{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T09:56:08Z","timestamp":1780394168691,"version":"3.54.1"},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"10","license":[{"start":{"date-parts":[[2019,9,3]],"date-time":"2019-09-03T00:00:00Z","timestamp":1567468800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,9,3]],"date-time":"2019-09-03T00:00:00Z","timestamp":1567468800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["61802102"],"award-info":[{"award-number":["61802102"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["61772168"],"award-info":[{"award-number":["61772168"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100003995","name":"Natural Science Foundation of Anhui Province","doi-asserted-by":"crossref","award":["1708085QA01"],"award-info":[{"award-number":["1708085QA01"]}],"id":[{"id":"10.13039\/501100003995","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100004543","name":"China Scholarship Council","doi-asserted-by":"crossref","award":["201806695004"],"award-info":[{"award-number":["201806695004"]}],"id":[{"id":"10.13039\/501100004543","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Quantum Inf Process"],"published-print":{"date-parts":[[2019,10]]},"DOI":"10.1007\/s11128-019-2440-1","type":"journal-article","created":{"date-parts":[[2019,9,3]],"date-time":"2019-09-03T17:40:51Z","timestamp":1567532451000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":26,"title":["Hermitian dual-containing narrow-sense constacyclic BCH codes and quantum codes"],"prefix":"10.1007","volume":"18","author":[{"given":"Liqi","family":"Wang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhonghua","family":"Sun","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shixin","family":"Zhu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2019,9,3]]},"reference":[{"key":"2440_CR1","doi-asserted-by":"crossref","unstructured":"Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: Primitive quantum BCH codes over finite fields. In: Proceedings of International Symposium on Information Theory (ISIT), pp. 1114\u20131118 (2006)","DOI":"10.1109\/ISIT.2006.261957"},{"issue":"3","key":"2440_CR2","doi-asserted-by":"publisher","first-page":"1183","DOI":"10.1109\/TIT.2006.890730","volume":"53","author":"SA Aly","year":"2007","unstructured":"Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: On quantum and classical BCH codes. IEEE Trans. Inf. Theory 53(3), 1183\u20131188 (2007)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"7","key":"2440_CR3","doi-asserted-by":"publisher","first-page":"3065","DOI":"10.1109\/18.959288","volume":"47","author":"A Ashikhmin","year":"2001","unstructured":"Ashikhmin, A., Kill, E.: Nonbinary quantum stabilizer codes. IEEE Trans. Inf. Theory 47(7), 3065\u20133072 (2001)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"2440_CR4","unstructured":"Beigi, S., Chen, J., Grassl, M., Ji, Z., Wang, Q., Zeng, B.: Symmetries of codeword stabilized quantum codes. e-print \n                    arXiv:1303.7020"},{"key":"2440_CR5","volume-title":"Algebraic Coding Theory","author":"ER Berlekamp","year":"1984","unstructured":"Berlekamp, E.R.: Algebraic Coding Theory. Aegean Park Press, Laguna Hills (1984)"},{"issue":"4","key":"2440_CR6","doi-asserted-by":"publisher","first-page":"1369","DOI":"10.1109\/18.681315","volume":"44","author":"AR Calderbank","year":"1998","unstructured":"Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction via codes over $${\\rm GF}(4)$$. IEEE Trans. Inf. Theory 44(4), 1369\u20131387 (1998)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"3","key":"2440_CR7","doi-asserted-by":"publisher","first-page":"1474","DOI":"10.1109\/TIT.2015.2388576","volume":"61","author":"B Chen","year":"2015","unstructured":"Chen, B., Ling, S., Zhang, G.: Application of constacyclic codes to quantum MDS codes. IEEE Trans. Inf. Theory 61(3), 1474\u20131484 (2015)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"2440_CR8","unstructured":"Faist, P., Nezami, S., Albert, V. V., Salton, G., Pastawski, F., Hayden, P., Preskill, J.: