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

Early Access articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Early Access publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Early Access articles via the “Early Access” tab for the selected journal.

New quantum codes from metacirculant graphs via self-dual additive $\mathbb{F}_4$-codes

  • * Corresponding author: Martianus Frederic Ezerman

    * Corresponding author: Martianus Frederic Ezerman

Nanyang Technological University Grant Number 04INS000047C230GRT01 supports the research carried out by M. F. Ezerman

Abstract Full Text(HTML) Figure(2) / Table(2) Related Papers Cited by
  • We use symplectic self-dual additive codes over $ \mathbb{F}_4 $ obtained from metacirculant graphs to construct, for the first time, $ \left[\kern-0.15em\left[ {\ell, 0, d} \right]\kern-0.15em\right] $ qubit codes with parameters $ (\ell,d) \in \{(78, 20), (90, 21), (91, 22), (93,21),(96,22)\} $. Secondary constructions applied to the qubit codes result in many new qubit codes that perform better than the previous best-known.

    Mathematics Subject Classification: Primary: 94B25, 81P73; Secondary: 05C75.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  The Petersen Graph as $ \Gamma(2,5,2,\{1,4\}, \{0\}) $

    Figure 2.  $ G_{12}: = \Gamma(2, 6, 5, \{ 3 \},\{ 0, 3, 4, 5 \}) $ of the dodecacode $ \mathcal{D} $.

    Table 1.  New codes from modifying $ Q_{78} $

    Code Parameters Propagation rule
    $ Q_{78, 1} $ $ \left[\kern-0.15em\left[ {77,0,19} \right]\kern-0.15em\right]_2 $ Puncture $ Q_{78} $ at $ \{78\} $
    $ Q_{78,2} $ $ \left[\kern-0.15em\left[ {77,1,19} \right]\kern-0.15em\right]_2 $ Shorten $ Q_{78} $ at $ \{78\} $
    $ Q_{78,3} $ $ \left[\kern-0.15em\left[ {78,1,19} \right]\kern-0.15em\right]_2 $ Lengthen $ Q_{78,2} $ by $ 1 $
    $ Q_{78,4} $ $ \left[\kern-0.15em\left[ {76,2,18} \right]\kern-0.15em\right]_2 $ Shorten $ Q_{78} $ at $ \{77, 78\} $
    $ Q_{78,5} $ $ \left[\kern-0.15em\left[ {76,1,18} \right]\kern-0.15em\right]_2 $ Subcode of $ Q_{78,4} $
    $ Q_{78,6} $ $ \left[\kern-0.15em\left[ {77,2,18} \right]\kern-0.15em\right]_2 $ Lengthen $ Q_{78,4} $ by $ 1 $
    $ Q_{78,7} $ $ \left[\kern-0.15em\left[ {75,3,17} \right]\kern-0.15em\right]_2 $ Shorten $ Q_{78} $ at $ \{76,77,78\} $
    $ Q_{78,8} $ $ \left[\kern-0.15em\left[ {76,3,17} \right]\kern-0.15em\right]_2 $ Lengthen $ Q_{78,7} $ by $ 1 $
    $ Q_{78,9} $ $ \left[\kern-0.15em\left[ {75, 2, 17} \right]\kern-0.15em\right]_2 $ Subcode of $ Q_{78,7} $
     | Show Table
    DownLoad: CSV

