{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T16:09:12Z","timestamp":1774022952376,"version":"3.50.1"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2024,2,27]],"date-time":"2024-02-27T00:00:00Z","timestamp":1708992000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,2,27]],"date-time":"2024-02-27T00:00:00Z","timestamp":1708992000000},"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 Science Foundation of China","doi-asserted-by":"crossref","award":["12071321"],"award-info":[{"award-number":["12071321"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Cryptogr. Commun."],"published-print":{"date-parts":[[2024,7]]},"DOI":"10.1007\/s12095-024-00704-3","type":"journal-article","created":{"date-parts":[[2024,2,27]],"date-time":"2024-02-27T08:03:05Z","timestamp":1709020985000},"page":"857-878","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["The [1,\u00a00]-twisted generalized Reed-Solomon code"],"prefix":"10.1007","volume":"16","author":[{"given":"Canze","family":"Zhu","sequence":"first","affiliation":[]},{"given":"Qunying","family":"Liao","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,2,27]]},"reference":[{"key":"704_CR1","unstructured":"Baicheva, T., Bouyukliev, I., Dodunekov, S., Willems, W.: On the $$[10,5,6]_9$$ Reed-Solomon and Glynn codes. In: Mathematica Balkanica, New Series, pp. 67\u201378 (2004)"},{"key":"704_CR2","doi-asserted-by":"crossref","unstructured":"Beelen, P., Puchinger, S., Nielsen, J.: Twisted Reed-Solomon codes. IEEE ISIT, 336-340 (2017)","DOI":"10.1109\/ISIT.2017.8006545"},{"issue":"5","key":"704_CR3","doi-asserted-by":"publisher","first-page":"3047","DOI":"10.1109\/TIT.2022.3146254","volume":"68","author":"P Beelen","year":"2022","unstructured":"Beelen, P., Puchinger, S., Nielsen, J.: Twisted Reed-Solomon codes. IEEE Trans. Inf. Theory 68(5), 3047\u20133061 (2022)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"704_CR4","doi-asserted-by":"crossref","unstructured":"Beelen, P., Bossert, M., Puchinger, S., Rosenkilde, J.: Structural properties of twisted Reed-Solomon codes with applications to code-based cryptography. IEEE ISIT, 946-950 (2018)","DOI":"10.1109\/ISIT.2018.8437923"},{"issue":"8","key":"704_CR5","doi-asserted-by":"publisher","first-page":"5776","DOI":"10.1109\/TIT.2017.2748955","volume":"64","author":"B Chen","year":"2018","unstructured":"Chen, B., Liu, H.: New constructions of MDS codes with complementary duals. IEEE Trans. Inf. Theory 64(8), 5776\u20135782 (2018)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"11","key":"704_CR6","doi-asserted-by":"publisher","first-page":"2605","DOI":"10.1007\/s10623-018-0463-8","volume":"86","author":"C Carlet","year":"2018","unstructured":"Carlet, C., Mesnager, S., Tang, C., Qi, Y.: Euclidean and Hermitian LCD MDS codes. Des. Codes Cryptogr. 86(11), 2605\u20132618 (2018)","journal-title":"Des. Codes Cryptogr."},{"issue":"1\u20132","key":"704_CR7","doi-asserted-by":"publisher","first-page":"30","DOI":"10.1007\/BF01222850","volume":"54","author":"S Doudunekov","year":"1995","unstructured":"Doudunekov, S., Landjev, I.: On near MDS codes. J. Geom. 54(1\u20132), 30\u201343 (1995)","journal-title":"J. Geom."},{"issue":"1","key":"704_CR8","doi-asserted-by":"publisher","first-page":"130","DOI":"10.1109\/TIT.2020.3020986","volume":"67","author":"X Fang","year":"2020","unstructured":"Fang, X., Liu, M., Luo, J.: New MDS Euclidean self-orthogonal codes. IEEE Trans. Inf. Theory 67(1), 130\u2013137 (2020)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"6","key":"704_CR9","doi-asserted-by":"publisher","first-page":"1127","DOI":"10.1007\/s10623-020-00734-x","volume":"88","author":"X Fang","year":"2020","unstructured":"Fang, X., Lebed, K., Liu, H., Luo, J.: New self-dual MDS codes over finite fields of odd characteristic. Des. Codes Cryptogr. 