{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T17:23:24Z","timestamp":1783099404485,"version":"3.54.6"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"7","license":[{"start":{"date-parts":[[2023,4,8]],"date-time":"2023-04-08T00:00:00Z","timestamp":1680912000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,4,8]],"date-time":"2023-04-08T00:00:00Z","timestamp":1680912000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"the National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12071001"],"award-info":[{"award-number":["12071001"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"2021 University Graduate Research Project","award":["Y020410077"],"award-info":[{"award-number":["Y020410077"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Des. Codes Cryptogr."],"published-print":{"date-parts":[[2023,7]]},"DOI":"10.1007\/s10623-023-01210-y","type":"journal-article","created":{"date-parts":[[2023,4,8]],"date-time":"2023-04-08T04:03:15Z","timestamp":1680926595000},"page":"2563-2583","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["An improved method for constructing formally self-dual codes with small hulls"],"prefix":"10.1007","volume":"91","author":[{"given":"Shitao","family":"Li","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4990-6271","authenticated-orcid":false,"given":"Minjia","family":"Shi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Juan","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,4,8]]},"reference":[{"issue":"2\u20133","key":"1210_CR1","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1016\/0012-365X(90)90003-Z","volume":"83","author":"EF Assmus Jr","year":"1990","unstructured":"Assmus E.F. Jr., Key J.D.: Affine and projective planes. Discret. Math. 83(2\u20133), 161\u2013187 (1990).","journal-title":"Discret. Math."},{"issue":"1","key":"1210_CR2","doi-asserted-by":"publisher","first-page":"270","DOI":"10.1016\/j.disc.2018.09.034","volume":"342","author":"M Araya","year":"2019","unstructured":"Araya M., Harada M.: On the classification of linear complementary dual codes. Discret. Math. 342(1), 270\u2013278 (2019).","journal-title":"Discret. Math."},{"issue":"2","key":"1210_CR3","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1007\/s12095-019-00402-5","volume":"12","author":"M Araya","year":"2020","unstructured":"Araya M., Harada M.: On the minimum weights of binary linear complementary dual codes. Cryptogr. Commun. 12(2), 285\u2013300 (2020).","journal-title":"Cryptogr. Commun."},{"key":"1210_CR4","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2021.101925","volume":"76","author":"M Araya","year":"2021","unstructured":"Araya M., Harada M., Saito K.: On the minimum weights of binary LCD codes and ternary LCD codes. Finite Fields Appl. 76, 101925 (2021).","journal-title":"Finite Fields Appl."},{"issue":"2","key":"1210_CR5","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1006\/ffta.1998.0221","volume":"5","author":"M Bhargava","year":"1999","unstructured":"Bhargava M., Zieve M.E.: Factoring Dickson polynomials over finite fields. Finite Fields Appl. 5(2), 103\u2013111 (1999).","journal-title":"Finite Fields Appl."},{"key":"1210_CR6","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1006\/jsco.1996.0125","volume":"24","author":"W Bosma","year":"1997","unstructured":"Bosma W., Cannon J., Playoust C.: The Magma algebra system I: the user language. J. Symbol. Comput. 24, 235\u2013265 (1997).","journal-title":"J. Symbol. Comput."},{"issue":"11","key":"1210_CR7","doi-asserted-by":"publisher","first-page":"2445","DOI":"10.1007\/s10623-021-00929-w","volume":"89","author":"S Bouyuklieva","year":"2021","unstructured":"Bouyuklieva S.: Optimal binary LCD codes. Des. Codes Cryptogr. 89(11), 2445\u20132461 (2021).","journal-title":"Des. Codes Cryptogr."},{"issue":"1","key":"1210_CR8","doi-asserted-by":"publisher","first-page":"131","DOI":"10.3934\/amc.2016.10.131","volume":"10","author":"C Carlet","year":"2016","unstructured":"Carlet C., Guilley S.: Complementary dual codes for counter-measures to side-channel attacks. Adv. Math. Commun. 