Continuous symmetries and approximate quantum error correction. e-print \n                    arXiv:1902.07714"},{"key":"2440_CR9","unstructured":"Grassl, M., Beth, T.: Quantum BCH codes. In: Proceedings of X. International Symposium on Theoretical Electrical Engineering, pp. 207\u2013212 (1999)"},{"key":"2440_CR10","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1007\/3-540-46796-3_23","volume-title":"Applied Algebra, Algebraic Algorithms and Error-Correcting Codes","author":"Markus Grassl","year":"1999","unstructured":"Grassl, M., Geiselmann, W., Beth, T.: Quantum Reed\u2013Solomon codes. In: Proceedings of AAECC-13, pp. 231\u2013244 (1999)"},{"key":"2440_CR11","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1007\/s11128-018-2049-9","volume":"17","author":"L Hu","year":"2018","unstructured":"Hu, L., Yue, Q., He, X.: Quantum MDS codes from BCH constacyclic codes. Quantum Inf. Process. 17, 323 (2018)","journal-title":"Quantum Inf. Process."},{"key":"2440_CR12","doi-asserted-by":"publisher","DOI":"10.1145\/1460833.1460872","volume-title":"A Programming Language","author":"KE Iverson","year":"1962","unstructured":"Iverson, K.E.: A Programming Language. Wiley, Amsterdam (1962)"},{"issue":"7","key":"2440_CR13","doi-asserted-by":"publisher","first-page":"1850059","DOI":"10.1142\/S0219749918500594","volume":"16","author":"X Kai","year":"2018","unstructured":"Kai, X., Li, P., Zhu, S.: Construction of quantum negacyclic BCH codes. Int. J. Quantum Inf. 16(7), 1850059 (2018)","journal-title":"Int. J. Quantum Inf."},{"key":"2440_CR14","doi-asserted-by":"publisher","first-page":"012326","DOI":"10.1103\/PhysRevA.88.012326","volume":"88","author":"X Kai","year":"2013","unstructured":"Kai, X., Zhu, S.: Quantum negacyclic codes. Phys. Rev. A 88, 012326 (2013)","journal-title":"Phys. Rev. A"},{"issue":"4","key":"2440_CR15","doi-asserted-by":"publisher","first-page":"2080","DOI":"10.1109\/TIT.2014.2308180","volume":"60","author":"X Kai","year":"2014","unstructured":"Kai, X., Zhu, S., Li, P.: Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inf. Theory 60(4), 2080\u20132085 (2014)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"2440_CR16","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1080\/00029890.1992.11995869","volume":"99","author":"DE Knuth","year":"1992","unstructured":"Knuth, D.E.: Two notes on notation. Am. Math. Monthly 99, 403\u2013422 (1992)","journal-title":"Am. Math. Monthly"},{"issue":"4","key":"2440_CR17","doi-asserted-by":"publisher","first-page":"880","DOI":"10.1109\/18.53751","volume":"36","author":"A Krishna","year":"1990","unstructured":"Krishna, A., Sarwate, D.V.: Pseudo-cyclic maximum-distance separable codes. IEEE Trans. Inf. Theory 36(4), 880\u2013884 (1990)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"4","key":"2440_CR18","doi-asserted-by":"publisher","first-page":"042331","DOI":"10.1103\/PhysRevA.80.042331","volume":"80","author":"GG La Guardia","year":"2009","unstructured":"La Guardia, G.G.: Constructions of new families of nonbinary quantum codes. Phys. Rev. A 80(4), 042331 (2009)","journal-title":"Phys. Rev. A"},{"issue":"8","key":"2440_CR19","doi-asserted-by":"publisher","first-page":"5551","DOI":"10.1109\/TIT.2011.2159039","volume":"57","author":"GG La Guardia","year":"2011","unstructured":"La Guardia, G.G.: New quantum MDS codes. IEEE Trans. Inf. Theory 57(8), 5551\u20135554 (2011)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"3","key":"2440_CR20","doi-asserted-by":"publisher","first-page":"1528","DOI":"10.1109\/TIT.2014.2298137","volume":"60","author":"GG La Guardia","year":"2014","unstructured":"La Guardia, G.G.: On the construction of nonbinary quantum BCH codes. IEEE Trans. Inf. Theory 60(3), 1528\u20131535 (2014)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"2440_CR21","doi-asserted-by":"publisher","first-page":"594","DOI":"10.1016\/j.laa.2016.02.014","volume":"496","author":"GG La Guardia","year":"2016","unstructured":"La Guardia, G.G.: On optimal constacyclic codes. Linear Algebra Appl. 496, 594\u2013610 (2016)","journal-title":"Linear Algebra Appl."