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Optimal constructions of quantum and synchronizable codes from repeated-root cyclic codes of length \(3p^s\)

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Abstract

In this paper, we use the CSS and Steane’s constructions to establish quantum error-correcting codes (briefly, QEC codes) from cyclic codes of length \(3p^s\) over \(\mathbb F_{p^m}\). We obtain several new classes of QEC codes in the sense that their parameters are different from all the previous constructions. Among them, we identify all quantum MDS (briefly, qMDS) codes, i.e., optimal quantum codes with respect to the quantum Singleton bound. In addition, we construct quantum synchronizable codes (briefly, QSCs) from cyclic codes of length \(3p^s\) over \(\mathbb F_{p^m}\). Furthermore, we give many new QSCs to enrich the variety of available QSCs. A lot of them are QSCs codes with shorter lengths and much larger minimum distances than known non-primitive narrow-sense BCH codes.

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References

  1. Ashikhmin, A., Knill, E.: Nonbinary quantum stablizer codes. IEEE Trans. Inf. Theory 47, 3065–3072 (2001)

    MATH  Google Scholar 

  2. Bennett, H., DiVincenzo, P., Smolin, A., Wooters, K.: Mixed state entanglement and quantum error correction. Phys. Rev. A 54, 3824 (1996)

    ADS  MathSciNet  MATH  Google Scholar 

  3. Berman, D.: Semisimple cyclic and Abelian codes. II, Kibernetika (Kiev), Vol 3, pp. 21-30, 1967 (Russian). English translation: Cybernetics, 3: 17-23, (1967)

  4. Bouwmeester, D., Pan, J., Daniell, M., Weinfurter, H., Zeilinger, A.: Observation of Three-Photon Greenberger-Horne-Zeilinger Entanglement. Phys. Rev. Lett. 82, 1345 (1999)

    ADS  MathSciNet  MATH  Google Scholar 

  5. Bregni, S.: Synchronization of Digital Telecommunications Networks. John Wiley Sons, New York (2002)

    Google Scholar 

  6. Brun, T., Devetak, I., Hsieh, H.: Correcting quantum errors with entanglement. Science 314, 436–439 (2006)

    ADS  MathSciNet  MATH  Google Scholar 

  7. Bush, K.A.: Orthogonal arrays of index unity. Ann. Math. Statist. 23, 426–434 (1952)

    MathSciNet  MATH  Google Scholar 

  8. Calderbank, A.R., Shor, P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54, 1098–1106 (1996)

    ADS  Google Scholar 

  9. Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction via codes over GF(4). IEEE Trans. Inform. Theory. 44, 1369–1387 (1998)

    MathSciNet  MATH  Google Scholar 

  10. Castagnoli, G., Massey, J.L., Schoeller, P.A., von Seemann, N.: On repeated-root cyclic codes. IEEE Trans. Inform. Theory. 37, 337–342 (1991)

    MathSciNet  MATH  Google Scholar 

  11. Cao, Y., Cao, Y., Dinh, H.Q., Fu, F., Gao, J., Sriboonchitta, S.: Constacyclic codes of length \(np^s\) over \(\mathbb{F} _{p^m}+u\mathbb{F} _{p^m}\). Adv. Math. Commun. 12, 231–262 (2018)

    MathSciNet  Google Scholar 

  12. Cao, Y., Cao, Y., Dinh, H. Q., Fu, F., Gao, J., Sriboonchitta, S.:“A class of repeated-root constacyclic codes over \({\mathbb{F}}_{p^m} [u]/\langle u^e\rangle \) of Type 2., Finite Fields & Appl., pp. 238-267, (2019)

  13. Cao, Y., Cao, Y., Dinh, H.Q., Jitman, S.: An explicit representation and enumeration for self-dual cyclic codes over \(\mathbb{F} _{2^m}+u\mathbb{F} _{2^m}\) of length \(2^s\). Discrete Math. 342, 2077–2091 (2019)

    MathSciNet  MATH  Google Scholar 

  14. Cao, Y., Cao, Y., Dinh, H.Q., Fu, F., Gao, J., Sriboonchitta, S.: A class of linear codes of length 2 over fnite chain rings. J. Algebra Appl. 19, 2050103 (2020)

