{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T19:14:58Z","timestamp":1757618098733,"version":"3.44.0"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2025,6,2]],"date-time":"2025-06-02T00:00:00Z","timestamp":1748822400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,6,2]],"date-time":"2025-06-02T00:00:00Z","timestamp":1748822400000},"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":["12441103"],"award-info":[{"award-number":["12441103"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Innovation Program for Quantum Science and Technology","award":["2021ZD0302902"],"award-info":[{"award-number":["2021ZD0302902"]}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["NP2024129"],"award-info":[{"award-number":["NP2024129"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Cryptogr. Commun."],"published-print":{"date-parts":[[2025,7]]},"DOI":"10.1007\/s12095-025-00811-9","type":"journal-article","created":{"date-parts":[[2025,6,1]],"date-time":"2025-06-01T23:11:31Z","timestamp":1748819491000},"page":"1149-1172","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Generator polynomials of cyclic expurgated or extended Goppa codes with odd characteristic"],"prefix":"10.1007","volume":"17","author":[{"given":"Xue","family":"Jia","sequence":"first","affiliation":[]},{"given":"Qin","family":"Yue","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,6,2]]},"reference":[{"key":"811_CR1","doi-asserted-by":"crossref","unstructured":"Berger, T.: Cyclic alternant codes induced by an automorphism of a GRS code. In: Finite Field Theory, Applications and Algorithm (Contemporary Mathematics), Mullin, R., and Mullen, G., Eds. Providence, RI, USA: American Mathematical Society. 225, 143-154 (1999)","DOI":"10.1090\/conm\/225\/03216"},{"issue":"4","key":"811_CR2","doi-asserted-by":"publisher","first-page":"1264","DOI":"10.1109\/18.761281","volume":"45","author":"T Berger","year":"1999","unstructured":"Berger, T.: New classes of cyclic extended Goppa codes. IEEE Trans. Inf. Theory. 45(4), 1264\u20131266 (1999)","journal-title":"IEEE Trans. Inf. Theory."},{"issue":"7","key":"811_CR3","doi-asserted-by":"publisher","first-page":"2628","DOI":"10.1109\/18.887871","volume":"46","author":"T Berger","year":"2000","unstructured":"Berger, T.: Goppa and related codes invariant under a prescribed permutation. IEEE Trans. Inf. Theory. 46(7), 2628\u20132633 (2000)","journal-title":"IEEE Trans. Inf. Theory."},{"issue":"3","key":"811_CR4","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1006\/ffta.2000.0277","volume":"6","author":"T Berger","year":"2000","unstructured":"Berger, T.: On the cyclicity of Goppa codes, parity-check subcodes of Goppa codes, and extended Goppa codes. Finite Fields Appl. 6(3), 255\u2013281 (2000)","journal-title":"Finite Fields Appl."},{"key":"811_CR5","unstructured":"Berger, T.: Quasi-cyclic Goppa codes. In: Proceedings of IEEE International Symposium on Information Theory. (2000)"},{"key":"811_CR6","doi-asserted-by":"publisher","first-page":"817","DOI":"10.1109\/TIT.1973.1055098","volume":"19","author":"E Berlekamp","year":"1973","unstructured":"Berlekamp, E., Moreno, O.: Extended double-error-correcting binary Goppa codes are cyclic. IEEE Trans. Inform. Theory. 19, 817\u2013818 (1973)","journal-title":"IEEE Trans. Inform. Theory."},{"key":"811_CR7","doi-asserted-by":"crossref","unstructured":"Cintas-Canto, A., Kermani, M., Azarderakhsh, R.: Reliable architectures for finite field multipliers using cyclic codes on FPGA utilized in classic and post-quantum cryptography. IEEE Trans. Very Large Scale Integr. Syst. 31(1), 157-161 (2023)","DOI":"10.1109\/TVLSI.2022.3224357"},{"issue":"8","key":"811_CR8","doi-asserted-by":"publisher","first-page":"5145","DOI":"10.1109\/TIT.2022.3166847","volume":"68","author":"B Chen","year":"2022","unstructured":"Chen, B., Zhang, G.: Enumeration of Extended Irreducible Binary Goppa Codes. IEEE Trans. Inf. Theory. 