{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,29]],"date-time":"2026-05-29T07:03:07Z","timestamp":1780038187849,"version":"3.53.1"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T00:00:00Z","timestamp":1776643200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T00:00:00Z","timestamp":1776643200000},"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":["Des. Codes Cryptogr."],"published-print":{"date-parts":[[2026,5]]},"DOI":"10.1007\/s10623-026-01834-w","type":"journal-article","created":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T12:06:20Z","timestamp":1776686780000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Codes over an infinite family of local rings of order $$2^{5^k}$$ with two Gray maps"],"prefix":"10.1007","volume":"94","author":[{"given":"Steven T.","family":"Dougherty","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Joe","family":"Gildea","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Adrian","family":"Korban","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Adam M.","family":"Roberts","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,4,20]]},"reference":[{"issue":"2","key":"1834_CR1","doi-asserted-by":"publisher","first-page":"122","DOI":"10.1016\/S0021-9800(69)80115-8","volume":"6","author":"EF Assmus Jr","year":"1969","unstructured":"Assmus E.F. Jr., Mattson H.F. Jr.: New 5-designs. J. Combin. Theory 6(2), 122\u2013151 (1969). https:\/\/doi.org\/10.1016\/S0021-9800(69)80115-8.","journal-title":"J. Combin. Theory"},{"key":"1834_CR2","doi-asserted-by":"publisher","first-page":"222","DOI":"10.1016\/j.laa.2014.06.045","volume":"459","author":"W Barrett","year":"2014","unstructured":"Barrett W., Butler S., Catral M., Fallat S.M., Hall H.T., Hogben L., van den Driessche P., Young M.: The principal rank characteristic sequence over various fields. Linear Algebra Appl. 459, 222\u2013236 (2014). https:\/\/doi.org\/10.1016\/j.laa.2014.06.045.","journal-title":"Linear Algebra Appl."},{"key":"1834_CR3","doi-asserted-by":"publisher","first-page":"825","DOI":"10.1007\/s10623-019-00713-x","volume":"88","author":"S Bhowmick","year":"2020","unstructured":"Bhowmick S., Fotue-Tabue A., Mart\u00ednez-Moro E., Bandi R., Bagchi S.: Do non-free LCD codes over finite commutative Frobenius rings exist? Des. Codes Cryptogr. 88, 825\u2013840 (2020). https:\/\/doi.org\/10.1007\/s10623-019-00713-x.","journal-title":"Des. Codes Cryptogr."},{"issue":"2","key":"1834_CR4","doi-asserted-by":"publisher","first-page":"115","DOI":"10.55630\/sjc.2007.1.115-130","volume":"1","author":"IG Bouyukliev","year":"2007","unstructured":"Bouyukliev I.G.: What is Q-Extension? Serdica J. Comput. 1(2), 115\u2013130 (2007).","journal-title":"Serdica J. Comput."},{"issue":"11","key":"1834_CR5","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). https:\/\/doi.org\/10.1007\/s10623-021-00929-w.","journal-title":"Des. Codes Cryptogr."},{"issue":"4","key":"1834_CR6","doi-asserted-by":"publisher","first-page":"1222","DOI":"10.1109\/18.86979","volume":"37","author":"RA Brualdi","year":"1991","unstructured":"Brualdi R.A., Pless V.S.: Weight enumerators of self-dual codes. IEEE Trans. Inform. Theory 37(4), 1222\u20131225 (1991). https:\/\/doi.org\/10.1109\/18.86979.","journal-title":"IEEE Trans. Inform. Theory"},{"key":"1834_CR7","doi-asserted-by":"crossref","unstructured":"Carlet C., Guilley S.: Complementary dual codes for counter-measures to side-channel attacks. In: Proceedings of the 4th International Castle Meeting on Coding Theory and Applications (ICMCTA), Palmela, Portugal, pp. 97\u2013105 (2014).","DOI":"10.1007\/978-3-319-17296-5_9"},{"issue":"3","key":"1834_CR8","doi-asserted-by":"publisher","first-page":"559","DOI":"10.1007\/s10623-012-9787-y","volume":"72","author":"Y Cengellenmis","year":"2014","unstructured":"Cengellenmis Y., Dertli A., Dougherty S.T.: Codes over an infinite family of rings with a Gray map. Des. Codes Cryptogr. 