    MathSciNet  MATH  Google Scholar 

  15. Cao, Y. , Cao, Y., Dinh, H. Q., Fu, F., Ma, F.: Construction and enumeration for self-dual cyclic codes of even length over \({\mathbb{F}}_{2^m} + u{\mathbb{F}}_{2^m}\)", Finite Fields & Appl., Vol 61 (in press) (2020)

  16. Cao, Y., Cao, Y., Dinh, H.Q., Fu, F., Ma, F.: On matrix-product structure of repeatedroot constacyclic codes over fnite felds. Discrete Math. 343, 111768 (2020). ((in press))

    MathSciNet  Google Scholar 

  17. Cao, Y., Cao, Y., Dinh, H. Q., Bandi, R., Fu, F.: An explicit representation and enumeration for negacyclic codes of length \(2^kn\) over \({\mathbb{Z}} _4+u{\mathbb{Z}} _4\), Adv. Math. Commun., (in press) (2020)

  18. Cao, Y., Cao, Y., Dinh, H.Q., Bag, T., Yamaka, W.: “Explicit representation and enumeration of repeated-root \(\delta ^2+\alpha u^2\)-constacyclic codes over \(\mathbb{F} _{2^m}[u]/\langle u^{2\lambda }\). IEEE Access 8, 55550–55562 (2020)

    Google Scholar 

  19. Cao, Y., Cao, Y., Dinh, H.Q., Jitman, S.: “An efcient method to construct self-dual cyclic codes of length \(p^s\) over \(\mathbb{F} _{p^m}+u\mathbb{F} _{p^m}\). Discrete Math. 343, 111868 (2020)

    MathSciNet  MATH  Google Scholar 

  20. Chen, B., Ling, S., Zhang, G.: Application of Constacyclic codes to Quantum MDS Codes. IEEE Trans. Inform. Theory 61, 1474–1478 (2014)

    ADS  MathSciNet  MATH  Google Scholar 

  21. Chen, J., Huang, Y., Feng, C., Chen, R.: Entanglement-assisted quantum MDS codes constructed from negacyclic codes. Quantum Inf. Process. 16, 303 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  22. Chen, B., Dinh, H.Q., Liu, H.: Repeated-root constacyclic codes of length \(\ell p^s\) and their duals. Discrete Appl. Math. 177, 60–70 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Cleve, R., Gottesman, D.: Efficient computations of encodings for quantum error correction. Phys. Rev. A 56, 76 (1997)

    ADS  Google Scholar 

  24. Deutsch, D.: Quantum theory, the Church-Turing principle and the universal quantum computer. Proc. Royal Soci. Lond. A 400, 97–117 (1985)

    ADS  MathSciNet  MATH  Google Scholar 

  25. Dinh, H.Q.: Constacyclic codes of length \(p^s\) over \(\mathbb{F} _{p^m}+u\mathbb{F} _{p^m}\). J. Algebra 324, 940–950 (2010)

    MathSciNet  Google Scholar 

  26. Dinh, H.Q.: Repeated-root constacyclic codes of length \(2p^s\). Finite Fields Appl. 18, 133–143 (2012)

    MathSciNet  MATH  Google Scholar 

  27. Dinh, H.Q.: Structure of repeated-root constacyclic codes of length \(3p^s\) and their duals. Discrete Appl. Math. 313, 983–991 (2013)

    MATH  Google Scholar 

  28. Dinh, H.Q.: Repeated-Root cyclic and negacyclic codes of length \(6p^s\). Contemp. Math. 609, 69–87 (2014)

    MATH  Google Scholar 

  29. Dinh, H.Q., Dhompongsa, S., Sriboonchitta, S.: On constacyclic codes of length \(4p^s\) over \(\mathbb{F} _{p^m} + u\mathbb{F} _{p^m}\). Discrete Math. 340, 832–849 (2017)

    MathSciNet  MATH  Google Scholar 

  30. Dinh, H.Q., Nguyen, H.D.T., Sriboonchitta, S., Vo, T.M.: Repeated-root constacyclic codes of prime power lengths over finite chain rings. Finite Fields Appl. 43, 22–41 (2017)

