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CN 34-1054/N

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(1+u)-constacyclic codes over the ring F2+uF2+u2F2

  • Department of Common Course, Anhui Xinhua University, Hefei 230088, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2015.01.007
  • Received Date: 12 March 2014
  • Accepted Date: 16 May 2014
  • Rev Recd Date: 16 May 2014
  • Publish Date: 30 January 2015
    Abstract

  • Abstract

    In view of the factorization of (xn－1) in F2［x］, the minimal generating set and rank of (1+u)-constacyclic codes with an arbitrary length over the ring R=F2+uF2+u2F2 were studied. A new Gray map from R to F42 was defined, the structures and generator polynomials of the Gray image of a linear (1+u)-constacyclic code with an arbitrary length were determined, and some optimal binary linear cyclic codes were obtained.

    Abstract

    In view of the factorization of (xn－1) in F2［x］, the minimal generating set and rank of (1+u)-constacyclic codes with an arbitrary length over the ring R=F2+uF2+u2F2 were studied. A new Gray map from R to F42 was defined, the structures and generator polynomials of the Gray image of a linear (1+u)-constacyclic code with an arbitrary length were determined, and some optimal binary linear cyclic codes were obtained.
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    • References(16)
  • [1]
    Roger H A, Vijay K P, Calderbank A R, et al. The -linearity of Kerdock, Preparata, Goethals, and related codes[J]. IEEE Transactions on Information Theory, 1994, 40 (2): 301-319.
    [2]
    Dougherty S T, Shiromoto K. Maximum distance codes over rings of order 4[J]. IEEE Transactions on Information Theory, 2001, 47(1): 400-404.
    [3]
    Qian J F, Zhang L N, Zhu S X. (1+u)-constacyclic and cyclic codes over F2+uF2 [J]. Applied Mathematics Letters, 2006, 19(8): 820-823.
    [4]
    Qian J F, Zhang L N. Constacyclic and cyclic codes over F2+uF2+u2F2 [J]. IEICE Transactions on Fundamentals, Communications and Computer Science, 2006, E89-A(6): 1 863-1 865.
    [5]
    Zhu S X, Shi M J. The ranks of cyclic and negacyclic codes over the finite ring R[J]. Journal of Electronics (China), 2008, 25(1): 97-101.
    [6]
    Abular T, Siap I. Constacyclic codes over F2+uF2 [J]. Journal of Franklin Institute, 2009, 346(5):520-529.
    [7]
    Shi M J, Zhu S X. Constacyclic codes over ring Fq+uFq+…+uk-1Fq[J]. Journal of University of Science and Technology of China, 2009, 39(6): 583-587.
    施敏加, 朱士信. 环Fq+uFq+…+uk-1Fq上的常循环码[J]. 中国科学技术大学学报, 2009, 39(6): 583-587.
    [8]
    Qian J F. Cyclic codes over finite rings[C]// Proceedings of 7th International Conference on Wireless Communications, Networking and Mobile Computing. Wuhan, China: IEEE Press, 2011:1-4.
    [9]
    Kai X S, Zhu S X, Li P. (1+λu)-constacyclic codes over[J]. Journal of Franklin Institute, 2010, 347(5): 751-762.
    [10]
    Ding J, Li H J, Li H X. On the equivalence of constacyclic codes over the ring Fpm+uFpm [J]. Joural of University of Science and Technology of China, 2013, 43(2): 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(2): 334-339.
    [11]
    Wang L Q, Zhu S X. A class of constacyclic codes over and its Gray image[J]. Journal of electronics and information technology, 2013, 35(2): 499-503.
    王立启, 朱士信. 环上的一类常循环码及其Gray象[J]. 电子与信息学报, 2013, 35(2): 499-503.
    [12]
    Hu Q, Li P. Cyclic codes of arbitrary lengths over the ring Fq+uFq+u2Fq [J]. Journal of Hefei university of technology, 2013, 36(2): 243-247.
    胡庆,李平. 环Fq+uFq+u2Fq上任意长度的循环码[J]. 合肥工业大学学报(自然科学版), 2013, 36(2): 243-247.
    [13]
    Liu J Q, Liu L, Kai X S. (1+uk)-cyclic codes over the ring F2+uF2+…ukF2 [J]. Journal of Hefei university of technology, 2013, 36(1): 124-128.
    刘金秋, 刘丽, 开晓山. 环F2+uF2+…ukF2上的(1+uk)常循环码[J]. 合肥工业大学学报(自然科学版), 2013, 36(1): 124-128.
