given, the dual codes of these negacyclic codes were studied, and the self-dual negacyclic codes over the ring were also studied. The results show that self-dual negacyclic codes over ring Fq+uFq+u2Fq exist if and only if p=2.

    Abstract

    According to the structure of negacyclic codes over ring Fq+uFq, the negacyclic codes of arbitrary lengths over ring Fq+uFq+u2Fq were studied. Bying ring homomorphism, the structure of negacyclic codes over ring Fq+uFq+u2Fq of length n were given, the dual codes of these negacyclic codes were studied, and the self-dual negacyclic codes over the ring were also studied. The results show that self-dual negacyclic codes over ring Fq+uFq+u2Fq exist if and only if p=2.
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    • References(16)
  • [1]
    Abualrub T,Oehmke R. On the generators of Z4 cyclic codes of length 2e[J]. IEEE Transactions on Information Theory, 2003,49 (9):2 126-2 133.
    [2]
    Blackford T. Cyclic codes over Z4 of odely even length[J]. Discrete Applied Mathematics, 2003,128 (1):27-46.
    [3]
    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):1 392-1 396.
    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):1 392-1 396.
    [4]
    Dinh H Q. Constacyclic codes of length ps over Fpm+uFpm[J]. Journal of Algebra, 2010,324(5):940-950.
    [5]
    Ding Jian, Li Hongju, Liu Jiabao. A class of constacyclic codes over the ring Fpm+uFpm[J]. Journal of Hefei University of Technology(Natural Science), 2011,34 (4): 634-640.
    丁健,李红菊,刘家保. 环Fpm+uFpm上的一类常循环码[J].合肥工业大学学报(自然科学版),2011,34 (4):634-640.
    [6]
    Ding Jian, Li Hongju, Li Haixia. On the equivalence of constacyclic codes over the ring Fpm+uFpm[J]. Journal of University of Science and Technology of China, 2013,43 (4):334-339.
    丁健,李红菊,李海霞. 关于环Fpm+uFpm上常循环码的等价性[J]. 中国科学技术大学学报,2013,43 (4):334-339.
    [7]
    Abualrub T,Siap I. Cyclic codes over the ring Z2+uZ2 and Z2+uZ2+u2Z2[J]. Designs Codes and Cryptography, 2007,42(3):273-287.
    [8]
    Yu Haifeng, Zhu Shixin. (1+u+u2)-cyclic codes over ring F2+uF2+u2F2[J]. Application Research of Computers, 2010,27(5):1 845-1 846.
    余海峰, 朱士信. 环F2+uF2+u2F2上的(1+u+u2)-循环码[J]. 计算机应用研究,2010,27(5):1 845- 1 846.
    [9]
    Shi Minjia. Constacyclic self-dual codes over ring F2+uF2+…+um－1F2[J]. Acta Electronica Sinica, 2013, 41(6): 1 088-1 092.
    施敏加. 环F2+uF2+…+um－1F2上的常循环自对偶码[J]. 电子学报, 2013，41（6）：1 088-1 092.
    [10]
    Shi Minjia, Zhu Shixin. Constacyclic codes over ring Fp+uFp+…+us－1Fp[J]. Journal of University of Science and Technology of China, 2009, 39(6): 583-587.
    施敏加，朱士信. 环Fp+uFp+…+us－1Fp上的常循环码[J].中国科学技术大学学报，2009，39（6）：583-587.
    [11]
    Dinh H Q, López-Permouth S R. Cyclic and negacyclic codes over finite chain rings[J]. IEEE Transactions on Information Theory, 2004, 50(8): 1 728-1 744.
    [12]
    Blackford T. Negacyclic codes over Z4 of even length[J]. IEEE Transactions on Information Theory, 2003, 49(6): 1 417-1 424.
    [13]
    Dinh H Q. Complete Distances of all negacyclic codes of length 2s over Z2a [J]. IEEE Transactions on Information Theory, 2007, 53(1): 147-161.
    [14]
    Zhu Shixin, Kai Xiaoshan. On the homogeneous distance of negacyclic codes over Z2a[J/OL]. Sciencepaper Online, [2012-02-28]. http://www.paper.edu.cn/releasepaper/content/201202-1048.
    朱士信，开晓山. 关于Z2a上的负循环码的齐次距离[J/OL]. 中国科技论文在线， [2012-02-28]. http://www.paper.edu.cn/releasepaper/content/201202-1048.
    [15]
    Xi Hongqi, Zheng Xiying, Kong Bo. Repeated-root negacyclic codes over Zpa[J]. Journal of Zhengzhou University (Natural Science Edition), 2012,44(3):26-28.