    MathSciNet  MATH  Google Scholar 

  31. Dinh, H.Q., Nguyen, B.T., Sriboonchitta, S., Vo, T.M.: On a class of constacyclic codes of length \(4p^s\) over \(\mathbb{F} _{p^m} + u\mathbb{F} _{p^m}\). J. Algebra Appl. 18, 1950022 (2019)

    MathSciNet  Google Scholar 

  32. Dinh, H.Q., Nguyen, B.T., Sriboonchitta, S., Vo, T.M.: On \((\alpha + u\beta )\)-constacyclic codes of length \(4p^s\) over \(\mathbb{F} _{p^m} + u\mathbb{F} _{p^m}\). J. Algebra Appl. 18, 1950023 (2019)

    MathSciNet  Google Scholar 

  33. Dinh, H.Q., Nguyen, B.A.C.T., Sriboonchitta, S.: Negacyclic codes of length \(4p^s\) over \({\mathbb{F} } _{p^m} + u{\mathbb{F} } _{p^m}\). Discrete Math. 341, 1055–1071 (2018)

    MathSciNet  Google Scholar 

  34. Dinh, H.Q., 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)

    Google Scholar 

  35. Dinh, H.Q., Wang, X., Liu, H., Sriboonchitta, S.: Hamming distances of constacyclic codes of length \(3p^s\) and optimal codes with respect to the Griesmer and Singleton bounds. Discrete Appl. Math. 70, 1–22 (2020)

    Google Scholar 

  36. Denes, J., Keedwell, A.D.: Latin squares and their applications. Academic Press, New York (1974)

    MATH  Google Scholar 

  37. El-Khamy, M., McEliece, R.J.: The partition weight enumerator of MDS codes and its applications. Proc. Int. Symp. Inf. Theory ISIT 4, 926–930 (2005)

    Google Scholar 

  38. Fang, W., Fu, F.: Two new classes of quantum MDS codes. Finite Fields Appl. 53, 85–98 (2018)

    MathSciNet  MATH  Google Scholar 

  39. Fang, W., Fu, F.: Some new constructions of quantum MDS codes. IEEE Trans. Inf. Theory 65, 7840–7847 (2019)

    MathSciNet  MATH  Google Scholar 

  40. Golomb, S.W., Posner, E.C.: Rook domains, Latin squares, affine planes, and error-distributing codes. IEEE Trans. Inform. Theory. 10, 196–208 (1964)

    MathSciNet  MATH  Google Scholar 

  41. Grassl, M., Beth, T., Rȯtteler, M.: On optimal quantum codes. Int. J. Quantum Inform. 2, 757–766 (2004)

    MATH  Google Scholar 

  42. Grassl, M., Klappenecker, A., Rotteler, M.: Graphs, quadratic forms, and quantum codes,In: Proceedings 2002 IEEE International Symposium on Information Theory, pp. 45, (2002)

  43. Grassl, M., Beth, T., Geiselmann, W.: Quantum reed-solomon codes, AAECC-13. Honolulu, HI, USA (1999)

  44. Grassl, M., Beth, T.: Quantum BCH codes. In Proc. X. Intl. Symp., pp. 207–212. Theoretical Electrical Engineering, Magdeburg (1999)

    Google Scholar 

  45. Grassl, M.: Bounds on the minimum distance of linear codes and quantum codes. available online at http://www.codetables.de, accessed 2021-04-14 (2007)

  46. Guardia, G.G.L.: Constructions of new families of nonbinary quantum codes. Phys. Rev. A 80, 042331–04233 (2009)

    ADS  Google Scholar 

  47. Guenda, K., Jitman, S., Gulliver, T.A.: Constructions of good entanglement-assisted quantum error correcting codes. Des. Codes Crypt. 86, 121–136 (2018)

    MathSciNet  MATH  Google Scholar 

  48. Fujiwara, Y.: Block synchronization for quantum information. Phys. Rev. A 87, 109–120 (2013)

    Google Scholar 

  49. Fujiwara, Y., Tonchev, D.: High-rate self-synchronizing codes. IEEE Trans. Inform. Theory. 59, 2328–2335 (2013)

    MathSciNet  MATH  Google Scholar 

  50. Fujiwara, Y., Tonchev, D., Wong, H.: Algebraic techniques in designing quantum synchronizable codes. Phys. Rev. A 88, 162–166 (2013)