72(3), 559\u2013580 (2014). https:\/\/doi.org\/10.1007\/s10623-012-9787-y.","journal-title":"Des. Codes Cryptogr."},{"issue":"6","key":"1834_CR9","doi-asserted-by":"publisher","first-page":"1406","DOI":"10.3934\/amc.2021056","volume":"17","author":"Y \u00c7engellenmi\u015f","year":"2021","unstructured":"\u00c7engellenmi\u015f Y., Dertli A., Dougherty S.T., Korban A., \u015eahinkaya S., Ustun D.: Reversible $$G$$-codes over the ring $$\\cal{F} _{j, k}$$ with applications to DNA codes. Adv. Math. Commun. 17(6), 1406\u20131421 (2021). https:\/\/doi.org\/10.3934\/amc.2021056.","journal-title":"Adv. Math. Commun."},{"key":"1834_CR10","unstructured":"\u00c7engellenmi\u015f Y., Dougherty S.T.: Cyclic codes over $$A_k$$. In: Proceedings of the Thirteenth International Workshop on Algebraic and Combinatorial Coding Theory (ACCT 2012), Pomorie, Bulgaria (2012)."},{"issue":"6","key":"1834_CR11","doi-asserted-by":"publisher","first-page":"1319","DOI":"10.1109\/18.59931","volume":"36","author":"JH Conway","year":"1990","unstructured":"Conway J.H., Sloane N.J.A.: A new upper bound on the minimal distance of self-dual codes. IEEE Trans. Inform. Theory 36(6), 1319\u20131333 (1990). https:\/\/doi.org\/10.1109\/18.59931.","journal-title":"IEEE Trans. Inform. Theory"},{"key":"1834_CR12","doi-asserted-by":"crossref","unstructured":"Dougherty S.T.: Algebraic Coding Theory Over Finite Commutative Rings. Springer Briefs in Mathematics. Springer, Cham (2017).","DOI":"10.1007\/978-3-319-59806-2"},{"issue":"6","key":"1834_CR13","doi-asserted-by":"publisher","first-page":"1827","DOI":"10.3934\/amc.2023023","volume":"18","author":"ST Dougherty","year":"2023","unstructured":"Dougherty S.T.: Dualities for codes over finite abelian groups. Adv. Math. Commun. 18(6), 1827\u20131841 (2023). https:\/\/doi.org\/10.3934\/amc.2023023.","journal-title":"Adv. Math. Commun."},{"issue":"4","key":"1834_CR14","doi-asserted-by":"publisher","first-page":"571","DOI":"10.3934\/amc.2011.5.571","volume":"5","author":"ST Dougherty","year":"2011","unstructured":"Dougherty S.T., Fern\u00e1ndez-C\u00f3rdoba C.: Codes over $$\\mathbb{Z} _{2^k}$$, Gray map and self-dual codes. Adv. Math. Commun. 5(4), 571\u2013588 (2011). https:\/\/doi.org\/10.3934\/amc.2011.5.571.","journal-title":"Adv. Math. Commun."},{"key":"1834_CR15","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2024.102384","author":"ST Dougherty","year":"2024","unstructured":"Dougherty S.T., Gildea J., Korban A., Roberts A.M.: Codes over a ring of order 32 with two Gray maps. Finite Fields Appl. (2024). https:\/\/doi.org\/10.1016\/j.ffa.2024.102384.","journal-title":"Finite Fields Appl."},{"issue":"7","key":"1834_CR16","doi-asserted-by":"publisher","first-page":"2345","DOI":"10.1109\/18.796375","volume":"45","author":"ST Dougherty","year":"1999","unstructured":"Dougherty S.T., Gaborit P., Harada M., Munemasa A., Sol\u00e9 P.: Type IV self-dual codes over rings. IEEE Trans. Inform. Theory 45(7), 2345\u20132360 (1999). https:\/\/doi.org\/10.1109\/18.796375.","journal-title":"IEEE Trans. Inform. Theory"},{"issue":"1","key":"1834_CR17","doi-asserted-by":"publisher","first-page":"32","DOI":"10.1109\/18.746770","volume":"45","author":"ST Dougherty","year":"1999","unstructured":"Dougherty S.T., Gaborit P., Harada M., Sol\u00e9 P.: Type II codes over $$\\mathbb{F} _2+u\\mathbb{F} _2$$. IEEE Trans. Inform. Theory 45(1), 32\u201345 (1999). https:\/\/doi.org\/10.1109\/18.746770.","journal-title":"IEEE Trans. Inform. Theory"},{"issue":"3","key":"1834_CR18","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1016\/j.ffa.2010.11.002","volume":"17","author":"ST Dougherty","year":"2011","unstructured":"Dougherty S.T., Yildiz B., Karadeniz S.: Codes over $$R_k$$, Gray maps and their binary images. Finite Fields Appl. 17(3), 205\u2013219 (2011). https:\/\/doi.org\/10.1016\/j.ffa.2010.11.002.","journal-title":"Finite Fields Appl."