68(8), 5145\u20135153 (2022)","journal-title":"IEEE Trans. Inf. Theory."},{"issue":"2","key":"811_CR9","doi-asserted-by":"publisher","first-page":"995","DOI":"10.1109\/TIT.2022.3208438","volume":"69","author":"B Chen","year":"2023","unstructured":"Chen, B., Zhang, G.: A Tight Upper Bound on the Number of Non-Zero Weights of a Cyclic Code. IEEE Trans. Inf. Theory. 69(2), 995\u20131004 (2023)","journal-title":"IEEE Trans. Inf. Theory."},{"issue":"12","key":"811_CR10","doi-asserted-by":"publisher","first-page":"7691","DOI":"10.1109\/TIT.2023.3317290","volume":"69","author":"B Chen","year":"2023","unstructured":"Chen, B., Zhang, G.: The number of extended irreducible binary Goppa codes. IEEE Trans. Inf. Theory. 69(12), 7691\u20137711 (2023)","journal-title":"IEEE Trans. Inf. Theory."},{"issue":"13","key":"811_CR11","doi-asserted-by":"publisher","first-page":"1478","DOI":"10.1049\/cmu2.12636","volume":"17","author":"Y Ding","year":"2023","unstructured":"Ding, Y., Huang, Z., Zhou, J.: Joint blind reconstruction of cyclic codes and self-synchronous scramblers. IET Commun. 17(13), 1478\u20131491 (2023)","journal-title":"IET Commun."},{"key":"811_CR12","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1016\/0012-365X(85)90035-4","volume":"56","author":"G Feng","year":"1985","unstructured":"Feng, G., Tzeng, K.: Further results about the cyclicity of generalized Goppa codes. Discr. Math. 56, 281\u2013290 (1985)","journal-title":"Discr. Math."},{"issue":"3","key":"811_CR13","first-page":"207","volume":"6","author":"V Goppa","year":"1970","unstructured":"Goppa, V.: A new class of linear error correcting codes. Probl. Peredachi Inf. 6(3), 207\u2013212 (1970)","journal-title":"Probl. Peredachi Inf."},{"issue":"1","key":"811_CR14","doi-asserted-by":"publisher","first-page":"230","DOI":"10.1109\/TIT.2021.3116659","volume":"68","author":"D Huang","year":"2022","unstructured":"Huang, D., Yue, Q.: Extended irreducible binary sextic Goppa codes. IEEE Trans. Inf. Theory. 68(1), 230\u2013237 (2022)","journal-title":"IEEE Trans. Inf. Theory."},{"key":"811_CR15","doi-asserted-by":"crossref","unstructured":"Huang, D., Yue, Q.: Upper bound on the number of inequivalent extended binary irreducible Goppa codes. Discr. Math. 347, 113703 (2024)","DOI":"10.1016\/j.disc.2023.113703"},{"key":"811_CR16","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":"11","key":"811_CR17","doi-asserted-by":"publisher","first-page":"7711","DOI":"10.1109\/TIT.2024.3462712","volume":"70","author":"X Jia","year":"2024","unstructured":"Jia, X., Li, F., Sun, H., Yue, Q.: Generator polynomials of cyclic expurgated or extended Goppa codes. IEEE Trans. Inf. Theory. 70(11), 7711\u20137722 (2024)","journal-title":"IEEE Trans. Inf. Theory."},{"key":"811_CR18","volume-title":"Finite Fields","author":"R Lidl","year":"2008","unstructured":"Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press, Cambridge (2008)"},{"issue":"7","key":"811_CR19","doi-asserted-by":"publisher","first-page":"1629","DOI":"10.1007\/s10623-022-01062-y","volume":"90","author":"X Li","year":"2022","unstructured":"Li, X., Yue, Q.: Non-binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes. Des. Codes Cryptogr. 90(7), 1629\u20131647 (2022)","journal-title":"Des. Codes Cryptogr."},{"issue":"10","key":"811_CR20","doi-asserted-by":"publisher","first-page":"6472","DOI":"10.1109\/TIT.2022.3178110","volume":"68","author":"X Li","year":"2022","unstructured":"Li, X., Yue, Q.: Construction of expurgated and extended Goppa codes with dihedral automorphism groups. IEEE Trans. Inf. Theory. 