{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T04:08:06Z","timestamp":1769486886352,"version":"3.49.0"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,12,20]],"date-time":"2025-12-20T00:00:00Z","timestamp":1766188800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,12,20]],"date-time":"2025-12-20T00:00:00Z","timestamp":1766188800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"Talent Research Foundation of Anhui Agricultural University, China","award":["rc482402"],"award-info":[{"award-number":["rc482402"]}]},{"DOI":"10.13039\/501100003995","name":"Natural Science Foundation of Anhui Province","doi-asserted-by":"publisher","award":["2508085QA033"],"award-info":[{"award-number":["2508085QA033"]}],"id":[{"id":"10.13039\/501100003995","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62201009"],"award-info":[{"award-number":["62201009"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["U21A20428"],"award-info":[{"award-number":["U21A20428"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Des. Codes Cryptogr."],"published-print":{"date-parts":[[2026,1]]},"DOI":"10.1007\/s10623-025-01759-w","type":"journal-article","created":{"date-parts":[[2025,12,20]],"date-time":"2025-12-20T14:27:24Z","timestamp":1766240844000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Binary duadic codes and their related codes with a square-root-like lower bound"],"prefix":"10.1007","volume":"94","author":[{"given":"Tingting","family":"Wu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lanqiang","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiuyu","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shixin","family":"Zhu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,12,20]]},"reference":[{"key":"1759_CR1","first-page":"1269","volume":"2","author":"E Assmus","year":"1998","unstructured":"Assmus E., Key J.: Polynomial codes and finite geometries. Handbook Coding Theory 2, 1269\u20131343 (1998).","journal-title":"Handbook Coding Theory"},{"issue":"4","key":"1759_CR2","doi-asserted-by":"publisher","first-page":"357","DOI":"10.1109\/TIT.1964.1053699","volume":"10","author":"R Chien","year":"1964","unstructured":"Chien R.: Cyclic decoding procedure for the Bose-Chaudhuri-Hocquenghem codes. IEEE Trans. Inf. Theory 10(4), 357\u2013363 (1964).","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1759_CR3","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2023.102308","volume":"92","author":"T Chen","year":"2023","unstructured":"Chen T., Ding C., Li C., Sun Z.: Four infinite families of ternary cyclic codes with a square-root-like lower bound. Finite Fields Appl. 92, 102308 (2023).","journal-title":"Finite Fields Appl."},{"issue":"1","key":"1759_CR4","doi-asserted-by":"publisher","first-page":"149","DOI":"10.3233\/FI-1999-381212","volume":"38","author":"C Ding","year":"1999","unstructured":"Ding C., Lam K., Xing C.: Enumeration and construction of all duadic codes of length $$p^m$$. Fundam. Inform. 38(1), 149\u2013161 (1999).","journal-title":"Fundam. Inform."},{"issue":"2","key":"1759_CR5","doi-asserted-by":"publisher","first-page":"453","DOI":"10.1109\/18.748995","volume":"45","author":"C Ding","year":"1999","unstructured":"Ding C., Pless V.: Cyclotomy and duadic codes of prime lengths. IEEE Trans. Inf. Theory 45(2), 453\u2013466 (1999).","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"4","key":"1759_CR6","doi-asserted-by":"publisher","first-page":"2231","DOI":"10.1109\/TIT.2011.2176915","volume":"58","author":"C Ding","year":"2012","unstructured":"Ding C.: Cyclotomic constructions of cyclic codes with length being the product of two primes. IEEE Trans. Inf. Theory 58(4), 2231\u20132236 (2012).","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1759_CR7","doi-asserted-by":"publisher","first-page":"8","DOI":"10.1016\/j.ffa.2013.03.006","volume":"23","author":"C Ding","year":"2013","unstructured":"Ding C.: Cyclic codes from cyclotomic sequences of order four. Finite Fields Appl. 23, 8\u201334 (2013).","journal-title":"Finite Fields Appl."