Abstract
Let \(R_{v,q}=\mathbb {F}_{q}+v\mathbb {F}_{q}\) with \(v^2=v\). In this paper, we provide three methods of constructing quantum synchronizable codes (QSCs) by using the Euclidean sums of \((1-2v)\)-constacyclic codes over \(R_{v,q}\). Concrete examples are presented to enrich the variety of available QSCs. In addition, two of methods in our research are easier than available methods for constructing QSCs and producing more QSCs.
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References
Bose, R.C., Caldwell, J.G.: Synchronizable error-correcting codes. Inf. Contr. 10, 616–630 (1967)
Bosma, W., Cannon, J., Playoust, C.: The MAGMA algebra system I: the user langua. J. Symbolic Compyt. 24, 235–265 (1997)
Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction via codes over \(GF(4)\). IEEE Trans. Inf. Theory 44, 1369–1387 (1998)
Dinh, H., Nguyen, B.T., Yamaka, W.: Quantum MDS and synchronizable codes from cyclic and negacyclic codes of length \(2p^s\) over \({\mathbb{F} }_{p^{m}}\). IEEE Access 8, 124608–124623 (2020)
Fujiwara, Y.: Block synchronization for quantum information. Phys. Rev. A 87, 109–120 (2013)
Fujiwara, Y., Tonchev, V.D.T., Wong, W.H.: Algebraic techniques in designing quantum synchronizable codes. Phy. Rev. A 88, 162–166 (2013)
Fujiwara, Y., Vandendriessche, P.: Quantum synchronizable codes from finite geometries. IEEE Trans. Infom. Theory 60(11), 7345–7354 (2014)
Guenda, K., La Guardia, G.G., Gulliver, T.A.: Algebraic quantum synchronizable codes. J. Appl. Math. Comput. 55, 393–407 (2017)
Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)
Krishna, A., Sarwate, D.V.: Pseudocyclic maximum-distance-separable codes. IEEE Trans. Inf. Theory 36, 880–884 (1990)
Luo, L., Ma, Z.: Non-binary quantum synchronizable codes from repeated-root cyclic codes. IEEE Trans. Inf. Theory 64(3), 1461–1470 (2018)
Luo, L., Ma, Z., Lin, D.: Two new families of quantum synchronizable codes. Quantum Inf. Process 18, 1–18 (2019)
Liu, Y., Shi, M., Sepasdar, Z., Solé, P.: Construction of Hermitian self-dual constacyclic codes over \({\mathbb{F} }_{q^{2}} + v{\mathbb{F} }_{q^{2}}\). Appl. Comput. Math. 15(3), 359–369 (2016)
Liu, H., Liu, X.: Quantum synchronizable codes from finite rings. Quantum Inf. Process 20, 215 (2021)
Dai,Y., H., Liu, X. : QScs from the Euclidean sums of cyclic codes over finite rings. Quantum Inf. Process, 21, 295(2022)
Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press, Cambridge (1997)
Shi, X., Yue, Q., Huang, X.: Quantum synchronizable codes from the Whiteman\(^{\prime }\) generalized cyclotomy. Cryptogr. Commun. https://doi.org/10.1007/s12095-021-005012
Xie, Y., Yang, L., Yuan, J.: \(q\)-ary chain-containing quantum synchronizable codes. IEEE Commun. Lett. 20(3), 414–417 (2016)
Wang, Y., Kai, X., Sun, Z., Zhu, S.: Quantum codes from Hermitian dual-containing constacyclic codes over \({\mathbb{F} }_{q^{2}}+ v{\mathbb{F} }_{q^{2}}\). Quantum Inf. Process 20, 122 (2021)
Acknowledgements
This work was supported by Research Funds of Hubei Province (Grant No. Q20164505) and the talent project of Hubei Polytechnic University of China (Grant No. 16xjzo8R).
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Liu, J., Hu, P. & Liu, X. Quantum synchronizable codes from the ring \(\mathbb {F}_{q} +\) \(\varvec{v}\) \(\mathbb {F}_{q}\). Quantum Inf Process 23, 44 (2024). https://doi.org/10.1007/s11128-023-04248-y
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DOI: https://doi.org/10.1007/s11128-023-04248-y