{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,6]],"date-time":"2025-07-06T04:01:15Z","timestamp":1751774475219,"version":"3.41.0"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2025,6,6]],"date-time":"2025-06-06T00:00:00Z","timestamp":1749168000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,6,6]],"date-time":"2025-06-06T00:00:00Z","timestamp":1749168000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["U21A20428","12171134"],"award-info":[{"award-number":["U21A20428","12171134"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Quantum Inf Process"],"DOI":"10.1007\/s11128-025-04788-5","type":"journal-article","created":{"date-parts":[[2025,6,6]],"date-time":"2025-06-06T06:35:25Z","timestamp":1749191725000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Quantum MDS codes with length $$n\\equiv 0,1($$mod$$\\,\\frac{q+1}{2})$$ or $$n\\equiv 0,1($$mod$$\\,\\frac{q-1}{2})$$"],"prefix":"10.1007","volume":"24","author":[{"given":"Ruhao","family":"Wan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mengchen","family":"Lian","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2845-3568","authenticated-orcid":false,"given":"Shixin","family":"Zhu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,6,6]]},"reference":[{"issue":"4","key":"4788_CR1","doi-asserted-by":"crossref","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 GF(4). IEEE Trans. Inf. Theory 44(4), 1369\u20131387 (1998)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"6","key":"4788_CR2","doi-asserted-by":"crossref","first-page":"1827","DOI":"10.1109\/18.782103","volume":"45","author":"EM Rains","year":"1999","unstructured":"Rains, E.M.: Nonbinary quantum codes. IEEE Trans. Inf. Theory 45(6), 1827\u20131832 (1999)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"7","key":"4788_CR3","doi-asserted-by":"crossref","first-page":"3065","DOI":"10.1109\/18.959288","volume":"47","author":"A Ashikhmin","year":"2001","unstructured":"Ashikhmin, A., Knill, E.: Nonbinary quantum stabilizer codes. IEEE Trans. Inf. Theory 47(7), 3065\u20133072 (2001)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"11","key":"4788_CR4","doi-asserted-by":"crossref","first-page":"4892","DOI":"10.1109\/TIT.2006.883612","volume":"52","author":"A Ketkar","year":"2006","unstructured":"Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theory 52(11), 4892\u20134914 (2006)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"3","key":"4788_CR5","doi-asserted-by":"crossref","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":"8","key":"4788_CR6","doi-asserted-by":"crossref","first-page":"5551","DOI":"10.1109\/TIT.2011.2159039","volume":"57","author":"GGL Guardia","year":"2011","unstructured":"Guardia, G.G.L.: New quantum MDS codes. IEEE Trans. Inf. Theory 57(8), 5551\u20135554 (2011)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"4788_CR7","doi-asserted-by":"crossref","unstructured":"Grassl, M., R\u00f6tteler, M.: Quantum MDS codes over small fields, In: Proc. Int. Symp. Inf. Theory (ISIT), 1104\u20131108 (2015)","DOI":"10.1109\/ISIT.2015.7282626"},{"key":"4788_CR8","doi-asserted-by":"crossref","unstructured":"R\u00f6ttler, M., Grassl, M., Beth, T.: On quantum MDS codes. In Proceedings of the International Symposium on Information Theory, Chicago, USA, 356 (2004)","DOI":"10.1109\/ISIT.2004.1365393"},{"issue":"1","key":"4788_CR9","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1142\/S0219749904000079","volume":"2","author":"M Grassl","year":"2004","unstructured":"Grassl, M., Beth, T., R\u00f6ttler, M.: On optimal quantum codes. Int. J. Quantum Inf. 2(1), 55\u201364 (2004)","journal-title":"Int. J. Quantum Inf."},{"issue":"6","key":"4788_CR10","doi-asserted-by":"crossref","first-page":"3942","DOI":"10.1109\/TIT.2022.3149291","volume":"68","author":"M Grassl","year":"2022","unstructured":"Grassl, M., Huber, F., Winter, A.: Entropic proofs of singleton bounds for quantum error-correcting codes. IEEE Trans. Inf. Theory 68(6), 3942\u20133950 (2022)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"2","key":"4788_CR11","doi-asserted-by":"crossref","first-page":"1193","DOI":"10.1109\/TIT.2012.2220519","volume":"59","author":"X Kai","year":"2013","unstructured":"Kai, X., Zhu, S.: New quantum MDS codes from negacyclic codes. IEEE Trans. Inf. Theory 59(2), 1193\u20131197 (2013)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"4","key":"4788_CR12","doi-asserted-by":"crossref","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\u20132086 (2014)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"4","key":"4788_CR13","doi-asserted-by":"crossref","first-page":"1450019","DOI":"10.1142\/S0219749914500191","volume":"12","author":"G Zhang","year":"2014","unstructured":"Zhang, G., Chen, B.: New quantum MDS codes. Int. J. Quantum Inf. 12(4), 1450019 (2014)","journal-title":"Int. J. Quantum Inf."