    Google Scholar 

  51. Fujiwara, Y., Vandendriessche, P.: Quantum synchronizable codes from finite geometries. IEEE Trans. Inform. Theory. 60, 7345–7354 (2014)

    MathSciNet  MATH  Google Scholar 

  52. Gottesman. D.: PhD Thesis (Caltech). quantph/9705052, (1997)

  53. Hu, D., Tang, W., Zhao, M., Chen, Q., Yu, S., Oh, C.: Graphical nonbinary quantum error-correcting codes. Phys. Rev. A 78, 1–11 (2008)

    Google Scholar 

  54. Joshi, D.: A Note on Upper Bounds for Minimum Distance Codes. Inf. Control 3, 289–295 (1958)

    MathSciNet  MATH  Google Scholar 

  55. Jin, L., Kan, H., Wen, J.: Quantum MDS codes with relatively large minimum distance from Hermitian self-orthogonal codes. Des. Codes Crypt. 84, 463–471 (2017)

    MathSciNet  MATH  Google Scholar 

  56. Jin, L., Ling, S., Luo, J., Xing, C.: Application of classical Hermitian self-orthogonal MDS codes to quantum MDS codes. IEEE Trans. Inform. Theory. 56, 4735–4740 (2010)

    MathSciNet  MATH  Google Scholar 

  57. Jin, L., Xing, C.: A construction of new quantum MDS codes. IEEE Trans. Inform. Theory. 60, 2921–2925 (2014)

    MathSciNet  MATH  Google Scholar 

  58. Komamiya, Y.: Application of logical mathematics to information theory. (Application of theory of groups to logical mathematics.). In: Proceedings of the Third Japan National Congress for Applied Mechanics, 1953, pages 437-442, Tokyo, Science Council of Japan (1954)

  59. Kai, X., Zhu, S.: New quantum MDS codes from negacyclic codes. IEEE Trans. Inform. Theory. 2, 1193–1197 (2013)

    MathSciNet  MATH  Google Scholar 

  60. Kai, X., Zhu, S., Li, P.: A construction of new MDS symbol-pair codes. IEEE Trans. Inform. Theory. 11, 5828–5834 (2015)

    MathSciNet  MATH  Google Scholar 

  61. Knill, E., Laflamme, R.: Theory of quantum error-correcting codes. Phys. Rev. A 55, 900–911 (1997)

    ADS  MathSciNet  Google Scholar 

  62. Knill, E., Laflamme, R.: A theory of quantum error-correcting codes. Phys. Rev. Lett. 84, 2525 (2000)

    ADS  MathSciNet  MATH  Google Scholar 

  63. Kwiat, G., Mattle, K., Weinfurter, H., Zeilinger, A., Sergienko, V., Shih, Y.: New high-intensity source of polarization-entangled photon pairs. Phys. Rev. Lett. 75, 4337 (1995)

    ADS  Google Scholar 

  64. Lidar, A., Brun, A.: Quantum Error Correction. Cambridge University Press, Cambridge (2013)

    Google Scholar 

  65. Liu, L., Li, L.Q., Kai, X., Zhu, S.: Repeated-root constacyclic codes of length \(3lp^n\) and their dual codes. Finite Fields Appl. 42, 269–295 (2016)

    MathSciNet  MATH  Google Scholar 

  66. Liu, X., Liu, H., Yu, L.: Entanglement-assisted quantum codes from matrix-product codes. Quantum Inf. Process. 18, 183 (2019)

    ADS  MathSciNet  MATH  Google Scholar 

  67. Liu, X., Yu, L., Hu, P.: New entanglement-assisted quantum codes from k-Galois dual codes. Finite Fields Appl. 55, 21–32 (2019)

    MathSciNet  MATH  Google Scholar 

  68. Luo, L., Ma, Z.: Non-binary quantum synchronizable codes from repeated-root cyclic codes. IEEE Trans. Inform. Theory. 14, 1–10 (2015)