10(1), 131\u2013150 (2016).","journal-title":"Adv. Math. Commun."},{"issue":"3","key":"1210_CR9","doi-asserted-by":"publisher","first-page":"830","DOI":"10.1016\/j.disc.2017.12.004","volume":"341","author":"C Carlet","year":"2018","unstructured":"Carlet C., G\u00fcneri C., \u00d6zbudak F., Sol\u00e9 P.: A new concatenated type construction for LCD codes and isometry codes. Discret. Math. 341(3), 830\u2013835 (2018).","journal-title":"Discret. Math."},{"issue":"12","key":"1210_CR10","doi-asserted-by":"publisher","first-page":"3063","DOI":"10.1007\/s10623-019-00663-4","volume":"87","author":"C Carlet","year":"2019","unstructured":"Carlet C., Li C., Mesnager S.: Linear codes with small hulls in semi-primitive case. Des. Codes Cryptogr. 87(12), 3063\u20133075 (2019).","journal-title":"Des. Codes Cryptogr."},{"issue":"1","key":"1210_CR11","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1109\/TIT.2018.2829873","volume":"65","author":"C Carlet","year":"2019","unstructured":"Carlet C., Mesnager S., Tang C., Qi Y.: New characterization and parametrization of LCD codes. IEEE Trans. Inf. Theory 65(1), 39\u201349 (2019).","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"4","key":"1210_CR12","doi-asserted-by":"publisher","first-page":"3010","DOI":"10.1109\/TIT.2018.2789347","volume":"64","author":"C Carlet","year":"2018","unstructured":"Carlet C., Mesnager S., Tang C., Qi Y., Pellikaan R.: Linear codes over $$ {\\mathbb{F} }_q $$ are equivalent to LCD codes for $$ q > 3 $$. IEEE Trans. Inf. Theory 64(4), 3010\u20133017 (2018).","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"2\u20133","key":"1210_CR13","first-page":"116","volume":"4","author":"ST Dougherty","year":"2017","unstructured":"Dougherty S.T., Kim J., \u00d6zkaya B., Sok L., Sol\u00e9 P.: The combinatorics of LCD codes: linear programming bound and orthogonal matrices. Int. J. Inf. Coding Theory 4(2\u20133), 116\u2013128 (2017).","journal-title":"Int. J. Inf. Coding Theory"},{"key":"1210_CR14","doi-asserted-by":"publisher","first-page":"1237","DOI":"10.1142\/S0129054119500242","volume":"30","author":"Q Fu","year":"2019","unstructured":"Fu Q., Li R., Fu F., Rao Y.: On the construction of binary optimal LCD codes with short length. Int. J. Found. Comput. Sci. 30, 1237\u20131245 (2019).","journal-title":"Int. J. Found. Comput. Sci."},{"issue":"4","key":"1210_CR15","doi-asserted-by":"publisher","first-page":"719","DOI":"10.1007\/s12095-017-0258-1","volume":"10","author":"L Galvez","year":"2018","unstructured":"Galvez L., Kim J.L., Lee N., Roe Y.G., Won B.S.: Some bounds on binary LCD codes. Cryptogr. Commun. 10(4), 719\u2013728 (2018).","journal-title":"Cryptogr. Commun."},{"key":"1210_CR16","unstructured":"Grassl M.: Bounds on the minimum distance of linear codes and quantum codes. http:\/\/www.codetables.de. Accessed 4 (2021)."},{"key":"1210_CR17","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1016\/j.ffa.2016.07.005","volume":"42","author":"C G\u00fcneri","year":"2016","unstructured":"G\u00fcneri C., \u00d6zkaya B., Sol\u00e9 P.: Quasi-cyclic complementary dual codes. Finite Fields Appl. 42, 67\u201380 (2016).","journal-title":"Finite Fields Appl."},{"key":"1210_CR18","doi-asserted-by":"publisher","first-page":"1950053","DOI":"10.1142\/S0219749919500539","volume":"17","author":"M Harada","year":"2019","unstructured":"Harada M.: Some optimal entanglement-assisted quantum codes constructed from quaternary Hermitian linear complementary dual codes. Int. J. Quant. Inf. 17, 1950053 (2019).","journal-title":"Int. J. Quant. Inf."},{"issue":"4","key":"1210_CR19","doi-asserted-by":"publisher","first-page":"677","DOI":"10.1007\/s12095-018-0319-0","volume":"11","author":"M Harada","year":"2019","unstructured":"Harada M., Saito K.: Binary linear complementary dual codes. Cryptogr. Commun. 11(4), 677\u2013696 (2019).","journal-title":"Cryptogr. Commun."},{"issue":"1","key":"1210_CR20","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1007\/BF01388559","volume":"4","author":"GT Kennedy","year":"1994","unstructured":"Kennedy G.T., Pless V.: On designs and formally self-dual codes. Des. Codes Cryptogr. 