{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T21:48:41Z","timestamp":1768686521787,"version":"3.49.0"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2022,12,17]],"date-time":"2022-12-17T00:00:00Z","timestamp":1671235200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2022,12,17]],"date-time":"2022-12-17T00:00:00Z","timestamp":1671235200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Cryptogr. Commun."],"published-print":{"date-parts":[[2023,5]]},"DOI":"10.1007\/s12095-022-00621-3","type":"journal-article","created":{"date-parts":[[2022,12,17]],"date-time":"2022-12-17T11:03:37Z","timestamp":1671275017000},"page":"565-587","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["On MDS codes with galois hulls of arbitrary dimensions"],"prefix":"10.1007","volume":"15","author":[{"given":"Yang","family":"Li","sequence":"first","affiliation":[]},{"given":"Shixin","family":"Zhu","sequence":"additional","affiliation":[]},{"given":"Ping","family":"Li","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,12,17]]},"reference":[{"key":"621_CR1","doi-asserted-by":"crossref","unstructured":"Assmus, Jr, E.F., Key, J.D.: Affine and projective planes. Discrete Math. 83(2-3), 161\u2013187 (1990)","DOI":"10.1016\/0012-365X(90)90003-Z"},{"issue":"5798","key":"621_CR2","doi-asserted-by":"publisher","first-page":"436","DOI":"10.1126\/science.1131563","volume":"314","author":"T Brun","year":"2006","unstructured":"Brun, T., Devetak, I., Hsieh, M.H.: Correcting quantum errors with entanglement. Science 314(5798), 436\u2013439 (2006)","journal-title":"Science"},{"issue":"2","key":"621_CR3","doi-asserted-by":"publisher","first-page":"1098","DOI":"10.1103\/PhysRevA.54.1098","volume":"54","author":"AR Calderbank","year":"1996","unstructured":"Calderbank, A.R., Shor, P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54(2), 1098 (1996)","journal-title":"Phys. Rev. A"},{"key":"621_CR4","doi-asserted-by":"crossref","unstructured":"Carlet, C., Guilley, S.: Complementary dual codes for counter-measures to side-channel attacks. In: Coding theory and applications, pp. 97\u2013105. Springer, Cham (2015)","DOI":"10.1007\/978-3-319-17296-5_9"},{"issue":"12","key":"621_CR5","doi-asserted-by":"publisher","first-page":"7964","DOI":"10.1109\/TIT.2021.3117562","volume":"67","author":"M Cao","year":"2021","unstructured":"Cao, M.: MDS codes with Galois hulls of arbitrary dimensions and the related entanglement-assisted quantum error correction. IEEE Trans. Inf. Theory 67(12), 7964\u20137984 (2021)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"621_CR6","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1007\/s11128-018-2044-1","volume":"17","author":"X Chen","year":"2018","unstructured":"Chen, X., Zhu, S., Kai, X.: Entanglement-assisted quantum MDS codes constructed from constacyclic codes. Quantum Inf Process. 17, 273 (2018)","journal-title":"Quantum Inf Process."},{"issue":"1","key":"621_CR7","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1007\/s12095-021-00525-8","volume":"14","author":"X Fang","year":"2022","unstructured":"Fang, X., Jin, R., Luo, J., Ma, W.: New Galois hulls of GRS codes and application to EAQECCs. Cryptogr. Commun. 14(1), 145\u2013159 (2022)","journal-title":"Cryptogr. Commun."},{"issue":"3","key":"621_CR8","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1007\/s10623-016-0282-8","volume":"84","author":"Y Fan","year":"2017","unstructured":"Fan, Y., Zhang, L.: Galois self-dual constacyclic codes. Des. Codes Cryptogr. 84(3), 473\u2013492 (2017)","journal-title":"Des. Codes Cryptogr."},{"key":"621_CR9","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1016\/j.ffa.2018.06.003","volume":"53","author":"W Fang","year":"2018","unstructured":"Fang, W., Fu, F.W.: Two new classes of quantum MDS codes. Finite Fields Appl. 53, 85\u201398 (2018)","journal-title":"Finite Fields Appl."},{"issue":"6","key":"621_CR10","doi-asserted-by":"publisher","first-page":"3527","DOI":"10.1109\/TIT.2019.2950245","volume":"66","author":"W Fang","year":"2019","unstructured":"Fang, W., Fu, F.W., Li, L., Zhu, S.: Euclidean and hermitian hulls of MDS codes and their applications to EAQECCs. IEEE Trans. Inf. Theory 66 (6), 3527\u20133537 (2019)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"85","key":"621_CR11","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1090\/S0025-5718-1964-0159774-6","volume":"18","author":"DB Gillies","year":"1964","unstructured":"Gillies, D.B.: Three new Mersenne primes and a statistical theory. Math. Comput. 