68(10), 6472\u20136480 (2022)","journal-title":"IEEE Trans. Inf. Theory."},{"key":"811_CR21","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)"},{"key":"811_CR22","unstructured":"Richard, E.: Groups and Group Actions. Hilary and Trinity Terms, (2014)"},{"key":"811_CR23","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511808968","volume-title":"Introduction to Coding Theory","author":"R Roth","year":"2006","unstructured":"Roth, R.: Introduction to Coding Theory. Cambridge University Press, Cambridge (2006)"},{"key":"811_CR24","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1016\/0097-3165(89)90045-9","volume":"51","author":"H Stichtenoth","year":"1989","unstructured":"Stichtenoth, H.: Which extended Goppa codes are cyclic? J. Comb. Theory. 51, 205\u2013220 (1989)","journal-title":"J. Comb. Theory."},{"key":"811_CR25","doi-asserted-by":"crossref","unstructured":"Thiong-Ly, J.: Automorphisms of two families of extended non binary cyclic Goppa codes. In: Lecture Notes in Computer Science. 229. New York\/Berlin: Springer-Verlag, 112-121 (1985)","DOI":"10.1007\/3-540-16767-6_55"},{"key":"811_CR26","doi-asserted-by":"publisher","first-page":"246","DOI":"10.1109\/TIT.1979.1056016","volume":"25","author":"K Tzeng","year":"1979","unstructured":"Tzeng, K., Yu, C.: Characterization theorems for extending Goppa codes to cyclic codes. IEEE Trans. Inform. Theory. 25, 246\u2013250 (1979)","journal-title":"IEEE Trans. Inform. Theory."},{"key":"811_CR27","doi-asserted-by":"publisher","first-page":"712","DOI":"10.1109\/TIT.1975.1055449","volume":"21","author":"K Tzeng","year":"1975","unstructured":"Tzeng, K., Zimmermann, K.: On extending Goppa codes to cyclic codes. IEEE Trans. Inform. Theory. 21, 712\u2013716 (1975)","journal-title":"IEEE Trans. Inform. Theory."},{"key":"811_CR28","first-page":"171","volume":"18","author":"A Vishnevetskii","year":"1982","unstructured":"Vishnevetskii, A.: Cyclicity of extended Goppa codes. Probl. Pered. Inform. 18, 171\u2013175 (1982)","journal-title":"Probl. Pered. Inform."},{"key":"811_CR29","doi-asserted-by":"crossref","unstructured":"Wu, Y., Yue, Q., Fan, S.: Self-reciprocal and self-conjugate-reciprocal irreducible factors of $$x^n-\\lambda $$ and their applications. Finite Fields Appl. 63, 101648 (2020)","DOI":"10.1016\/j.ffa.2020.101648"}],"container-title":["Cryptography and Communications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-025-00811-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12095-025-00811-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-025-00811-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,6]],"date-time":"2025-09-06T17:20:39Z","timestamp":1757179239000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12095-025-00811-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,2]]},"references-count":29,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,7]]}},"alternative-id":["811"],"URL":"https:\/\/doi.org\/10.1007\/s12095-025-00811-9","relation":{},"ISSN":["1936-2447","1936-2455"],"issn-type":[{"type":"print","value":"1936-2447"},{"type":"electronic","value":"1936-2455"}],"subject":[],"published":{"date-parts":[[2025,6,2]]},"assertion":[{"value":"10 December 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 May 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 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 no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing Interests"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical Approval and Consent to Participate"}},{"value":"Not applicable.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for Publication"}}]}}