4(1), 43\u201355 (1994).","journal-title":"Des. Codes Cryptogr."},{"issue":"1","key":"1210_CR21","doi-asserted-by":"publisher","first-page":"622","DOI":"10.1109\/TIT.2017.2711601","volume":"64","author":"C Lai","year":"2018","unstructured":"Lai C., Ashikhmin A.: Linear programming bounds for entanglement-assisted quantum error-correcting codes by split weight enumerators. IEEE Trans. Inf. Theory 64(1), 622\u2013639 (2018).","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"4\u20135","key":"1210_CR22","first-page":"533","volume":"12","author":"J Leon","year":"1982","unstructured":"Leon J.: Permutation group algorithms based on partition I: theory and algorithms. J. Symb. Comput. 12(4\u20135), 533\u2013583 (1982).","journal-title":"J. Symb. Comput."},{"issue":"3","key":"1210_CR23","doi-asserted-by":"publisher","first-page":"1668","DOI":"10.1109\/TIT.2018.2863693","volume":"65","author":"C Li","year":"2019","unstructured":"Li C., Zeng P.: Constructions of linear codes with one-dimensional hull. IEEE Trans. Inf. Theory 65(3), 1668\u20131676 (2019).","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1210_CR24","doi-asserted-by":"publisher","first-page":"104","DOI":"10.1016\/j.ffa.2019.05.005","volume":"59","author":"Z Liu","year":"2019","unstructured":"Liu Z., Wang J.: Further results on Euclidean and Hermitian linear complementary dual codes. Finite Fields Appl. 59, 104\u2013133 (2019).","journal-title":"Finite Fields Appl."},{"key":"1210_CR25","unstructured":"Lu L., Zhan X., Yang S., Cao H.: Optimal quaternary Hermitian LCD codes. https:\/\/arxiv.org\/pdf\/2010.10166.pdf."},{"key":"1210_CR26","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1016\/0012-365X(92)90563-U","volume":"106\u2013107","author":"J Massey","year":"1992","unstructured":"Massey J.: Linear codes with complementary duals. Discret. Math. 106\u2013107, 337\u2013342 (1992).","journal-title":"Discret. Math."},{"key":"1210_CR27","volume-title":"The Theory of Error Correcting Codes, Amsterdam","author":"FJ MacWilliams","year":"1977","unstructured":"MacWilliams F.J., Sloane N.J.A.: The Theory of Error Correcting Codes, Amsterdam. North-Holland, The Netherlands (1977)."},{"issue":"2","key":"1210_CR28","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1007\/s12095-020-00462-y","volume":"13","author":"L Qian","year":"2021","unstructured":"Qian L., Cao X., Mesnager S.: Linear codes with one-dimensional hull associated with Gaussian sums. Cryptogr. Commun. 13(2), 225\u2013243 (2021).","journal-title":"Cryptogr. Commun."},{"issue":"11","key":"1210_CR29","doi-asserted-by":"publisher","first-page":"2663","DOI":"10.1007\/s10623-021-00940-1","volume":"90","author":"L Qian","year":"2022","unstructured":"Qian L., Cao X., Lu W., Sol\u00e9 P.: A new method for constructing linear codes with small hulls. Des. Codes Cryptogr. 90(11), 2663\u20132682 (2022).","journal-title":"Des. Codes Cryptogr."},{"issue":"4","key":"1210_CR30","doi-asserted-by":"publisher","first-page":"717","DOI":"10.1007\/s12095-018-0322-5","volume":"11","author":"L Qian","year":"2019","unstructured":"Qian L., Shi M., Sol\u00e9 P.: On self-dual and LCD quasi-twisted codes of index two over a special chain ring. Cryptogr. Commun. 11(4), 717\u2013734 (2019).","journal-title":"Cryptogr. Commun."},{"issue":"1","key":"1210_CR31","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1016\/j.disc.2004.05.005","volume":"285","author":"N Sendrier","year":"2004","unstructured":"Sendrier N.: Linear codes with complementary duals meet the Gilbert-Varshamov bound. Discret. Math. 285(1), 345\u2013347 (2004).","journal-title":"Discret. Math."},{"issue":"4","key":"1210_CR32","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 codes: the support splitting algorithm. IEEE Trans. Inf. Theory 46(4), 1193\u20131203 (2000).","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1210_CR33","unstructured":"Sendrier N., Skersys G.: On the computation of the automorphism group of a linear code. In: Proc. IEEE Int. Symp. Inf. Theory, Washington, DC, p. 13 (2001)."