    MATH  Google Scholar 

  69. Luo, L., Ma, Z., Lin, D.: Two new families of quantum synchronizable codes. Quantum Inf. Process. 18, 1–18 (2019)

    ADS  MathSciNet  MATH  Google Scholar 

  70. Laflamme, R., Miquel, C., Paz, P., Zurek, H.: Perfect quantum error correcting code. Phys. Rev. Lett. 77, 198 (1996)

    ADS  Google Scholar 

  71. van Lint, J.H.: Repeated-root cyclic codes. IEEE Trans. Inform. Theory. 37, 343–345 (1991)

    MathSciNet  MATH  Google Scholar 

  72. Li, Z., Xing, J., Wang, M.: Quantum generalized Reed-Solomon codes: unified framework for quantum maximum-distanceseparable codes. Phys. Rev. A 77, 1–4 (2008)

    Google Scholar 

  73. MacWilliams, J., Sloane, J.A.: The theory of error-correcting Codes, \(10^{th}\) impression. North-Holland, Amsterdam (1998)

    Google Scholar 

  74. Maneri, C., Silverman, R.: A combinatorial problem with applications to geometry. J. Comb. Theory Ser. A 11, 118–121 (1966)

    MathSciNet  MATH  Google Scholar 

  75. Massey, L., Costello, J., Justesen, J.: Polynomial weights and code constructions. IEEE Trans. Inform. Theory. 19, 101–110 (1973)

    MathSciNet  MATH  Google Scholar 

  76. Matthews, F., Politi, A., Stefanov, A., O’Brien, J.L.: Manipulation of multiphoton entanglement in waveguide quantum circuits. Nat. Photon. 3, 346–350 (2009)

    ADS  Google Scholar 

  77. Nielsen, A., Chuang, L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  78. Ozadam, H., Ozbudak , F.: The minimum Hamming distance of cyclic codes of length \(2p^s\). In: International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, pp. 92-100, (2009)

  79. Lopez-Permouth, S.R., Ozadam, H., Ozbudak, F., Szabo, S.: Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes. Finite Fields Appl. 19, 16–38 (2013)

    MathSciNet  MATH  Google Scholar 

  80. Pless, V., Huffman, W.C.: Handbook of Coding Theory. Elsevier, Amsterdam (1998)

    MATH  Google Scholar 

  81. Polyanskiy, Y.: Asynchronous communication: Exact synchronization, universality, and dispersion. IEEE Trans. Inform. Theory. 59, 1256–1270 (2013)

    MathSciNet  MATH  Google Scholar 

  82. Prange, E.: Cyclic error-correcting codes in two symbols, TN-57-103 (1957)

  83. Prevedel, R., Cronenberg, G., Tame, M.S., Paternostro, M., Walther, P., Kim, M.S., Zeilinger, A.: Experimental realization of dicke states of up to six qubits for multiparty quantum networking. Phys. Rev. Lett. 103, 020503 (2009)

    ADS  Google Scholar 

  84. Sklar, B.: Digital communications: fundamentals and applications, 2nd edn. Prentice Hall, Upper Saddle River (2001)

    MATH  Google Scholar 

  85. Radmark, M., Zukowski, M., Bourennane, M.: Experimental test of fidelity limits in six-photon interferometry and of rotational invariance properties of the photonic six-qubit entanglement singlet State. Phys. Rev. Lett. 103, 150501 (2009)

    ADS  Google Scholar 

  86. Rains, M.: Quantum weight Enumerators. IEEE Trans. Inform. Theory. 4, 1388–1394 (1998)

    MathSciNet  MATH  Google Scholar 

  87. Roman, S.: Coding and Information Theory, GTM, 134, Springer-Verlag, ISBN 0-387-97812-7, (1992)

  88. Roth, M., Seroussi, G.: On cyclic MDS codes of length \(q\) over \(GF(q)\). IEEE Trans. Inform. Theory. 32, 284–285 (1986)

    MathSciNet  MATH  Google Scholar 

  89. Schlingemann, D., Werner, F.: Quantum error-correcting codes associated with graphs. Phys. Rev. A 65, 012308 (2001)

    ADS  Google Scholar 

  90. Schlingemann, D.: Stabilizer codes can be realized as graph codes. Quantum Inf. Comput. 2, 307–323 (2002)

    MathSciNet  MATH  Google Scholar 

  91. Shi, X., Yue, Q., Zhu, X.: Construction of some new quantum MDS codes. Finite Fields Appl. 46, 347–362 (2017)

    MathSciNet  MATH  Google Scholar 

  92. Shor, W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, 2493 (1995)

    ADS  Google Scholar 

  93. Silverman, R.: A metrization for power-sets with applications to combinatorial analysis. Canad. J. Math. 12, 158–176 (1960)