},{"issue":"3","key":"4788_CR14","doi-asserted-by":"crossref","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":"9","key":"4788_CR15","doi-asserted-by":"crossref","first-page":"5224","DOI":"10.1109\/TIT.2015.2450235","volume":"61","author":"T Zhang","year":"2015","unstructured":"Zhang, T., Ge, G.: Some new classes of quantum MDS codes from constacyclic codes. IEEE Trans. Inf. Theory 61(9), 5224\u20135228 (2015)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"4788_CR16","doi-asserted-by":"crossref","first-page":"891","DOI":"10.1007\/s11401-016-1043-8","volume":"37","author":"L Hu","year":"2016","unstructured":"Hu, L., Yue, Q., Zhu, X.: New quantum MDS codes from constacyclic codes. Chin. Ann. Math. Ser B. 37, 891\u2013898 (2016)","journal-title":"Chin. Ann. Math. Ser B."},{"issue":"3","key":"4788_CR17","doi-asserted-by":"crossref","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"},{"issue":"4","key":"4788_CR18","doi-asserted-by":"crossref","first-page":"1703","DOI":"10.1109\/TIT.2016.2535180","volume":"62","author":"S Li","year":"2016","unstructured":"Li, S., Xiong, M., Ge, G.: Pseudo-cyclic codes and the construction of quantum MDS codes. IEEE Trans. Inf. Theory 62(4), 1703\u20131710 (2016)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"1","key":"4788_CR19","first-page":"012308-1","volume":"77","author":"Z Li","year":"2008","unstructured":"Li, Z., Xing, L., Wang, X.: Quantum generalized Reed-Solomon codes: unified framework for quantum MDS codes. Phys. Rev. A 77(1), 012308-1\u2013012308-4 (2008)","journal-title":"Phys. Rev. A"},{"issue":"9","key":"4788_CR20","doi-asserted-by":"crossref","first-page":"4735","DOI":"10.1109\/TIT.2010.2054174","volume":"56","author":"L Jin","year":"2010","unstructured":"Jin, L., Ling, S., Luo, J., Xing, C.: Application of classical Hermitian self-orthogonal MDS codes to quantum MDS codes. IEEE. Trans. Inf. Theory 56(9), 4735\u20134740 (2010)","journal-title":"IEEE. Trans. Inf. Theory"},{"issue":"8","key":"4788_CR21","doi-asserted-by":"crossref","first-page":"5484","DOI":"10.1109\/TIT.2011.2177066","volume":"58","author":"L Jin","year":"2012","unstructured":"Jin, L., Xing, C.: Euclidean and Hermitian self-orthogonal algebraic geometry codes and their application to quantum codes. IEEE Trans. Inf. Theory 58(8), 5484\u20135489 (2012)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"5","key":"4788_CR22","doi-asserted-by":"crossref","first-page":"2921","DOI":"10.1109\/TIT.2014.2299800","volume":"60","author":"L Jin","year":"2014","unstructured":"Jin, L., Xing, C.: A construction of new quantum MDS codes. IEEE Trans. Inf. Theory 60(5), 2921\u20132925 (2014)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"3","key":"4788_CR23","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1007\/s10623-016-0281-9","volume":"84","author":"L Jin","year":"2017","unstructured":"Jin, L., Kan, H., Wen, J.: Quantum MDS codes with relatively large minimum distance from Hermitian self-orthogonal codes. Des. Codes Cryptogr. 84(3), 463\u2013471 (2017)","journal-title":"Des. Codes Cryptogr."},{"issue":"7","key":"4788_CR24","doi-asserted-by":"crossref","first-page":"2745","DOI":"10.1007\/s11128-016-1311-2","volume":"15","author":"X He","year":"2016","unstructured":"He, X., Xu, L., Chen, H.: New $$q$$-ary quantum MDS codes with distances bigger than $$\\frac{q}{2}$$. Quantum Inf. Process. 15(7), 2745\u20132758 (2016)","journal-title":"Quantum Inf. Process."},{"issue":"3","key":"4788_CR25","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1007\/s10623-016-0245-0","volume":"83","author":"T Zhang","year":"2017","unstructured":"Zhang, T., Ge, G.: Quantum MDS codes with large minimum distance. Des. Codes Cryptogr. 83(3), 503\u2013517 (2017)","journal-title":"Des. Codes Cryptogr."