# American Institute of Mathematical Sciences

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
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##### 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. 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.  Google Scholar [4] A. R. Calderbank, E. M. Rains, P. 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. Einstein, B. 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. Li, S. 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
The Petersen Graph as $\Gamma(2,5,2,\{1,4\}, \{0\})$
$G_{12}: = \Gamma(2, 6, 5, \{ 3 \},\{ 0, 3, 4, 5 \})$ of the dodecacode $\mathcal{D}$.
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 $Q_{78,7}$
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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