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Quantum codes from cyclic codes over ring Fq+uFq

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

Funds:  Supported by the National Natural Science Foundation of China (61370089), the Fundamental Research Funds for the Central Universities (JZ2014HGBZ0349).
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2015.03.004
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  • Author Bio:

    LI Jin, female, born in 1987, PhD. Research field: algebraic coding theory. E-mail: lijin_0102@126.com

  • Received Date: 27 September 2013
  • Accepted Date: 05 January 2014
  • Rev Recd Date: 05 January 2014
  • Publish Date: 30 March 2015
    Abstract

  • Abstract

    A method for constructing quantum codes from cyclic codes over the ring Fq+uFq was given, where q is a prime power and u2=0. First, symplectic self-orthogonal codes over Fq were obtained as images of cyclic codes over Fq+uFq. Then, these codes were used to construct quantum codes. Examples including quantum MDS codes were given.

    Abstract

    A method for constructing quantum codes from cyclic codes over the ring Fq+uFq was given, where q is a prime power and u2=0. First, symplectic self-orthogonal codes over Fq were obtained as images of cyclic codes over Fq+uFq. Then, these codes were used to construct quantum codes. Examples including quantum MDS codes were given.
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    • References(15)
  • [1]
    Shor P W. Scheme for reducing decoherence in quantum computer memory[J]. Phys Rev A, 1995, 52: 2 493-2 496.
    [2]
    Calderbank A R, Rains E M, Shor P M, et al. Quantum error correction via codes over GF(4) [J]. IEEE Trans Inform Theory, 1998, 44: 1 369-1 387.
    [3]
    Cohen G, Encheva S, Litsyn S. On binary constructions of quantum codes[J]. IEEE Trans Inform Theory, 1999, 45: 2 495-2 498.
    [4]
    Li R, Li X. Binary construction of quantum codes of minimum distance three and four[J]. IEEE Trans Inform Theory, 2004, 50: 1 331-1 336.
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    Steane A M. Quantum Reed-Muller codes[J]. IEEE Trans Inform Theory, 1999, 45: 1 701-1 703.
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    [8]
    Thangaraj A, McLaughlin S W. Quantum codes from cyclic codes over GF(4m)[J]. IEEE Trans Inform Theory, 2001, 47: 1 176-1 178.
    [9]
    Kai X, Zhu S. Quaternary construction of quantum codes from cyclic codes over F4+uF4[J]. Int J Quant Inform, 2011, 9: 689-700.
    [10]
    Qian J, Ma W, Guo W. Quantum codes from cyclic codes over finite ring[J]. Int J Quant Inform, 2009, 7: 1 277-1 283.
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    Ashikhmin A, Knill E. Nonbinary quantum stabilizer codes[J]. IEEE Trans Inform Theory, 2001, 47(7): 3 065-3 072.
    [12]
    Bachoc C. Applications of coding theory to the construction of modular lattices[J]. J Combin, Theory Ser A, 1997, 78(1): 92-119.
    [13]
    Gaborit P. Mass formulas for self-dual codes over Z4 and Fq+uFq rings[J]. IEEE Trans Inform Theory, 1996, 42: 1 222-1 228.
    [14]
    McDonald B R. Finite Rings With Identity[M]. New York: Dekker, 1974.
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    Dinh H Q, Lpez-Permouth S R. Cyclic and negacyclic codes over finite chain rings[J]. IEEE Trans Inform Theory, 2004, 50: 1 728-1 744.
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    [1]
    Shor P W. Scheme for reducing decoherence in quantum computer memory[J]. Phys Rev A, 1995, 52: 2 493-2 496.
    [2]
    Calderbank A R, Rains E M, Shor P M, et al. Quantum error correction via codes over GF(4) [J]. IEEE Trans Inform Theory, 1998, 44: 1 369-1 387.
    [3]
    Cohen G, Encheva S, Litsyn S. On binary constructions of quantum codes[J]. IEEE Trans Inform Theory, 1999, 45: 2 495-2 498.
    [4]
    Li R, Li X. Binary construction of quantum codes of minimum distance three and four[J]. IEEE Trans Inform Theory, 2004, 50: 1 331-1 336.
    [5]
    Li R, Li X. Quantum codes constructed from binary cyclic codes[J]. Int J Quant Inform, 2004, 2: 265-272.
    [6]
    Steane A M. Quantum Reed-Muller codes[J]. IEEE Trans Inform Theory, 1999, 45: 1 701-1 703.
    [7]
    Lin X. Quantum cyclic and constacyclic codes[J]. IEEE Trans Inform Theory, 2004, 50: 547-549.
    [8]
    Thangaraj A, McLaughlin S W. Quantum codes from cyclic codes over GF(4m)[J]. IEEE Trans Inform Theory, 2001, 47: 1 176-1 178.
    [9]
    Kai X, Zhu S. Quaternary construction of quantum codes from cyclic codes over F4+uF4[J]. Int J Quant Inform, 2011, 9: 689-700.
    [10]
    Qian J, Ma W, Guo W. Quantum codes from cyclic codes over finite ring[J]. Int J Quant Inform, 2009, 7: 1 277-1 283.
    [11]
    Ashikhmin A, Knill E. Nonbinary quantum stabilizer codes[J]. IEEE Trans Inform Theory, 2001, 47(7): 3 065-3 072.
    [12]
    Bachoc C. Applications of coding theory to the construction of modular lattices[J]. J Combin, Theory Ser A, 1997, 78(1): 92-119.
    [13]
    Gaborit P. Mass formulas for self-dual codes over Z4 and Fq+uFq rings[J]. IEEE Trans Inform Theory, 1996, 42: 1 222-1 228.
    [14]
    McDonald B R. Finite Rings With Identity[M]. New York: Dekker, 1974.
    [15]
    Dinh H Q, Lpez-Permouth S R. Cyclic and negacyclic codes over finite chain rings[J]. IEEE Trans Inform Theory, 2004, 50: 1 728-1 744.
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