},{"key":"4788_CR26","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1016\/j.ffa.2017.04.002","volume":"46","author":"X Shi","year":"2017","unstructured":"Shi, X., Yue, Q., Zhu, X.: Construction of some new quantum MDS codes. Finite Fields Appl. 46, 347\u2013362 (2017)","journal-title":"Finite Fields Appl."},{"issue":"6","key":"4788_CR27","doi-asserted-by":"crossref","first-page":"1165","DOI":"10.1007\/s12095-017-0274-1","volume":"10","author":"X Shi","year":"2018","unstructured":"Shi, X., Yue, Q., Chang, Y.: Some quantum MDS codes with large minimum distance from generalized Reed-Solomon codes. Cryptogr. Commun. 10(6), 1165\u20131182 (2018)","journal-title":"Cryptogr. Commun."},{"issue":"12","key":"4788_CR28","doi-asserted-by":"crossref","first-page":"111593","DOI":"10.1016\/j.disc.2019.07.009","volume":"342","author":"F Tian","year":"2019","unstructured":"Tian, F., Zhu, S.: Some new quantum MDS codes from generalized Reed-Solomon codes. Discrete Math. 342(12), 111593 (2019)","journal-title":"Discrete Math."},{"issue":"7","key":"4788_CR29","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1007\/s11128-024-04487-7","volume":"23","author":"F Tian","year":"2024","unstructured":"Tian, F., Li, L., Wu, T., Chen, X.: Some constructions of quantum MDS codes and EAQMDS codes from GRS codes. Quantum Inf. Process. 23(7), 277 (2024)","journal-title":"Quantum Inf. Process."},{"key":"4788_CR30","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.ffa.2018.06.003","volume":"53","author":"W Fang","year":"2018","unstructured":"Fang, W., Fu, F.: Two new classes of quantum MDS codes. Finite Fields Appl. 53, 85\u201398 (2018)","journal-title":"Finite Fields Appl."},{"issue":"6","key":"4788_CR31","doi-asserted-by":"crossref","first-page":"3527","DOI":"10.1109\/TIT.2019.2950245","volume":"66","author":"W Fang","year":"2020","unstructured":"Fang, W., Fu, F., Li, L., Zhu, S.: Euclidean and Hermitian hulls of MDS codes and their applications to EAQECCs. IEEE Trans. Inf. Theory 66(6), 3527\u20133537 (2020)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"12","key":"4788_CR32","doi-asserted-by":"crossref","first-page":"7840","DOI":"10.1109\/TIT.2019.2939114","volume":"65","author":"W Fang","year":"2019","unstructured":"Fang, W., Fu, F.: Some new constructions of quantum MDS codes. IEEE Trans. Inf. Theory 65(12), 7840\u20137847 (2019)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"1","key":"4788_CR33","doi-asserted-by":"crossref","first-page":"113662","DOI":"10.1016\/j.disc.2023.113662","volume":"347","author":"W Fang","year":"2024","unstructured":"Fang, W., Wen, J., Fu, F.: Quantum MDS codes with new length and large minimum distance. Discrete Math. 347(1), 113662 (2024)","journal-title":"Discrete Math."},{"issue":"1","key":"4788_CR34","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1007\/s11128-019-2506-0","volume":"19","author":"X Fang","year":"2020","unstructured":"Fang, X., Luo, J.: New quantum MDS codes over finite fields. Quantum Inf. Process. 19(1), 16 (2020)","journal-title":"Quantum Inf. Process."},{"issue":"11","key":"4788_CR35","doi-asserted-by":"crossref","first-page":"3525","DOI":"10.1007\/s10773-018-3867-3","volume":"57","author":"G Guo","year":"2018","unstructured":"Guo, G., Li, R., Guo, L.: On the construction of quantum MDS codes. Int. J. Theor. Phys. 57(11), 3525\u20133539 (2018)","journal-title":"Int. J. Theor. Phys."},{"key":"4788_CR36","doi-asserted-by":"crossref","first-page":"101901","DOI":"10.1016\/j.ffa.2021.101901","volume":"76","author":"G Guo","year":"2021","unstructured":"Guo, G., Li, R., Liu, Y.: Application of Hermitian self-orthogonal GRS codes to some quantum MDS codes. Finite Fields Appl. 76, 101901 (2021)","journal-title":"Finite Fields Appl."},{"issue":"5","key":"4788_CR37","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1007\/s10623-021-00846-y","volume":"89","author":"S Ball","year":"2021","unstructured":"Ball, S.: Some constructions of quantum MDS codes. Des. Codes Cryptogr. 89(5), 811\u2013821 (2021)","journal-title":"Des. Codes Cryptogr."},{"issue":"6","key":"4788_CR38","doi-asserted-by":"crossref","first-page":"3796","DOI":"10.1109\/TIT.2022.3150277","volume":"68","author":"S Ball","year":"2022","unstructured":"Ball, S., Vilar, R.: Determining when a truncated generalised Reed-Solomon code is Hermitian self-orthogonal. IEEE Trans. Inf. Theory 68(6), 3796\u20133805 (2022)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"1","key":"4788_CR39","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1007\/s11128-020-02949-2","volume":"20","author":"H Liu","year":"2021","unstructured":"Liu, H., Liu, X.: Constructions of quantum MDS codes. Quantum Inf. Process. 