},{"key":"2440_CR22","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1007\/s11128-019-2242-5","volume":"18","author":"R Li","year":"2019","unstructured":"Li, R., Wang, J., Liu, Y., Guo, G.: New quantum constacyclic codes. Quantum Inf. Process. 18, 127 (2019)","journal-title":"Quantum Inf. Process."},{"key":"2440_CR23","first-page":"0021","volume":"12","author":"R Li","year":"2013","unstructured":"Li, R., Zuo, F., Liu, Y., Xu, Z.: Hermitian dual-containing BCH codes and construction of new quantum codes. Quantum Inf. Comput. 12, 0021\u20130035 (2013)","journal-title":"Quantum Inf. Comput."},{"issue":"3","key":"2440_CR24","first-page":"5551","volume":"50","author":"X Lin","year":"2004","unstructured":"Lin, X.: Quantum cyclic and constacyclic codes. IEEE Trans. Inf. Theory 50(3), 5551\u20135554 (2004)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"2440_CR25","doi-asserted-by":"publisher","first-page":"1910","DOI":"10.1016\/j.disc.2017.04.001","volume":"340","author":"H Liu","year":"2017","unstructured":"Liu, H., Ding, C., Li, C.: Dimensions of three types of BCH codes over $${\\rm GF}(q)$$. Discrete Math. 340, 1910\u20131927 (2017)","journal-title":"Discrete Math."},{"key":"2440_CR26","doi-asserted-by":"publisher","first-page":"66","DOI":"10.1007\/s11128-017-1533-y","volume":"16","author":"Y Liu","year":"2017","unstructured":"Liu, Y., Li, R., Lv, L., Ma, Y.: A class of constacyclic BCH codes and new quantum codes. Quantum Inf. Process. 16, 66 (2017)","journal-title":"Quantum Inf. Process."},{"key":"2440_CR27","doi-asserted-by":"publisher","first-page":"675","DOI":"10.1007\/11750321_63","volume":"3959","author":"Z Ma","year":"2006","unstructured":"Ma, Z., Lu, X., Feng, K., Feng, D.: On non-binary quantum BCH codes. Lect. Notes Comput. Sci. 3959, 675\u2013683 (2006)","journal-title":"Lect. Notes Comput. Sci."},{"key":"2440_CR28","volume-title":"The theory of error-correcting codes","author":"FJ MacWilliams","year":"1997","unstructured":"MacWilliams, F.J., Sloane, N.J.A.: The theory of error-correcting codes. North-Holland Pub. Co, Amsterdam (1997)"},{"issue":"6","key":"2440_CR29","doi-asserted-by":"publisher","first-page":"3117","DOI":"10.1109\/TIT.2014.2313577","volume":"60","author":"Y Ouyang","year":"2014","unstructured":"Ouyang, Y.: Concatenated quantum codes can attain the quantum Gilbert\u2013Varshamov bound. IEEE Trans. Inf. Theory 60(6), 3117\u20133122 (2014)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"2440_CR30","doi-asserted-by":"publisher","first-page":"062317","DOI":"10.1103\/PhysRevA.90.062317","volume":"90","author":"Y Ouyang","year":"2014","unstructured":"Ouyang, Y.: Permutation-invariant quantum codes. Phys. Rev. A 90, 062317 (2014)","journal-title":"Phys. Rev. A"},{"key":"2440_CR31","unstructured":"Ouyang, Y.: Permutation-invariant constant-excitation quantum codes for amplitude damping. e-print \n                    arXiv:1809.09801"},{"key":"2440_CR32","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1016\/j.laa.2004.06.014","volume":"392","author":"H Pollatsek","year":"2004","unstructured":"Pollatsek, H., Ruskai, M.B.: Permutationally invariant codes for quantum error correction. Linear Algebra Appl. 392, 255\u2013288 (2004)","journal-title":"Linear Algebra Appl."},{"key":"2440_CR33","doi-asserted-by":"publisher","first-page":"1355","DOI":"10.1007\/s10773-017-3277-y","volume":"56","author":"J Qian","year":"2017","unstructured":"Qian, J., Zhang, L.: Improved constructions for nonbinary quantum BCH codes. Int. J. Theor. Phys. 56, 1355\u20131363 (2017)","journal-title":"Int. J. Theor. Phys."},{"key":"2440_CR34","doi-asserted-by":"publisher","first-page":"194","DOI":"10.1103\/PhysRevLett.85.194","volume":"85","author":"MB Ruskai","year":"2000","unstructured":"Ruskai, M.B.: Pauli exchange errors in quantum computation. Phys. Rev. Lett. 85, 194\u2013197 (2000)","journal-title":"Phys. Rev. Lett."