},{"issue":"4","key":"1759_CR8","doi-asserted-by":"publisher","first-page":"549","DOI":"10.1109\/TIT.1965.1053825","volume":"11","author":"G Forney","year":"1995","unstructured":"Forney G.: On decoding BCH codes. IEEE Trans. Inf. Theory 11(4), 549\u2013557 (1995).","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"1","key":"1759_CR9","doi-asserted-by":"publisher","first-page":"402","DOI":"10.1109\/TIT.2004.839522","volume":"51","author":"P Gaborit","year":"2005","unstructured":"Gaborit P., Nedeloaia C.S., Wassermann A.: On the weight enumerators of duadic and quadratic residue codes. IEEE Trans. Inf. Theory 51(1), 402\u2013407 (2005).","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1759_CR10","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511807077","volume-title":"Fundamentals Error-Correcting Codes","author":"W Huffman","year":"2003","unstructured":"Huffman W., Pless V.: Fundamentals Error-Correcting Codes. Cambridge University Press, Cambridge (2003)."},{"issue":"5","key":"1759_CR11","doi-asserted-by":"publisher","first-page":"709","DOI":"10.1109\/TIT.1984.1056944","volume":"30","author":"J Leon","year":"1984","unstructured":"Leon J., Masley J., Pless V.: Duadic codes. IEEE Trans. Inf. Theory 30(5), 709\u2013714 (1984).","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"5","key":"1759_CR12","doi-asserted-by":"publisher","first-page":"1354","DOI":"10.1109\/18.21270","volume":"34","author":"J Leon","year":"1988","unstructured":"Leon J.: A probabilistic algorithm for computing minimum weight of large error-correcting codes. IEEE Trans. Inf. Theory 34(5), 1354\u20131359 (1988).","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1759_CR13","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2024.114390","volume":"348","author":"L Li","year":"2025","unstructured":"Li L., Shi M., Tao S., Sun Z., Zhu S., Kim J., Sol\u00e9 P.: A generalization of the Tang-Ding binary cyclic codes. Discret. Math. 348, 114390 (2025).","journal-title":"Discret. Math."},{"key":"1759_CR14","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2023.102270","volume":"91","author":"H Liu","year":"2023","unstructured":"Liu H., Li C., Ding C.: Five infinite families of binary cyclic codes and their related codes with good parameters. Finite Field Appl. 91, 102270 (2023).","journal-title":"Finite Field Appl."},{"key":"1759_CR15","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10623-023-01285-7","volume":"92","author":"H Liu","year":"2024","unstructured":"Liu H., Li C., Qian H.: Parameters of several families of binary duadic codes and their related codes. Designs Codes Cryptogr. 92, 1\u201312 (2024).","journal-title":"Designs Codes Cryptogr."},{"key":"1759_CR16","volume-title":"The Theory of Error-Correcting Codes","author":"F Macwilliams","year":"1977","unstructured":"Macwilliams F., Sloane N.: The Theory of Error-Correcting Codes. Elsevier, North Holland (1977)."},{"key":"1759_CR17","unstructured":"Prange, E.: Some cyclic error-correcting codes with simple decoding algorithms Cambridge, MA, USA, Air Force Cambridge Research Center-TN-58-156 (1958)."},{"key":"1759_CR18","doi-asserted-by":"publisher","first-page":"6","DOI":"10.1016\/0097-3165(87)90056-2","volume":"44","author":"V Pless","year":"1987","unstructured":"Pless V., Masley J., Leon J.: On weights in duadic codes. J. Comb. Theory 44, 6\u201321 (1987).","journal-title":"J. Comb. Theory"},{"issue":"6","key":"1759_CR19","doi-asserted-by":"publisher","first-page":"3735","DOI":"10.1109\/TIT.2021.3069772","volume":"67","author":"M Shi","year":"2021","unstructured":"Shi M., \u00d6zbudak F., Sol\u00e9 P.: Geometric Approach to $$b$$-symbol Hamming weights of cyclic codes. IEEE Trans. Inf. Theory 67(6), 3735\u20133751 (2021).","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1759_CR20","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1007\/s10801-020-01006-6","volume":"55","author":"M Shi","year":"2022","unstructured":"Shi M., Helleseth T., Sol\u00e9 P.: Strongly regular graphs from reducible cyclic codes. J. Algebraic Comb. 55, 173\u2013184 (2022).","journal-title":"J. Algebraic Comb."