20(1), 14 (2021)","journal-title":"Quantum Inf. Process."},{"issue":"11","key":"4788_CR40","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1007\/s11128-022-03742-z","volume":"21","author":"R Jin","year":"2022","unstructured":"Jin, R., Luo, J., Fang, X., Qu, L.: New constructions of quantum MDS codes over finite fields. Quantum Inf. Process. 21(11), 395 (2022)","journal-title":"Quantum Inf. Process."},{"issue":"3","key":"4788_CR41","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/s11128-024-04319-8","volume":"23","author":"Yang Li","year":"2024","unstructured":"Li, Yang, Zhu, S., Zhang, Y.: Several families of MDS QECCs and MDS EAQECCs from Hermitian self-orthogonal GRS codes. Quantum Inf. Process. 23(3), 111 (2024)","journal-title":"Quantum Inf. Process."},{"key":"4788_CR42","doi-asserted-by":"publisher","DOI":"10.1007\/s12095-024-00752-9","author":"R Wan","year":"2024","unstructured":"Wan, R., Zheng, X., Zhu, S.: Construction of quantum MDS codes from Hermitian self-orthogonal generalized Reed-Solomon codes. Cryptogr. Commun. (2024). https:\/\/doi.org\/10.1007\/s12095-024-00752-9","journal-title":"Cryptogr. Commun."},{"issue":"1","key":"4788_CR43","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1007\/s10623-017-0330-z","volume":"86","author":"K Guenda","year":"2018","unstructured":"Guenda, K., Jitman, S., Gulliver, T.A.: Constructions of good entanglement-assisted quantum error correcting codes. Des. Codes Cryptogr. 86(1), 121\u2013136 (2018)","journal-title":"Des. Codes Cryptogr."},{"issue":"8","key":"4788_CR44","doi-asserted-by":"crossref","first-page":"2665","DOI":"10.1007\/s10623-023-01232-6","volume":"91","author":"H Chen","year":"2023","unstructured":"Chen, H.: New MDS entanglement-assisted quantum codes from MDS Hermitian self-orthogonal codes. Des. Codes Cryptogr. 91(8), 2665\u20132676 (2023)","journal-title":"Des. Codes Cryptogr."},{"issue":"7","key":"4788_CR45","doi-asserted-by":"crossref","first-page":"1845","DOI":"10.1007\/s10623-024-01369-y","volume":"92","author":"H Chen","year":"2024","unstructured":"Chen, H.: Large Hermitian hull GRS codes of any given length. Des. Codes Cryptogr. 92(7), 1845\u20131853 (2024)","journal-title":"Des. Codes Cryptogr."},{"key":"4788_CR46","unstructured":"Luo, G., Ezerman, M.F., Ling, S.: Entanglement-assisted and subsystem quantum codes: new propagation rules and constructions. arXiv:2206.09782 (2022)"},{"key":"4788_CR47","doi-asserted-by":"crossref","first-page":"102267","DOI":"10.1016\/j.ffa.2023.102267","volume":"91","author":"Y Li","year":"2023","unstructured":"Li, Y., Su, Y., Zhu, S., Li, S., Shi, M.: Several classes of Galois self-orthogonal MDS codes and related applications. Finite Field Appl. 91, 102267 (2023)","journal-title":"Finite Field Appl."},{"key":"4788_CR48","doi-asserted-by":"crossref","unstructured":"Wan, R., Zhu, S.: Three classes of propagation rules for generalized Reed\u2013Solomon codes and their applications to EAQECCs. arXiv:2401.08195v2 (2024)","DOI":"10.1016\/j.disc.2025.114405"}],"container-title":["Quantum Information Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-025-04788-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11128-025-04788-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-025-04788-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,5]],"date-time":"2025-07-05T04:02:33Z","timestamp":1751688153000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11128-025-04788-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,6]]},"references-count":48,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["4788"],"URL":"https:\/\/doi.org\/10.1007\/s11128-025-04788-5","relation":{},"ISSN":["1573-1332"],"issn-type":[{"type":"electronic","value":"1573-1332"}],"subject":[],"published":{"date-parts":[[2025,6,6]]},"assertion":[{"value":"30 August 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 May 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 June 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that there is no possible conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"165"}}