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The Gray image of a class of constacyclic codes over the ring Fpm[u]/

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

Funds:  Supported by National Natural Science Foundation of China(61370089), Anhui Province Natural Science Research (KJ2015A308, KJ2016A307, KJ2017A623) and Anhui Province Colleges Outstanding Young Talents Program (gxyqZD2016389).
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2017.07.009
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  • Author Bio:

    DING Jian, male, born in 1982, Master/Associate Professor. Research field: Algebraic coding. E-mail: dingjian_happy@163.com.

  • Corresponding author: LI Hongju
  • Received Date: 27 October 2015
  • Rev Recd Date: 24 May 2016
  • Publish Date: 31 July 2017
    Abstract

  • Abstract

    Let R(pm,k)=Fpm［u］/, where pj－1+1≤k≤pj and uk=0 for some positive prime number p and positive integer j. A new Gray map from R(pm,k) to Fpmpj was defined. It was proved that the Gray image of a linear (1+u+…+uk－1) constacyclic code of an arbitrary length N over R(pm,k) is a distance invariant linear cyclic code of length pjN over Fpm. Moreover, the generator polynomial of the Gray image of such a constacyclic code was determined, and some optimal linear cyclic codes over F3, F5 and F7 were constructed via the Gray map.

    Abstract

    Let R(pm,k)=Fpm［u］/, where pj－1+1≤k≤pj and uk=0 for some positive prime number p and positive integer j. A new Gray map from R(pm,k) to Fpmpj was defined. It was proved that the Gray image of a linear (1+u+…+uk－1) constacyclic code of an arbitrary length N over R(pm,k) is a distance invariant linear cyclic code of length pjN over Fpm. Moreover, the generator polynomial of the Gray image of such a constacyclic code was determined, and some optimal linear cyclic codes over F3, F5 and F7 were constructed via the Gray map.
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    • References(9)
  • [1]
    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.
    [2]
    AMARRA M C V, NEMENZO F R. On (1－u)cyclic codes over Fpk+uFpk[J]. Applied Mathematics Letters, 2008, 21 (11): 1129-1133.
    [3]
    SOBHANI R, ESMAEILI E. Some constacyclic and cyclic codes over Fq[u]/ [J]. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 2010, 93 (4): 808-813.
    [4]
    ABULAR T, SIAP I. Constacyclic codes over F2+uF2[J]. Journal of the Franklin Institute, 2009, 346(5): 520-529.
    [5]
    DING J, LI H J. The gray images of (1+u) constacyclic codes over F2m[u]/ [J]. Journal of Applied Mathematics and Computing, 2015, 49(1): 1-13.
    [6]
    DING J, LI H J. The gray image of a class of constacyclic codes over polynomial residue rings[J]. Journal of the Franklin Institute, 2014, 351(12): 5467-5479.
    [7]
    KAI X S, ZHU S X, LI P. (1+λu)-constacyclic codes over Fp[u]/ . Journal of the Franklin Institute, 2010, 347(5): 751-762.
    [8]
    DING J, LI H J. Construction of optimal codes with Homogeneous distance[J]. Journal of University of Science and Technology of China, 2015, 45(7): 588-593.
    [9]
    李岩，朱士信. 环Fpm+uFpm+…+uk－1Fpm上的一类常循环码[J]. 合肥工业大学学报，2013，35(3)：408-411.



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    [1]
    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.
    [2]
    AMARRA M C V, NEMENZO F R. On (1－u)cyclic codes over Fpk+uFpk[J]. Applied Mathematics Letters, 2008, 21 (11): 1129-1133.
    [3]
    SOBHANI R, ESMAEILI E. Some constacyclic and cyclic codes over Fq[u]/ [J]. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 2010, 93 (4): 808-813.
    [4]
    ABULAR T, SIAP I. Constacyclic codes over F2+uF2[J]. Journal of the Franklin Institute, 2009, 346(5): 520-529.
    [5]
    DING J, LI H J. The gray images of (1+u) constacyclic codes over F2m[u]/ [J]. Journal of Applied Mathematics and Computing, 2015, 49(1): 1-13.
    [6]
    DING J, LI H J. The gray image of a class of constacyclic codes over polynomial residue rings[J]. Journal of the Franklin Institute, 2014, 351(12): 5467-5479.
    [7]
    KAI X S, ZHU S X, LI P. (1+λu)-constacyclic codes over Fp[u]/ . Journal of the Franklin Institute, 2010, 347(5): 751-762.
    [8]
    DING J, LI H J. Construction of optimal codes with Homogeneous distance[J]. Journal of University of Science and Technology of China, 2015, 45(7): 588-593.
    [9]
    李岩，朱士信. 环Fpm+uFpm+…+uk－1Fpm上的一类常循环码[J]. 合肥工业大学学报，2013，35(3)：408-411.



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