    Table 2.  Properties of the Graphs

    $ G $ $ d_{\rm min}(G) $ $ \nu(G) $ $ \gamma(G) $ $ |{\rm Aut}(G)| $ $ G $ $ d_{\rm min}(G) $ $ \nu(G) $ $ \gamma(G) $ $ |{\rm Aut}(G)| $
    $ G_{12} $ $ 6 $ $ 5 $ $ 4 $ $ 24 $ $ G_{78} $ $ 20 $ $ 41 $ $ 7 $ $ 78 $
    $ G_{27,1} $ $ 9 $ $ 16 $ $ 6 $ $ 27 $ $ G_{90} $ $ 21 $ $ 42 $ $ 7 $ $ 90 $
    $ G_{27,2} $ $ 9 $ $ 10 $ $ 4 $ $ 27 $ $ G_{91} $ $ 22 $ $ 44 $ $ 7 $ $ 546 $
    $ G_{36,1} $ $ 12 $ $ 13 $ $ 6 $ $ 72 $ $ G_{93} $ $ 22 $ $ 28 $ $ 4 $ $ 186 $
    $ G_{36,2} $ $ 12 $ $ 13 $ $ 4 $ $ 72 $ $ G_{96} $ $ 22 $ $ 35 $ $ 6 $ $ 96 $
     | Show Table
    DownLoad: CSV
  • [1] B. Alspach and T. D. Parsons, A construction for vertex-transitive graphs, Canad. J. Math., 34 (1982), 307-318.  doi: 10.4153/CJM-1982-020-8.
    [2] Z. R. Bogdanowicz, Pancyclicity of connected circulant graphs, J. Graph Theory, 22 (1996), 167-174.  doi: 10.1002/(SICI)1097-0118(199606)22:2<167::AID-JGT7>3.0.CO;2-L.
    [3] W. BosmaJ. 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.
    [4] A. R. CalderbankE. M. RainsP. M. 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] L. E. Danielsen and M. G. Parker, On the classification of all self-dual additive codes over $GF(4)$ of length up to $12$, J. Combin. Theory, Ser. A, 113 (2006), 1351-1367.  doi: 10.1016/j.jcta.2005.12.004.
    [6] A. EinsteinB. Podolsky and N. Rosen, Can quantum-mechanical description of physical reality be considered complete?, Phys. Rev., 47 (1935), 777-780.  doi: 10.1103/PhysRev.47.777.
    [7] M. F. Ezerman, Quantum Error-Control Codes, Chapter 27 in W. C. Huffman, J.-L. Kim and P Solé (Eds.), Concise Encyclopedia of Coding Theory, 1$^st$ edition, Chapman and Hall (CRC Press), Boca Raton, 2021. doi: 10.1201/9781315147901.
    [8] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, Online available at http://www.codetables.de, 2007, accessed on 2021-04-30.
    [9] M. Grassl, Algebraic quantum codes: Linking quantum mechanics and discrete mathematics, Int. J. Comput. Math.: Comput. Syst. Theory, 6 (2021), 243-259.  doi: 10.1080/23799927.2020.1850530.
    [10] M. Grassl and M. Harada, New self-dual additive $\mathbb{F}_4$-codes constructed from circulant graphs, Discrete Math., 340 (2017), 399-403.  doi: 10.1016/j.disc.2016.08.023.
    [11] T. A. Gulliver and J-L. Kim, Circulant based extremal additive self-dual codes over $GF(4)$, IEEE Trans. Inform. Theory, 50 (2004), 359-366.  doi: 10.1109/TIT.2003.822616.
    [12] J. Hackl, TikZ-network manual, preprint, arXiv: 1709.06005. Source code at https://github.com/hackl/tikz-network.
    [13] C. H. LiS. J. Song and D. J. Wang, A characterization of metacirculants, J. Combin. Theory, Ser. A, 120 (2013), 39-48.  doi: 10.1016/j.jcta.2012.06.010.
    [14] D. Marušič, On $2$-arc-transitivity of Cayley graphs, J. Combin. Theory, Ser. B, 96 (2006), 761-764.  doi: 10.1016/j.jctb.2006.01.003.
    [15] È. A. Monakhova, A survey on undirected circulant graphs, Discrete Math. Algorithms Appl., 4 (2012), 1250002, 30pp. doi: 10.1142/S1793830912500024.
    [16] G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, Heidelberg, 2006. doi: 10.1007/3-540-30731-1.
    [17] K. Saito, Self-dual additive $\mathbb{F}_4$-codes of lengths up to $40$ represented by circulant graphs, Adv. Math. Commun., 13 (2019), 213-220.  doi: 10.3934/amc.2019014.
    [18] D. Schlingemann, Stabilizer codes can be realized as graph codes, Quantum Info. Comput., 2 (2002), 307-323.  doi: 10.26421/QIC2.4-4.
    [19] Z. Varbanov, Additive circulant graph codes over $GF(4)$, Math. Maced., 6 (2008), 73-79. 
    [20] A. Vardy, The intractability of computing the minimum distance of a code, IEEE Trans. Inform. Theory, 43 (1997), 1757-1766.  doi: 10.1109/18.641542.
  • 加载中

Figures(2)

Tables(2)

SHARE

Article Metrics

HTML views(515) PDF downloads(397) Cited by(0)

Access History

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return