},{"key":"1759_CR21","doi-asserted-by":"publisher","first-page":"795","DOI":"10.1007\/s11424-014-2188-8","volume":"27","author":"M Shi","year":"2014","unstructured":"Shi M., Wang Y.: Optimal binary codes from one-Lee weight codes and two-Lee weight projective codes over $${\\mathbb{Z} }_{4}^{*}$$. J. Syst. Sci. Complex. 27, 795\u2013810 (2014).","journal-title":"J. Syst. Sci. Complex."},{"key":"1759_CR22","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2023.102200","volume":"89","author":"Z Sun","year":"2023","unstructured":"Sun Z.: Several families of binary cyclic codes with good parameters. Finite Fields Appl. 89, 102200 (2023).","journal-title":"Finite Fields Appl."},{"issue":"4","key":"1759_CR23","doi-asserted-by":"publisher","first-page":"2411","DOI":"10.1109\/TIT.2023.3307732","volume":"70","author":"Z Sun","year":"2024","unstructured":"Sun Z., Li C., Ding C.: An infinite family of binary cyclic codes with best parameters. IEEE Trans. Inf. Theory 70(4), 2411\u20132418 (2024).","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"12","key":"1759_CR24","doi-asserted-by":"publisher","first-page":"7842","DOI":"10.1109\/TIT.2022.3193715","volume":"68","author":"C Tang","year":"2022","unstructured":"Tang C., Ding C.: Binary $$[n, \\frac{n+1}{2}]$$ cyclic codes with good minimum distances. IEEE Trans. Inf. Theory 68(12), 7842\u20137849 (2022).","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"1","key":"1759_CR25","doi-asserted-by":"publisher","first-page":"4","DOI":"10.1016\/j.ffa.2009.11.003","volume":"16","author":"H Tada","year":"2010","unstructured":"Tada H., Nishimura S., Hiramatsu T.: Cyclotomy and its application to duadic codes. Finite Fields Appl. 16(1), 4\u201313 (2010).","journal-title":"Finite Fields Appl."},{"issue":"6","key":"1759_CR26","doi-asserted-by":"publisher","first-page":"4110","DOI":"10.1109\/TIT.2023.3310500","volume":"70","author":"Y Wu","year":"2024","unstructured":"Wu Y., Sun Z., Ding C.: Another infinite family of binary cyclic codes with best parameters known. IEEE Trans. Inf. Theory 70(6), 4110\u20134116 (2024).","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"9","key":"1759_CR27","doi-asserted-by":"publisher","first-page":"6305","DOI":"10.1109\/TIT.2018.2829763","volume":"64","author":"M Xiong","year":"2018","unstructured":"Xiong M.: On cyclic codes of composite length and the minimum distance. IEEE Trans. Inf. Theory 64(9), 6305\u20136314 (2018).","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"8","key":"1759_CR28","doi-asserted-by":"publisher","first-page":"5097","DOI":"10.1109\/TIT.2021.3066008","volume":"67","author":"M Xiong","year":"2021","unstructured":"Xiong M., Zhang A.: On cyclic codes of composite length and the minimum distance II. IEEE Trans. Inf. Theory 67(8), 5097\u20135103 (2021).","journal-title":"IEEE Trans. Inf. Theory"}],"container-title":["Designs, Codes and Cryptography"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-025-01759-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10623-025-01759-w","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-025-01759-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,26]],"date-time":"2026-01-26T17:13:04Z","timestamp":1769447584000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10623-025-01759-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,20]]},"references-count":28,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,1]]}},"alternative-id":["1759"],"URL":"https:\/\/doi.org\/10.1007\/s10623-025-01759-w","relation":{},"ISSN":["0925-1022","1573-7586"],"issn-type":[{"value":"0925-1022","type":"print"},{"value":"1573-7586","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,20]]},"assertion":[{"value":"31 March 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 October 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 October 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 December 2025","order":4,"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 Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"12"}}