    席红旗, 郑喜英, 孔波. 环Zpa上的重根负循环码[J]. 郑州大学学报(理学版), 2012,44(3):26-28.
    [16]
    Al-Ashker M M, Chen J Z. Cyclic codes of arbitrary length over Fp+uFp+…+ukFp [J]. Palestine Journal of Mathermatics, 2013,2(1):72-80.
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    [1]
    Abualrub T,Oehmke R. On the generators of Z4 cyclic codes of length 2e[J]. IEEE Transactions on Information Theory, 2003,49 (9):2 126-2 133.
    [2]
    Blackford T. Cyclic codes over Z4 of odely even length[J]. Discrete Applied Mathematics, 2003,128 (1):27-46.
    [3]
    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):1 392-1 396.
    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):1 392-1 396.
    [4]
    Dinh H Q. Constacyclic codes of length ps over Fpm+uFpm[J]. Journal of Algebra, 2010,324(5):940-950.
    [5]
    Ding Jian, Li Hongju, Liu Jiabao. A class of constacyclic codes over the ring Fpm+uFpm[J]. Journal of Hefei University of Technology(Natural Science), 2011,34 (4): 634-640.
    丁健,李红菊,刘家保. 环Fpm+uFpm上的一类常循环码[J].合肥工业大学学报(自然科学版),2011,34 (4):634-640.
    [6]
    Ding Jian, Li Hongju, Li Haixia. On the equivalence of constacyclic codes over the ring Fpm+uFpm[J]. Journal of University of Science and Technology of China, 2013,43 (4):334-339.
    丁健,李红菊,李海霞. 关于环Fpm+uFpm上常循环码的等价性[J]. 中国科学技术大学学报,2013,43 (4):334-339.
    [7]
    Abualrub T,Siap I. Cyclic codes over the ring Z2+uZ2 and Z2+uZ2+u2Z2[J]. Designs Codes and Cryptography, 2007,42(3):273-287.
    [8]
    Yu Haifeng, Zhu Shixin. (1+u+u2)-cyclic codes over ring F2+uF2+u2F2[J]. Application Research of Computers, 2010,27(5):1 845-1 846.
    余海峰, 朱士信. 环F2+uF2+u2F2上的(1+u+u2)-循环码[J]. 计算机应用研究,2010,27(5):1 845- 1 846.
    [9]
    Shi Minjia. Constacyclic self-dual codes over ring F2+uF2+…+um－1F2[J]. Acta Electronica Sinica, 2013, 41(6): 1 088-1 092.
    施敏加. 环F2+uF2+…+um－1F2上的常循环自对偶码[J]. 电子学报, 2013，41（6）：1 088-1 092.
    [10]
    Shi Minjia, Zhu Shixin. Constacyclic codes over ring Fp+uFp+…+us－1Fp[J]. Journal of University of Science and Technology of China, 2009, 39(6): 583-587.
    施敏加，朱士信. 环Fp+uFp+…+us－1Fp上的常循环码[J].中国科学技术大学学报，2009，39（6）：583-587.
    [11]
    Dinh H Q, López-Permouth S R. Cyclic and negacyclic codes over finite chain rings[J]. IEEE Transactions on Information Theory, 2004, 50(8): 1 728-1 744.
    [12]
    Blackford T. Negacyclic codes over Z4 of even length[J]. IEEE Transactions on Information Theory, 2003, 49(6): 1 417-1 424.
    [13]
    Dinh H Q. Complete Distances of all negacyclic codes of length 2s over Z2a [J]. IEEE Transactions on Information Theory, 2007, 53(1): 147-161.
    [14]
    Zhu Shixin, Kai Xiaoshan. On the homogeneous distance of negacyclic codes over Z2a[J/OL]. Sciencepaper Online, [2012-02-28]. http://www.paper.edu.cn/releasepaper/content/201202-1048.
    朱士信，开晓山. 关于Z2a上的负循环码的齐次距离[J/OL]. 中国科技论文在线， [2012-02-28]. http://www.paper.edu.cn/releasepaper/content/201202-1048.
    [15]
    Xi Hongqi, Zheng Xiying, Kong Bo. Repeated-root negacyclic codes over Zpa[J]. Journal of Zhengzhou University (Natural Science Edition), 2012,44(3):26-28.
    席红旗, 郑喜英, 孔波. 环Zpa上的重根负循环码[J]. 郑州大学学报(理学版), 2012,44(3):26-28.
    [16]
    Al-Ashker M M, Chen J Z. Cyclic codes of arbitrary length over Fp+uFp+…+ukFp [J]. Palestine Journal of Mathermatics, 2013,2(1):72-80.
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