    MathSciNet  MATH  Google Scholar 

  94. Steane, M.: Error correcting codes in quantum theory. Phys. Rev. Lett. 77, 793 (1996)

    ADS  MathSciNet  MATH  Google Scholar 

  95. Steane, M.: Enlargement of Calderbank-Shor-Steane quantum codes. IEEE Trans. Inform. Theory. 45, 2492–2495 (1999)

    MathSciNet  MATH  Google Scholar 

  96. Tolhuizen, M.: On Maximum distance separable codes over alphabets of arbitrary size. In: Proc. Int. Symp. Inf. Theory ISIT, pp. 431, (1994)

  97. Wang, J., Li, R., Lv, J., Song, H.: Entanglement-assisted quantum codes from cyclic codes and negacyclic codes. Quantum Inf. Process. 5, 1–18 (2020)

    MathSciNet  MATH  Google Scholar 

  98. Wang, L., Zhu, S., Sun, Z.: Entanglement-assisted quantum MDS codes from cyclic codes. Quantum Inf. Process. 19, 1–18 (2020)

    ADS  MathSciNet  MATH  Google Scholar 

  99. Wieczorek, W., Krischek, R., Kiesel, N., Michelberger, P., Tóth, G., Weinfurter, H.: Experimental entanglement of a six-photon symmetric Dicke State. Phys. Rev. Lett. 103, 020504 (2009)

    ADS  Google Scholar 

  100. Xie, Y., Yuan, J., Fujiwara, Y.: Quantum synchronizable codes from augmentation of cyclic codes. PLoS ONE 6, e14641 (2014)

    Google Scholar 

  101. Xie, Y., Yang, L., Yuan, J.: q-ary chain-containing quantum synchronizable codes. IEEE Commun. Lett. 20, 414–417 (2016)

    Google Scholar 

  102. Yao, C., Wang, X., Xu, P., Lu, H., Pan, S., Bao, H., Peng, Z., Lu, Y., Chen, A., Pan, W.: Observation of eight-photon entanglement. Nat. Photon. 6, 225–228 (2012)

    ADS  Google Scholar 

  103. Zhou, X., Song, L., Zhang, Y.: Physical layer security in wireless communications. CRC Press, Inc., USA (2013)

    Google Scholar 

  104. Zhang, T., Ge, G.: Some new classes of quantum MDS codes from constacyclic codes. IEEE Trans. Inf. Theory 61, 5224–5228 (2015)

    ADS  MathSciNet  MATH  Google Scholar 

  105. Zhang, T., Ge, G.: Quantum MDS codes with large minimum distance. Des. Codes Cryptogr. 83, 503–517 (2017)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

H.Q. Dinh and W. Yamaka are grateful to the Centre of Excellence in Econometrics, Faculty of Economics, Chiang Mai University, for partial financial support.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Kent State University, Kent, USA

    Hai Q. Dinh

  2. Institute of Fundamental and Applied Sciences, Duy Tan University, Ho Chi Minh City, 700000, Vietnam

    Bac T. Nguyen

  3. Faculty of Natural Sciences, Duy Tan University, Da Nang, 550000, Vietnam

    Bac T. Nguyen

  4. Centre of Excellence in Econometrics, Faculty of Economics, Chiang Mai University, Chiang Mai, 52000, Thailand

    M. Paravee

  5. Faculty of Education, Thu Dau Mot University, Binh Duong, Vietnam

    Hiep L. Thi

  6. Department of General Education, Industrial University of Vinh, Vinh City, Nghe An, Vietnam

    Thang M. Vo

  7. Mathematics Department, Ohio University, Athens, OH, 45701, Greece

    Thang M. Vo

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Dinh, H.Q., Nguyen, B.T., Paravee, M. et al. Optimal constructions of quantum and synchronizable codes from repeated-root cyclic codes of length \(3p^s\). Quantum Inf Process 22, 257 (2023). https://doi.org/10.1007/s11128-023-03958-7

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