    [14]
    Zhang X Y. The Gray images of the liner codes and their dual codes over F2m+uF2m+u2F2m+u3F2m [J]. Journal of Mathematics, 2013, 33(4): 661-664.
    张晓燕. 环F2m+uF2m+u2F2m+u3F2m上的线性码及其对偶码的Gray象[J]. 数学杂志，2013, 33(4): 661-664.
    [15]
    Li S S, Li P. Cyclic codes over the ring Z4+uZ4 [J]. Journal of Hefei university of technology, 2013, 36(8): 1 006-1 009.
    李珊珊，李平. 环Z4+uZ4上的循环码[J]. 合肥工业大学学报, 2013, 36(8): 1 006-1 009.
    [16]
    Liang H, Tang Y S. The gray images of cyclic codes over Zp[u](uk+1)[J]. Mathematics in Practice and Theory, 2013, 43(5): 200-203.
    梁华, 唐元生. 环Zp[u](uk+1)上循环码的Gray象[J]. 数学的实践与认识, 2013, 43(5): 200-203.
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    [1]
    Roger H A, Vijay K P, Calderbank A R, et al. The -linearity of Kerdock, Preparata, Goethals, and related codes[J]. IEEE Transactions on Information Theory, 1994, 40 (2): 301-319.
    [2]
    Dougherty S T, Shiromoto K. Maximum distance codes over rings of order 4[J]. IEEE Transactions on Information Theory, 2001, 47(1): 400-404.
    [3]
    Qian J F, Zhang L N, Zhu S X. (1+u)-constacyclic and cyclic codes over F2+uF2 [J]. Applied Mathematics Letters, 2006, 19(8): 820-823.
    [4]
    Qian J F, Zhang L N. Constacyclic and cyclic codes over F2+uF2+u2F2 [J]. IEICE Transactions on Fundamentals, Communications and Computer Science, 2006, E89-A(6): 1 863-1 865.
    [5]
    Zhu S X, Shi M J. The ranks of cyclic and negacyclic codes over the finite ring R[J]. Journal of Electronics (China), 2008, 25(1): 97-101.
    [6]
    Abular T, Siap I. Constacyclic codes over F2+uF2 [J]. Journal of Franklin Institute, 2009, 346(5):520-529.
    [7]
    Shi M J, Zhu S X. Constacyclic codes over ring Fq+uFq+…+uk-1Fq[J]. Journal of University of Science and Technology of China, 2009, 39(6): 583-587.
    施敏加, 朱士信. 环Fq+uFq+…+uk-1Fq上的常循环码[J]. 中国科学技术大学学报, 2009, 39(6): 583-587.
    [8]
    Qian J F. Cyclic codes over finite rings[C]// Proceedings of 7th International Conference on Wireless Communications, Networking and Mobile Computing. Wuhan, China: IEEE Press, 2011:1-4.
    [9]
    Kai X S, Zhu S X, Li P. (1+λu)-constacyclic codes over[J]. Journal of Franklin Institute, 2010, 347(5): 751-762.
    [10]
    Ding J, Li H J, Li H X. On the equivalence of constacyclic codes over the ring Fpm+uFpm [J]. Joural of University of Science and Technology of China, 2013, 43(2): 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(2): 334-339.
    [11]
    Wang L Q, Zhu S X. A class of constacyclic codes over and its Gray image[J]. Journal of electronics and information technology, 2013, 35(2): 499-503.
    王立启, 朱士信. 环上的一类常循环码及其Gray象[J]. 电子与信息学报, 2013, 35(2): 499-503.
    [12]
    Hu Q, Li P. Cyclic codes of arbitrary lengths over the ring Fq+uFq+u2Fq [J]. Journal of Hefei university of technology, 2013, 36(2): 243-247.
    胡庆,李平. 环Fq+uFq+u2Fq上任意长度的循环码[J]. 合肥工业大学学报(自然科学版), 2013, 36(2): 243-247.
    [13]
    Liu J Q, Liu L, Kai X S. (1+uk)-cyclic codes over the ring F2+uF2+…ukF2 [J]. Journal of Hefei university of technology, 2013, 36(1): 124-128.
    刘金秋, 刘丽, 开晓山. 环F2+uF2+…ukF2上的(1+uk)常循环码[J]. 合肥工业大学学报(自然科学版), 2013, 36(1): 124-128.
    [14]
    Zhang X Y. The Gray images of the liner codes and their dual codes over F2m+uF2m+u2F2m+u3F2m [J]. Journal of Mathematics, 2013, 33(4): 661-664.
    张晓燕. 环F2m+uF2m+u2F2m+u3F2m上的线性码及其对偶码的Gray象[J]. 数学杂志，2013, 33(4): 661-664.
    [15]
    Li S S, Li P. Cyclic codes over the ring Z4+uZ4 [J]. Journal of Hefei university of technology, 2013, 36(8): 1 006-1 009.
    李珊珊，李平. 环Z4+uZ4上的循环码[J]. 合肥工业大学学报, 2013, 36(8): 1 006-1 009.
    [16]
    Liang H, Tang Y S. The gray images of cyclic codes over Zp[u](uk+1)[J]. Mathematics in Practice and Theory, 2013, 43(5): 200-203.
    梁华, 唐元生. 环Zp[u](uk+1)上循环码的Gray象[J]. 数学的实践与认识, 2013, 43(5): 200-203.
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