88(6), 1127\u20131138 (2020)","journal-title":"Des. Codes Cryptogr."},{"issue":"2","key":"704_CR10","doi-asserted-by":"publisher","first-page":"383","DOI":"10.1007\/s12095-022-00605-3","volume":"15","author":"G Guo","year":"2023","unstructured":"Guo, G., Li, R., Liu, Y., Song, H.: Duality of generalized twisted Reed-Solomon codes and Hermitian self-dual MDS or NMDS codes. Cryptogr. Commun. 15(2), 383\u2013395 (2023)","journal-title":"Cryptogr. Commun."},{"issue":"4","key":"704_CR11","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1016\/S1071-5797(02)90353-9","volume":"8","author":"S Georgiou","year":"2002","unstructured":"Georgiou, S., Koukouvinos, C.: self-dual MDS codes over large prime fields. Finite Fields Appl. 8(4), 455\u2013470 (2002)","journal-title":"Finite Fields Appl."},{"key":"704_CR12","doi-asserted-by":"crossref","unstructured":"Grass, M., Gulliver, T.: On self-duals MDS codes. IEEE. ISIT, 1954\u20131957 (2008)","DOI":"10.1109\/ISIT.2008.4595330"},{"issue":"1","key":"704_CR13","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1007\/s10623-011-9489-x","volume":"62","author":"K Guenda","year":"2012","unstructured":"Guenda, K.: New self-dual MDS codes over finite fields. Des. Codes Cryptogr. 62(1), 31\u201342 (2012)","journal-title":"Des. Codes Cryptogr."},{"issue":"9","key":"704_CR14","doi-asserted-by":"publisher","first-page":"4354","DOI":"10.1109\/TIT.2008.928297","volume":"54","author":"T Gulliver","year":"2008","unstructured":"Gulliver, T., Kim, J., Lee, Y.: New MDS or near- self-dual MDS codes. IEEE Trans. Inf. Theory 54(9), 4354\u20134360 (2008)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"704_CR15","first-page":"57","volume":"35","author":"M Harada","year":"2006","unstructured":"Harada, M., Kharaghani, H.: Orthogonal designs and self-dual MDS codes. Austral. J. Comb. 35, 57\u201367 (2006)","journal-title":"Austral. J. Comb."},{"key":"704_CR16","unstructured":"Huang, Z., Fang, W., Fu, W.: New constructions of MDS self-dual and self-orthogonal codes via GRS codes, (2021). https:\/\/arxiv.53yu.com\/abs\/2103.11665"},{"issue":"9","key":"704_CR17","doi-asserted-by":"publisher","first-page":"2195","DOI":"10.1007\/s10623-021-00910-7","volume":"89","author":"D Huang","year":"2021","unstructured":"Huang, D., Yue, Q., Niu, Y., Li, X.: MDS or NMDS self-dual codes from twisted generalized Reed-Solomon codes. Des. Codes Cryptogr. 89(9), 2195\u20132209 (2021)","journal-title":"Des. Codes Cryptogr."},{"key":"704_CR18","unstructured":"Huang, D., Yue, Q., Niu, Y.: MDS or NMDS LCD codes from twisted Reed-Solomon codes (2020). https:\/\/seta-2020.etu.ru\/assets\/files\/program\/papers\/paper-43.pdf"},{"key":"704_CR19","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511807077","volume-title":"Fundamentals of Error Correcting Codes","author":"W Huffman","year":"2003","unstructured":"Huffman, W., Pless, V.: Fundamentals of Error Correcting Codes. Cambridge University Press, Cambridge (2003)"},{"issue":"5","key":"704_CR20","first-page":"2843","volume":"63","author":"L Jin","year":"2017","unstructured":"Jin, L.: Construction of MDS codes with complementary duals. IEEE Trans. Inf. Theory 63(5), 2843\u20132847 (2017)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"3","key":"704_CR21","doi-asserted-by":"publisher","first-page":"1434","DOI":"10.1109\/TIT.2016.2645759","volume":"63","author":"L