18(85), 93\u201397 (1964)","journal-title":"Math. Comput."},{"issue":"4","key":"621_CR12","doi-asserted-by":"publisher","first-page":"1062","DOI":"10.1109\/LCOMM.2020.3044893","volume":"25","author":"G Guo","year":"2020","unstructured":"Guo, G., Li, R.: Hermitian self-dual GRS and extended GRS codes. IEEE Commun. Lett. 25(4), 1062\u20131065 (2020)","journal-title":"IEEE Commun. Lett."},{"key":"621_CR13","doi-asserted-by":"publisher","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":"1","key":"621_CR14","doi-asserted-by":"publisher","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":"3","key":"621_CR15","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10773-022-04972-8","volume":"61","author":"N Gao","year":"2022","unstructured":"Gao, N., Li, J., Huang, S.: Hermitian hulls of constacyclic codes and their applications to quantum codes. Int. J. Theor. Phys. 61(3), 1\u201314 (2022)","journal-title":"Int. J. Theor. Phys."},{"key":"621_CR16","volume-title":"Unsolved Problems in Number Theory","author":"RK Guy","year":"2007","unstructured":"Guy, R.K.: Unsolved Problems in Number Theory. Science Press, Beijing (2007)"},{"issue":"1-2","key":"621_CR17","first-page":"61","volume":"40","author":"GH Jo","year":"2022","unstructured":"Jo, G.H., Kim, D.: Mersenne prime factor and sum of binomial coefficients. J. Appl. Math. Comput. 40(1-2), 61\u201368 (2022)","journal-title":"J. Appl. Math. Comput."},{"issue":"5","key":"621_CR18","doi-asserted-by":"publisher","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":"9","key":"621_CR19","doi-asserted-by":"publisher","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":"7","key":"621_CR20","doi-asserted-by":"publisher","first-page":"4344","DOI":"10.1109\/TIT.2017.2672961","volume":"63","author":"C Li","year":"2017","unstructured":"Li, C., Ding, C., Li, S.: LCD cyclic codes over finite fields. IEEE Trans. Inf. Theory 63(7), 4344\u20134356 (2017)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"621_CR21","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1007\/s11128-019-2207-8","volume":"18","author":"G Luo","year":"2019","unstructured":"Luo, G., Cao, X.: Two new families of entanglement-assisted quantum MDS codes from generalized Reed-Solomon codes. Quantum Inf Process. 18, 89 (2019)","journal-title":"Quantum Inf Process."},{"issue":"5","key":"621_CR22","doi-asserted-by":"publisher","first-page":"2944","DOI":"10.1109\/TIT.2018.2874953","volume":"65","author":"G Luo","year":"2018","unstructured":"Luo, G., Cao, X., Chen, X.: MDS codes with hulls of arbitrary dimensions and their quantum error correction. IEEE Trans. Inf. Theory 65(5), 2944\u20132952 (2018)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"2","key":"621_CR23","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1007\/s10623-019-00681-2","volume":"88","author":"H Liu","year":"2020","unstructured":"Liu, H., Pan, X.: Galois hulls of linear codes over finite fields. Des. Codes Cryptogr. 88(2), 241\u2013255 (2020)","journal-title":"Des. Codes Cryptogr."},{"key":"621_CR24","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1007\/s11128-019-2269-7","volume":"18","author":"L Li","year":"2019","unstructured":"Li, L., Zhu, S., Liu, L., Kai, X.: Entanglement-assisted quantum MDS codes from generalized Reed-Solomon codes. Quantum Inf Process. 18, 153 (2019)","journal-title":"Quantum Inf Process."},{"issue":"3","key":"621_CR25","doi-asserted-by":"publisher","first-page":"496","DOI":"10.1109\/TIT.1982.1056498","volume":"28","author":"J Leon","year":"1982","unstructured":"Leon, J.: Computing automorphism groups of error-correcting codes. IEEE Trans. Inf. Theory 28(3), 496\u2013511 (1982)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"4-5","key":"621_CR26","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1016\/S0747-7171(08)80103-4","volume":"12","author":"JS Leon","year":"1991","unstructured":"Leon, J.S.: Permutation group algorithms based on partitions, I: theory and algorithms. J. Symb. Comput. 12(4-5), 533\u2013583 (1991)","journal-title":"J. Symb. Comput."},{"issue":"1","key":"621_CR27","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11128-019-2515-z","volume":"19","author":"X Liu","year":"2020","unstructured":"Liu, X., Liu, H., Yu, L.: New EAQEC codes constructed from Galois LCD codes. Quantum Inf Process. 19(1), 1\u201315 (2020)","journal-title":"Quantum Inf Process."