\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

From skew-cyclic codes to asymmetric quantum codes

Abstract Related Papers Cited by
  • We introduce an additive but not $\mathbb F$4-linear map $S$ from $\mathbb F$4n to $\mathbb F$42n and exhibit some of its interesting structural properties. If $C$ is a linear $[n,k,d]$4-code, then $S(C)$ is an additive $(2n,2$2k,$2d)$4-code. If $C$ is an additive cyclic code then $S(C)$ is an additive quasi-cyclic code of index $2$. Moreover, if $C$ is a module $\theta$-cyclic code, a recently introduced type of code which will be explained below, then $S(C)$ is equivalent to an additive cyclic code if $n$ is odd and to an additive quasi-cyclic code of index $2$ if $n$ is even. Given any $(n,M,d)$4-code $C$, the code $S(C)$ is self-orthogonal under the trace Hermitian inner product. Since the mapping $S$ preserves nestedness, it can be used as a tool in constructing additive asymmetric quantum codes.
    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symb. Comput., 24 (1997), 235-265.doi: 10.1006/jsco.1996.0125.

    [2]

    D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, Applied Algebra Engin. Commun. Comput., 18 (2007), 379-389.doi: 10.1007/s00200-007-0043-z.

    [3]

    D. Boucher and F. Ulmer, Codes as modules over skew polynomial rings, in "Proceedings of the 12th IMA Conference on Cryptography and Coding,'' Cirencester, (2009), 38-55.doi: 10.1007/978-3-642-10868-6_3.

    [4]

    A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over $GF(4)$, IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.doi: 10.1109/18.681315.

    [5]

    M. F. Ezerman, M. Grassl and P. Solé, The weights in MDS codes, IEEE Trans. Inform. Theory, 57 (2011), 392-396.doi: 10.1109/TIT.2010.2090246.

    [6]

    M. F. Ezerman, S. Ling and P. SoléAdditive asymmetric quantum codes, preprint, arXiv:1002.4088

    [7]

    K. Feng, S. Ling and C. Xing, Asymptotic bounds on quantum codes from algebraic geometry codes, IEEE Trans. Inform. Theory, 52 (2006), 986-991.doi: 10.1109/TIT.2005.862086.

    [8]

    E. M. Gabidulin, Theory of codes with maximum rank distance, Probl. Peredach. Inform. (in Russian), 21 (1985), 3-16; English translation, 1-12.

    [9]

    M. GrasslBounds on the minimum distance of linear codes and quantum codes, available online at http://www.codetables.de

    [10]

    W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge University Press, Cambridge, 2003.

    [11]

    J. L. Kim and V. Pless, Designs in additive codes over $GF(4)$, Des. Codes Crypt., 30 (2003), 187-199.doi: 10.1023/A:1025484821641.

    [12]

    S. Ling and C. P. Xing, "Coding Theory. A First Course,'' Cambridge University Press, Cambridge, 2004.doi: 10.1017/CBO9780511755279.

    [13]

    F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland, Amsterdam, 1977.

    [14]

    G. Nebe, E. M. Rains and N. J. A. Sloane, "Self-Dual Codes and Invariant Theory,'' Springer-Verlag, Berlin, Heidelberg, 2006.

    [15]

    E. M. Rains and N. J. A. Sloane, Self-dual codes, in "Handbook of Coding Theory I'' (eds. V.S. Pless and W.C. Huffman), Elsevier, (1998), 177-294.

    [16]

    P. K. Sarvepalli, A. Klappenecker and M. Rötteler, Asymmetric quantum codes: constructions, bounds and performance, Proc. Royal Soc. A, 465 (2009), 1645-1672.doi: 10.1098/rspa.2008.0439.

    [17]

    L. Wang, K. Feng, S. Ling and C. Xing, Asymmetric quantum codes: characterization and constructions, IEEE Trans. Inform. Theory, 56 (2010), 2938-2945.doi: 10.1109/TIT.2010.2046221.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(151) Cited by(0)

Access History

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return