Jin","year":"2017","unstructured":"Jin, L., Xing, C.: New self-dual MDS codes from generalized Reed-Solomon codes. IEEE Trans. Inf. Theory 63(3), 1434\u20131438 (2017)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"1","key":"704_CR22","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1016\/j.jcta.2003.10.003","volume":"105","author":"J Kim","year":"2004","unstructured":"Kim, J., Lee, Y.: Hermitian self-dual MDS codes over large finite fields. J. Comb. Theory Ser. A 105(1), 79\u201395 (2004)","journal-title":"J. Comb. Theory Ser. A"},{"key":"704_CR23","unstructured":"Kim, J., Lee, Y.: Self-dual MDS codes. IEEE ISIT 526 (2004)"},{"issue":"7","key":"704_CR24","doi-asserted-by":"publisher","first-page":"1285","DOI":"10.1007\/s10623-020-00747-6","volume":"88","author":"J Lavauzelle","year":"2020","unstructured":"Lavauzelle, J., Renner, J.: Crypt analysis of a system based on twisted Reed-Solomon codes. Des. Codes Cryptogr. 88(7), 1285\u20131300 (2020)","journal-title":"Des. Codes Cryptogr."},{"key":"704_CR25","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1016\/j.ffa.2019.05.007","volume":"59","author":"K Lebed","year":"2019","unstructured":"Lebed, K., Liu, H., Luo, J.: Construction of self-dual MDS codes over finite fields. Finite Fields Appl. 59, 199\u2013207 (2019)","journal-title":"Finite Fields Appl."},{"issue":"4","key":"704_CR26","doi-asserted-by":"publisher","first-page":"911","DOI":"10.1016\/j.ffa.2008.05.003","volume":"14","author":"J Li","year":"2008","unstructured":"Li, J., Wan, D.: On the subset sum problem over finite fields. Finite Fields Appl. 14(4), 911\u2013929 (2008)","journal-title":"Finite Fields Appl."},{"key":"704_CR27","doi-asserted-by":"crossref","unstructured":"Liu, H., Liu, S.: Construction of MDS twisted Reed-Solomon codes and LCD MDS codes. Des. Codes Cryptogr. 1\u201315 (2021)","DOI":"10.1007\/s10623-021-00899-z"},{"key":"704_CR28","volume-title":"Finite Fields","author":"R Lidl","year":"1997","unstructured":"Lidl, R., Niederreiter, H., Cohn, F.M.: Finite Fields. Cambridge University Press, Cambridge (1997)"},{"key":"704_CR29","volume-title":"The Theory of Error-Corrcting Codes","author":"F MacWilliams","year":"1977","unstructured":"MacWilliams, F., Sloane, N.: The Theory of Error-Corrcting Codes. North Holland, Amsterdam (1977)"},{"issue":"3","key":"704_CR30","doi-asserted-by":"publisher","first-page":"475","DOI":"10.3934\/amc.2016019","volume":"10","author":"J Sheekey","year":"2016","unstructured":"Sheekey, J.: A new family of linear maximum rank distance codes. Adv. Math. Commun. 10(3), 475\u2013488 (2016)","journal-title":"Adv. Math. Commun."},{"key":"704_CR31","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1016\/j.ffa.2018.08.011","volume":"54","author":"M Shi","year":"2018","unstructured":"Shi, M., Sok, L., Sole, P.: Self-dual codes and orthogonal matrices over large finite fields. Finite Fields Appl. 54, 297\u2013314 (2018)","journal-title":"Finite Fields Appl."},{"issue":"11","key":"704_CR32","doi-asserted-by":"publisher","first-page":"2485","DOI":"10.1007\/s10623-017-0455-0","volume":"86","author":"M Shi","year":"2018","unstructured":"Shi, M., Qian, L., Sole, P.: On self-dual negacirculant codes of index two and four. Des. Codes Cryptogr. 86(11), 2485\u20132494 (2018)","journal-title":"Des. Codes Cryptogr."