},{"issue":"8","key":"621_CR28","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11128-018-1978-7","volume":"17","author":"Y Liu","year":"2018","unstructured":"Liu, Y., Li, R., Lv, L., Ma, Y.: Application of constacyclic codes to entanglement-assisted quantum maximum distance separable codes. Quantum Inf Process. 17(8), 1\u201319 (2018)","journal-title":"Quantum Inf Process."},{"key":"621_CR29","first-page":"209","volume":"36","author":"L Murata","year":"2004","unstructured":"Murata, L., Pomerance, C.: On the largest prime factor of a Mersenne number. Number Theory 36, 209\u2013218 (2004)","journal-title":"Number Theory"},{"key":"621_CR30","unstructured":"Qian, L., Cao, X., Wu, X., Lu, W.: Entanglement-assisted quantum codes from l-Galois hulls MDS codes of arbitrary dimensions. preprint (2019)"},{"issue":"5","key":"621_CR31","doi-asserted-by":"publisher","first-page":"793","DOI":"10.1103\/PhysRevLett.77.793","volume":"77","author":"AM Steane","year":"1996","unstructured":"Steane, A. M.: Error correcting codes in quantum theory. Phys. Rev. Lett. 77(5), 793\u2013797 (1996)","journal-title":"Phys. Rev. Lett."},{"key":"621_CR32","doi-asserted-by":"crossref","unstructured":"Sendrier, N.: Finding the permutation between equivalent binary codes. In: Proceedings of IEEE international symposium on information theory, pp. 367 (1997)","DOI":"10.1109\/ISIT.1997.613303"},{"issue":"4","key":"621_CR33","doi-asserted-by":"publisher","first-page":"1193","DOI":"10.1109\/18.850662","volume":"46","author":"N Sendrier","year":"2000","unstructured":"Sendrier, N.: Finding the permutation between equivalent linear codes: the support splitting algorithm. IEEE Trans. Inf. Theory 46(4), 1193\u20131203 (2000)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"8","key":"621_CR34","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11128-022-03622-6","volume":"21","author":"G Wang","year":"2022","unstructured":"Wang, G., Tang, C.: Some constructions of optimal subsystem codes derived from GRS codes. Quantum Inf Process. 21(8), 1\u201316 (2022)","journal-title":"Quantum Inf Process."},{"issue":"8","key":"621_CR35","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11128-022-03635-1","volume":"21","author":"G Wang","year":"2022","unstructured":"Wang, G., Tang, C.: Some entanglement-assisted quantum MDS codes with large minimum distance. Quantum Inf Process. 21(8), 1\u201320 (2022)","journal-title":"Quantum Inf Process."},{"issue":"3","key":"621_CR36","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."},{"key":"621_CR37","doi-asserted-by":"publisher","first-page":"102084","DOI":"10.1016\/j.ffa.2022.102084","volume":"83","author":"Y Wu","year":"2022","unstructured":"Wu, Y., Li, C., Yang, S.: New galois hulls of generalized Reed-Solomon codes. Finite Fields Appl. 83, 102084 (2022)","journal-title":"Finite Fields Appl."}],"container-title":["Cryptography and Communications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-022-00621-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12095-022-00621-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-022-00621-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,10]],"date-time":"2024-10-10T16:38:25Z","timestamp":1728578305000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12095-022-00621-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,17]]},"references-count":37,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,5]]}},"alternative-id":["621"],"URL":"https:\/\/doi.org\/10.1007\/s12095-022-00621-3","relation":{},"ISSN":["1936-2447","1936-2455"],"issn-type":[{"value":"1936-2447","type":"print"},{"value":"1936-2455","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,12,17]]},"assertion":[{"value":"8 September 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 December 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 December 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"They have no known competing financial interests or personal relationships that might influence the work reported herein. All authors gave their informed consent.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Ethics approval and consent to participate"}},{"value":"All authors have seen and approved the final version of the submitted manuscript. They guarantee that this article is the authors\u2019 original work and has not been previously published or considered for publication elsewhere.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Consent for Publication"}},{"value":"The authors have declared that no completing interests exist.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Competing interests"}}]}}