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On self-dual and LCD double circulant codes over Fq+uFq+vFq+uvFq

Funds:  Supported by National Natural Science Foundation of China (61672036), Excellent Youth Foundation of Natural Science Foundation of Anhui Province(1808085J20).
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https://doi.org/10.3969/j.issn.0253-2778.2018.11.004
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  • Author Bio:

    LU Yaqi, female, born in 1994, master. Research field: Algebraic coding. E-mail: lyqSunshine8@163.com

  • Corresponding author: SHI Minjia
  • Received Date: 04 February 2018
  • Accepted Date: 11 April 2018
  • Rev Recd Date: 11 April 2018
  • Publish Date: 30 November 2018
    Abstract

  • Abstract

    Double circulant codes of length 2n over a non-chain ring Fq+uFq+vFq+uvFq, u2=v2=0, uv=vu, were studied when q was a prime power. Exact enumerations of self-dual and LCD double circulant codes for a positive integer n were given. Using a distance-preserving Gray map, self-dual and LCD codes of length 8n over Fq were constructed when q was even. Using random coding and the Artin conjecture, the modified Varshamov-Gilbert bounds were derived on the relative distance of the codes considered, building on exact enumeration results for given n and q.

    Abstract

    Double circulant codes of length 2n over a non-chain ring Fq+uFq+vFq+uvFq, u2=v2=0, uv=vu, were studied when q was a prime power. Exact enumerations of self-dual and LCD double circulant codes for a positive integer n were given. Using a distance-preserving Gray map, self-dual and LCD codes of length 8n over Fq were constructed when q was even. Using random coding and the Artin conjecture, the modified Varshamov-Gilbert bounds were derived on the relative distance of the codes considered, building on exact enumeration results for given n and q.
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    • References(14)
  • [1]
    MASSEY J L. Linear codes with complementary duals[J]. Discrete Mathematics, 1992，106-107： 337-342.
    [2]
    GNERI C, ZKAYA B, SOL P. Quasi-cyclic complementary dual codes[J]. Finite Fields and Their Applications, 2016, 42: 67-80.
    [3]
    ALAHMADI A, GNERI C, ZKAYA B, et al. On self-dual double negacirculant codes[J]. Discrete Applied Mathematics, 2017, 222: 205-212.
    [4]
    ALAHMADI A, OZDEMIR F, SOL P. On self-dual double circulant codes[J]. Designs, Codes and Cryptography, 2018, 86:1257-1265.
    [5]
    SHI M J, HUANG D T, SOK L, et al. Double circulant self-dual and LCD codes over Galois rings[EB/OL]. [2018-02-01] https://arxiv.org/abs/1801.06624.
    [6]
    SHI M J, QIAN L Q, SOL P. On self-dual negacirculant codes of index two and four[J]. Designs, Codes and Cryptography, 2018, 86: 2485-2494.
    [7]
    LIU Y, SHI M J, SOL P. Two-weight and three-weight codes from trace codes over Fp+ uFp+ vFp+ uvFp [J]. Discrete Mathematics, 2018, 341: 350-357.
    [8]
    ZHU S X, KAI X S. (1-uv)-constacyclic codes over Fp +uFp+vFp+uvFp[J]. Journal of Systems Science Complexity, 2014, 27(4): 811-816.
    [9]
    LING S, SOL P. On the algebraic structure of quasi-cyclic codes Ⅰ: Finite fields[J]. IEEE Transactions on Information Theory, 2001, 47: 2751-2760.
    [10]
    JIA Y. On quasi-twisted codes over finite fields[J]. Finite Fields and Their Applications, 2012, 18: 237-257.
    [11]
    LING S, SOL P. On the algebraic structure of quasi-cyclic codes Ⅱ: Chain rings[J]. Designs, Codes and Cryptography, 2003, 30(1): 113-130.
    [12]
    MOREE P. Artin’s primitive root conjecture a survey[J]. Integers, 2012, 10(6): 1305-1416.
    [13]
    HOOLEY C. On Artin’s conjecture[J]. Journal Für Die Reine Und Angewandte Mathematik, 1967, 225: 209-220.
    [14]
    HUFFMAN W C, PLESS V. Fundamentals of Error-Correcting Codes [M]. Cambridge: Cambridge University Press, 2003.)
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    [1]
    MASSEY J L. Linear codes with complementary duals[J]. Discrete Mathematics, 1992，106-107： 337-342.
    [2]
    GNERI C, ZKAYA B, SOL P. Quasi-cyclic complementary dual codes[J]. Finite Fields and Their Applications, 2016, 42: 67-80.
    [3]
    ALAHMADI A, GNERI C, ZKAYA B, et al. On self-dual double negacirculant codes[J]. Discrete Applied Mathematics, 2017, 222: 205-212.
    [4]
    ALAHMADI A, OZDEMIR F, SOL P. On self-dual double circulant codes[J]. Designs, Codes and Cryptography, 2018, 86:1257-1265.
    [5]
    SHI M J, HUANG D T, SOK L, et al. Double circulant self-dual and LCD codes over Galois rings[EB/OL]. [2018-02-01] https://arxiv.org/abs/1801.06624.
    [6]
    SHI M J, QIAN L Q, SOL P. On self-dual negacirculant codes of index two and four[J]. Designs, Codes and Cryptography, 2018, 86: 2485-2494.
    [7]
    LIU Y, SHI M J, SOL P. Two-weight and three-weight codes from trace codes over Fp+ uFp+ vFp+ uvFp [J]. Discrete Mathematics, 2018, 341: 350-357.
    [8]
    ZHU S X, KAI X S. (1-uv)-constacyclic codes over Fp +uFp+vFp+uvFp[J]. Journal of Systems Science Complexity, 2014, 27(4): 811-816.
    [9]
    LING S, SOL P. On the algebraic structure of quasi-cyclic codes Ⅰ: Finite fields[J]. IEEE Transactions on Information Theory, 2001, 47: 2751-2760.
    [10]
    JIA Y. On quasi-twisted codes over finite fields[J]. Finite Fields and Their Applications, 2012, 18: 237-257.
    [11]
    LING S, SOL P. On the algebraic structure of quasi-cyclic codes Ⅱ: Chain rings[J]. Designs, Codes and Cryptography, 2003, 30(1): 113-130.
    [12]
    MOREE P. Artin’s primitive root conjecture a survey[J]. Integers, 2012, 10(6): 1305-1416.
    [13]
    HOOLEY C. On Artin’s conjecture[J]. Journal Für Die Reine Und Angewandte Mathematik, 1967, 225: 209-220.
    [14]
    HUFFMAN W C, PLESS V. Fundamentals of Error-Correcting Codes [M]. Cambridge: Cambridge University Press, 2003.)
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