},{"key":"704_CR33","doi-asserted-by":"crossref","unstructured":"Shi X., Yue Q., Yang S., New LCD MDS codes constructed from generalized Reed-Solomon codes, J. Alg. Appl. 1950150 (2018)","DOI":"10.1142\/S0219498819501500"},{"key":"704_CR34","volume-title":"MDS codes","author":"D Simos","year":"2018","unstructured":"Simos, D., Varbanov, Z.: MDS codes. NMDS codes and their secret-sharing schems, Accessed April (2018)"},{"key":"704_CR35","doi-asserted-by":"publisher","first-page":"101627","DOI":"10.1016\/j.ffa.2019.101627","volume":"62","author":"R Wu","year":"2020","unstructured":"Wu, R., Shi, M.: A modified Gilbert-Varshamov bound for self-dual quasi-twisted codes of index four. Finite Fields Appl. 62, 101627 (2020)","journal-title":"Finite Fields Appl."},{"issue":"2","key":"704_CR36","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1007\/s12095-018-0288-3","volume":"11","author":"H Yan","year":"2019","unstructured":"Yan, H.: A note on the constructions of self-dual MDS codes. Cryptogr. Commun. 11(2), 259\u2013268 (2019)","journal-title":"Cryptogr. Commun."},{"issue":"6","key":"704_CR37","doi-asserted-by":"publisher","first-page":"3650","DOI":"10.1109\/TIT.2020.2963975","volume":"66","author":"A Zhang","year":"2020","unstructured":"Zhang, A., Feng, K.: A unified approach to construct self-dual MDS codes via Reed-Solomon codes. IEEE Trans. Inf. Theory 66(6), 3650\u20133656 (2020)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"5","key":"704_CR38","first-page":"2181","volume":"66","author":"A Thomas","year":"2018","unstructured":"Thomas, A., Rajan, B.: Binary Informed Source Codes and Index Codes Using Certain near MDS Codes. IEEE Trans. Comput. 66(5), 2181\u20132190 (2018)","journal-title":"IEEE Trans. Comput."},{"key":"704_CR39","unstructured":"Zhu, C., Liao, Q.: Self-dual twisted generalized Reed-Solomon codes (2021). https:\/\/arxiv.org\/abs\/2111.11901"},{"key":"704_CR40","unstructured":"Zhu, C., Liao, Q.: Self-orthogonal generalized twisted Reed-Solomon codes (2022). https:\/\/doi.org\/10.48550\/arXiv.2201.02758"},{"key":"704_CR41","doi-asserted-by":"publisher","unstructured":"Zhang, J., Zhou, Z., Tang ,C.: A class of twisted generalized Reed-Solomon codes (2022). https:\/\/doi.org\/10.48550\/arXiv.2202.09011","DOI":"10.48550\/arXiv.2202.09011"}],"container-title":["Cryptography and Communications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-024-00704-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12095-024-00704-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-024-00704-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,11,12]],"date-time":"2024-11-12T23:31:01Z","timestamp":1731454261000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12095-024-00704-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,27]]},"references-count":41,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2024,7]]}},"alternative-id":["704"],"URL":"https:\/\/doi.org\/10.1007\/s12095-024-00704-3","relation":{},"ISSN":["1936-2447","1936-2455"],"issn-type":[{"value":"1936-2447","type":"print"},{"value":"1936-2455","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,2,27]]},"assertion":[{"value":"30 August 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 February 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 February 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Not applicable.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent to participate"}},{"value":"Not applicable.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}},{"value":"The authors declare no competing interests.","order":5,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}},{"value":"The authors have no conflicts of interest to declare that are relevant to the content of this article.","order":6,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}