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Quadratic residue codes over Fp+u Fp+ v Fp+uv Fp+v2 Fp+uv2 Fp

  • 1.

    Key Laboratory of Intelligent Computing Signal Processing, Ministry of Education, Hefei 230039, China

  • 2.

    National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China

  • 3.

    School of Mathematical Sciences, Anhui University, Hefei 230026, China

  • 4.

    Department of Mathematics, Royal University of Phnom Penh, Cambodia

  • 5.

    School of Finance, Huainan Normal University, Huainan 232001, China

Funds:  Supported by National Natural Science Foundation of China (61672036), the Open Research Fund of National Mobile Communications Research Laboratory (2015D11), Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China (05015133), Key Projects of Support Program for Outstanding Young Talents in Colleges and Universities (gxyqZD2016008).
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2017.07.008
More Information
  • Author Bio:

    QIAN Liqin, female, born in 1991, Master candidate. Research field: Algebraic coding. E-mail: qianliqin_1108@163.com

  • Corresponding author: SHI Minjia
  • Received Date: 30 April 2016
  • Rev Recd Date: 30 December 2016
  • Publish Date: 31 July 2017
    Abstract

  • Abstract

    Let R=Fp+u Fp+v Fp+uv Fp+v2 Fp+uv2 Fp, where u2=1, v3=v, and p is an odd prime. Quadratic residue codes of prime length n=q over the ring R was investigated, where q (q≠p) is an odd prime such that p is a quadratic residue modulo q. The cyclic codes of length n over R were studied, and then the quadratic residue codes over R in terms of idempotent generators were difined. Moreover, the relation between these codes and their extended codes are discussed. Finally, two specific forms of idempotent generators of quadratic residue codes over  Fp+u Fp+v Fp+uv Fp+v2 Fp+uv2 Fp were given to illustrate some results.

    Abstract

    Let R=Fp+u Fp+v Fp+uv Fp+v2 Fp+uv2 Fp, where u2=1, v3=v, and p is an odd prime. Quadratic residue codes of prime length n=q over the ring R was investigated, where q (q≠p) is an odd prime such that p is a quadratic residue modulo q. The cyclic codes of length n over R were studied, and then the quadratic residue codes over R in terms of idempotent generators were difined. Moreover, the relation between these codes and their extended codes are discussed. Finally, two specific forms of idempotent generators of quadratic residue codes over  Fp+u Fp+v Fp+uv Fp+v2 Fp+uv2 Fp were given to illustrate some results.
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    • References(9)
  • [1]
    MACWILLIAMS J, SLOANE N J A. The theory of Error Correcting Codes[M]// Amsterdam: North- Holland Publishing, 1977, 68(1):185-186.
    [2]
    BONNECAZE A, SOL P, CALDERBANK A R. Quaternary quadratic residue codes and unimodular lattices [J]. IEEE Transactions on Information Theory, 1995, 41(2): 366-377.
    [3]
    GAO J, MA F H. Some results on quadratic residue codes over the ring Fp+vFp+v2Fp+v3Fp[J]. Discrete Mathematics Algorithms & Applications, 2017, 9(3): 1750035(1-9).
    [4]
    KAYA A, YILDIZ B, SIAP I. New extremal binary self-dual codes of length 68 from quadratic residue codes over F2 +uF2 +u2F2[J]. Finte Fields and Their Applications, 2014, 29(1): 160-177.
    [5]
    LIU Y, SHI M J, SOL P. Quadratic residue codes over Fp+vFp+v2Fp[J]. Lecture Notes in Computer Science, 2015, 9061: 204-211.
    [6]
    PLESS V S, QIAN Z Q. Cyclic codes and quadratic residue codes over Z4[J]. IEEE Transactions on Information Theory, 1996, 42(5): 1594-1600.
    [7]
    RAKA M, KATHURIA L, GOYAL M. (1—2u3)-constacyclic codes and quadratic residue codes over Fp[u]/<u4—u>[J]. Cryptography and Communications, 2017, 9(4): 459-473.
    [8]
    张涛, 朱士信. 环Fl + vFl上的二次剩余码[J]. 中国科学技术大学学报，2012, 42(3): 207-213.
    ZHANG T, ZHU S X. Quadratic residue codes over Fl + vFl[J]. Journal of University of Science and Technology of China, 2012, 42(3): 207-213.
    [9]
    BOSMA W, CANNON J. Handbook of Magma Functions [Z]. Sydney, 1995.
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    [1]
    MACWILLIAMS J, SLOANE N J A. The theory of Error Correcting Codes[M]// Amsterdam: North- Holland Publishing, 1977, 68(1):185-186.
    [2]
    BONNECAZE A, SOL P, CALDERBANK A R. Quaternary quadratic residue codes and unimodular lattices [J]. IEEE Transactions on Information Theory, 1995, 41(2): 366-377.
    [3]
    GAO J, MA F H. Some results on quadratic residue codes over the ring Fp+vFp+v2Fp+v3Fp[J]. Discrete Mathematics Algorithms & Applications, 2017, 9(3): 1750035(1-9).
    [4]
    KAYA A, YILDIZ B, SIAP I. New extremal binary self-dual codes of length 68 from quadratic residue codes over F2 +uF2 +u2F2[J]. Finte Fields and Their Applications, 2014, 29(1): 160-177.
    [5]
    LIU Y, SHI M J, SOL P. Quadratic residue codes over Fp+vFp+v2Fp[J]. Lecture Notes in Computer Science, 2015, 9061: 204-211.
    [6]
    PLESS V S, QIAN Z Q. Cyclic codes and quadratic residue codes over Z4[J]. IEEE Transactions on Information Theory, 1996, 42(5): 1594-1600.
    [7]
    RAKA M, KATHURIA L, GOYAL M. (1—2u3)-constacyclic codes and quadratic residue codes over Fp[u]/<u4—u>[J]. Cryptography and Communications, 2017, 9(4): 459-473.
    [8]
    张涛, 朱士信. 环Fl + vFl上的二次剩余码[J]. 中国科学技术大学学报，2012, 42(3): 207-213.
    ZHANG T, ZHU S X. Quadratic residue codes over Fl + vFl[J]. Journal of University of Science and Technology of China, 2012, 42(3): 207-213.
    [9]
    BOSMA W, CANNON J. Handbook of Magma Functions [Z]. Sydney, 1995.
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