},{"key":"2440_CR35","doi-asserted-by":"publisher","first-page":"270","DOI":"10.1007\/s11128-018-2042-3","volume":"17","author":"H Song","year":"2018","unstructured":"Song, H., Li, R., Wang, J., Liu, Y.: Two families of BCH codes and new quantum codes. Quantum Inf. Process. 17, 270 (2018)","journal-title":"Quantum Inf. Process."},{"issue":"7","key":"2440_CR36","doi-asserted-by":"publisher","first-page":"2492","DOI":"10.1109\/18.796388","volume":"45","author":"AM Steane","year":"1999","unstructured":"Steane, A.M.: Enlargement of Calderbank\u2013Shor\u2013Steane quantum codes. IEEE Trans. Inf. Theory 45(7), 2492\u20132495 (1999)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"1","key":"2440_CR37","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1587\/transfun.E102.A.303","volume":"E102\u2013A","author":"N Tang","year":"2019","unstructured":"Tang, N., Li, Z., Xing, L., Zhang, M., Zhao, F.: Some improved constructions for nonbinary quantum BCH codes. IEICE Trans. Fund. E102\u2013A(1), 303\u2013306 (2019)","journal-title":"IEICE Trans. Fund."},{"issue":"3","key":"2440_CR38","doi-asserted-by":"publisher","first-page":"881","DOI":"10.1007\/s11128-014-0903-y","volume":"14","author":"L Wang","year":"2015","unstructured":"Wang, L., Zhu, S.: New quantum MDS codes derived from constacyclic codes. Quantum Inf. Process. 14(3), 881\u2013889 (2015)","journal-title":"Quantum Inf. Process."},{"issue":"1","key":"2440_CR39","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1007\/s10623-016-0296-2","volume":"85","author":"J Yuan","year":"2017","unstructured":"Yuan, J., Zhu, S., Kai, X., Li, P.: On the construction of quantum constacyclic codes. Des. Codes Cryptogr. 85(1), 179\u2013190 (2017)","journal-title":"Des. Codes Cryptogr."},{"key":"2440_CR40","doi-asserted-by":"publisher","first-page":"36122","DOI":"10.1109\/ACCESS.2018.2851075","volume":"4","author":"M Zhang","year":"2018","unstructured":"Zhang, M., Li, Z., Xing, L., Tang, N.: Construction of some new quantum BCH codes. IEEE Access 4, 36122 (2018)","journal-title":"IEEE Access"},{"issue":"2","key":"2440_CR41","doi-asserted-by":"publisher","first-page":"615","DOI":"10.1007\/s10773-018-3959-0","volume":"58","author":"M Zhang","year":"2019","unstructured":"Zhang, M., Li, Z., Xing, L., Tang, N.: Some families of quantum BCH codes. Int. J. Theor. Phys. 58(2), 615\u2013630 (2019)","journal-title":"Int. J. Theor. Phys."},{"issue":"9","key":"2440_CR42","doi-asserted-by":"publisher","first-page":"5224","DOI":"10.1109\/TIT.2015.2450235","volume":"61","author":"T Zhang","year":"2015","unstructured":"Zhang, T., Ge, G.: Some new class of quantum MDS codes from constacyclic codes. IEEE Trans. Inf. Theory 61(9), 5224\u20135228 (2015)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"10","key":"2440_CR43","doi-asserted-by":"publisher","first-page":"2139","DOI":"10.1007\/s10623-017-0441-6","volume":"86","author":"S Zhu","year":"2018","unstructured":"Zhu, S., Sun, Z., Li, P.: A class of negacyclic BCH codes and its application to quantum codes. Des. Codes Cryptogr. 86(10), 2139\u20132165 (2018)","journal-title":"Des. Codes Cryptogr."}],"container-title":["Quantum Information Processing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-019-2440-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11128-019-2440-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-019-2440-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,9,1]],"date-time":"2020-09-01T23:38:27Z","timestamp":1599003507000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11128-019-2440-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,9,3]]},"references-count":43,"journal-issue":{"issue":"10","published-print":{"date-parts":[[2019,10]]}},"alternative-id":["2440"],"URL":"https:\/\/doi.org\/10.1007\/s11128-019-2440-1","relation":{},"ISSN":["1570-0755","1573-1332"],"issn-type":[{"value":"1570-0755","type":"print"},{"value":"1573-1332","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,9,3]]},"assertion":[{"value":"11 May 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 August 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 September 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"323"}}