},{"key":"1210_CR34","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1016\/j.ffa.2019.04.001","volume":"58","author":"M Shi","year":"2019","unstructured":"Shi M., Huang D., Sok L., Sol\u00e9 P.: Double circulant LCD codes over $${\\mathbb{Z} }_4$$. Finite Fields Appl. 58, 133\u2013144 (2019).","journal-title":"Finite Fields Appl."},{"issue":"1","key":"1210_CR35","doi-asserted-by":"publisher","first-page":"171","DOI":"10.3934\/amc.2019011","volume":"13","author":"M Shi","year":"2019","unstructured":"Shi M., Huang D., Sok L., Sol\u00e9 P.: Double circulant self-dual and LCD codes over Galois rings. Adv. Math. Commun. 13(1), 171\u2013183 (2019).","journal-title":"Adv. Math. Commun."},{"issue":"3","key":"1210_CR36","doi-asserted-by":"publisher","first-page":"627","DOI":"10.1007\/s12095-021-00545-4","volume":"14","author":"M Shi","year":"2022","unstructured":"Shi M., Li S., Kim J., Sol\u00e9 P.: LCD and ACD codes over a noncommutative non-unital ring with four elements. Cryptogr. Commun. 14(3), 627\u2013640 (2022).","journal-title":"Cryptogr. Commun."},{"key":"1210_CR37","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2021.101892","volume":"75","author":"M Shi","year":"2021","unstructured":"Shi M., \u00d6zbudak F., Xu L., Sol\u00e9 P.: LCD codes from tridiagonal Toeplitz matrices. Finite Fields Appl. 75, 101892 (2021).","journal-title":"Finite Fields Appl."},{"issue":"11","key":"1210_CR38","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., Sol\u00e9 P.: On self-dual negacirculant codes of index two and four. Des. Codes Cryptogr. 86(11), 2485\u20132494 (2018).","journal-title":"Des. Codes Cryptogr."},{"issue":"12","key":"1210_CR39","doi-asserted-by":"publisher","first-page":"2547","DOI":"10.1007\/s10623-020-00799-8","volume":"88","author":"M Shi","year":"2020","unstructured":"Shi M., Xu L., Sol\u00e9 P.: Construction of isodual codes from polycirculant matrices. Des. Codes Cryptogr. 88(12), 2547\u20132560 (2020).","journal-title":"Des. Codes Cryptogr."},{"key":"1210_CR40","unstructured":"Shi M., Xu L., Sol\u00e9 P.: On isodual double Toeplitz codes. https:\/\/arxiv.org\/abs\/2102.09233.pdf."},{"key":"1210_CR41","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1007\/s12095-019-00363-9","volume":"12","author":"M Shi","year":"2020","unstructured":"Shi M., Zhu H., Qian L., Sok L., Sol\u00e9 P.: On self-dual and LCD double circulant and double negacirculant codes over $${\\mathbb{F} }_q+u{\\mathbb{F} }_q$$. Cryptogr. Commun. 12, 53\u201370 (2020).","journal-title":"Cryptogr. Commun."},{"issue":"1","key":"1210_CR42","doi-asserted-by":"publisher","first-page":"138","DOI":"10.1016\/j.ffa.2017.11.007","volume":"50","author":"L Sok","year":"2018","unstructured":"Sok L., Shi M., Sol\u00e9 P.: Construction of optimal LCD codes over large finite fields. Finite Fields Appl. 50(1), 138\u2013153 (2018).","journal-title":"Finite Fields Appl."}],"container-title":["Designs, Codes and Cryptography"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-023-01210-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10623-023-01210-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-023-01210-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,12]],"date-time":"2023-06-12T17:13:53Z","timestamp":1686590033000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10623-023-01210-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,8]]},"references-count":42,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2023,7]]}},"alternative-id":["1210"],"URL":"https:\/\/doi.org\/10.1007\/s10623-023-01210-y","relation":{},"ISSN":["0925-1022","1573-7586"],"issn-type":[{"value":"0925-1022","type":"print"},{"value":"1573-7586","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,8]]},"assertion":[{"value":"1 July 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 January 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 March 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 April 2023","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 have no conflicts of interest to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}