doi: 10.3934/amc.2021073
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New quantum codes from metacirculant graphs via self-dual additive $\mathbb{F}_4$-codes

1. 

Department of Mathematics, Texas A&M University-Commerce, 2600 South Neal Street, Commerce TX 75428, USA

2. 

School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371

* Corresponding author: Martianus Frederic Ezerman

Received  May 2021 Revised  December 2021 Early access January 2022

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

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.

Citation: Padmapani Seneviratne, Martianus Frederic Ezerman. New quantum codes from metacirculant graphs via self-dual additive $\mathbb{F}_4$-codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2021073
References:
[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. Hackl, TikZ-network manual, preprint, arXiv: 1709.06005. Source code at https://github.com/hackl/tikz-network. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

È. A. Monakhova, A survey on undirected circulant graphs, Discrete Math. Algorithms Appl., 4 (2012), 1250002, 30pp. doi: 10.1142/S1793830912500024.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

D. Schlingemann, Stabilizer codes can be realized as graph codes, Quantum Info. Comput., 2 (2002), 307-323.  doi: 10.26421/QIC2.4-4.  Google Scholar

[19]

Z. Varbanov, Additive circulant graph codes over $GF(4)$, Math. Maced., 6 (2008), 73-79.   Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. Hackl, TikZ-network manual, preprint, arXiv: 1709.06005. Source code at https://github.com/hackl/tikz-network. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

È. A. Monakhova, A survey on undirected circulant graphs, Discrete Math. Algorithms Appl., 4 (2012), 1250002, 30pp. doi: 10.1142/S1793830912500024.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

D. Schlingemann, Stabilizer codes can be realized as graph codes, Quantum Info. Comput., 2 (2002), 307-323.  doi: 10.26421/QIC2.4-4.  Google Scholar

[19]

Z. Varbanov, Additive circulant graph codes over $GF(4)$, Math. Maced., 6 (2008), 73-79.   Google Scholar

[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.  Google Scholar

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} $
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} $
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 $
$ 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 $
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