},{"issue":"1","key":"1834_CR19","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1007\/s10623-011-9539-4","volume":"63","author":"ST Dougherty","year":"2012","unstructured":"Dougherty S.T., Karadeniz S., Yildiz B.: Cyclic codes over $$R_k$$. Des. Codes Cryptogr. 63(1), 113\u2013126 (2012). https:\/\/doi.org\/10.1007\/s10623-011-9539-4.","journal-title":"Des. Codes Cryptogr."},{"issue":"1","key":"1834_CR20","first-page":"89","volume":"6","author":"ST Dougherty","year":"2013","unstructured":"Dougherty S.T., Yildiz B., Karadeniz S.: Self-dual codes over $$R_k$$ and binary self-dual codes. Eur. J. Pure Appl. Math. 6(1), 89\u2013106 (2013).","journal-title":"Eur. J. Pure Appl. Math."},{"issue":"5","key":"1834_CR21","doi-asserted-by":"publisher","first-page":"435","DOI":"10.1007\/s00200-016-0288-5","volume":"27","author":"ST Dougherty","year":"2016","unstructured":"Dougherty S.T., Kaya A., Salt\u00fcrk E.: Constructions of self-dual codes and formally self-dual codes over rings. Appl. Algebra Engrg. Comm. Comput. 27(5), 435\u2013449 (2016). https:\/\/doi.org\/10.1007\/s00200-016-0288-5.","journal-title":"Appl. Algebra Engrg. Comm. Comput."},{"issue":"2\u20133","key":"1834_CR22","doi-asserted-by":"publisher","first-page":"116","DOI":"10.1504\/IJICOT.2017.083827","volume":"4","author":"ST Dougherty","year":"2017","unstructured":"Dougherty S.T., Kim J.-L., Ozkaya 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). https:\/\/doi.org\/10.1504\/IJICOT.2017.083827.","journal-title":"Int. J. Inf. Coding Theory"},{"issue":"5","key":"1834_CR23","doi-asserted-by":"publisher","first-page":"601","DOI":"10.1007\/s12095-021-00487-x","volume":"13","author":"ST Dougherty","year":"2021","unstructured":"Dougherty S.T., Gildea J., Korban A., \u015eahinkaya S.: $$G$$-codes, self-dual $$G$$-codes and reversible $$G$$-codes over the ring $$\\cal{B} _{j, k}$$. Cryptogr. Commun. 13(5), 601\u2013616 (2021). https:\/\/doi.org\/10.1007\/s12095-021-00487-x.","journal-title":"Cryptogr. Commun."},{"issue":"Part 3","key":"1834_CR24","doi-asserted-by":"publisher","first-page":"512","DOI":"10.1016\/j.dam.2016.09.025","volume":"217","author":"ST Dougherty","year":"2017","unstructured":"Dougherty S.T., Salt\u00fcrk E.: Codes over a family of local Frobenius rings, Gray maps and self-dual codes. Discret. Appl. Math. 217(Part 3), 512\u2013524 (2017). https:\/\/doi.org\/10.1016\/j.dam.2016.09.025.","journal-title":"Discret. Appl. Math."},{"issue":"4","key":"1834_CR25","doi-asserted-by":"publisher","first-page":"922","DOI":"10.3934\/amc.2022036","volume":"18","author":"ST Dougherty","year":"2024","unstructured":"Dougherty S.T., Korban A., \u015eahinkaya S., Ustun D.: Binary self-dual and LCD codes from generator matrices constructed from two group ring elements by a heuristic search scheme. Adv. Math. Commun. 18(4), 922\u2013934 (2024). https:\/\/doi.org\/10.3934\/amc.2022036.","journal-title":"Adv. Math. Commun."},{"key":"1834_CR26","doi-asserted-by":"publisher","DOI":"10.5281\/zenodo.16438894","author":"ST Dougherty","year":"2025","unstructured":"Dougherty S.T., Gildea J., Korban A., Roberts A.M.: Generator matrices. Zenodo (2025). https:\/\/doi.org\/10.5281\/zenodo.16438894.","journal-title":"Zenodo"},{"key":"1834_CR27","unstructured":"Eaton J.W., Bateman D., Hauberg S., Wehbring R.: GNU Octave version 9.3.0 manual: a high-level interactive language for numerical computations (2024). https:\/\/www.gnu.org\/software\/octave\/doc\/v9.3.0\/."},{"key":"1834_CR28","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, 719\u2013728 (2018). https:\/\/doi.org\/10.1007\/s12095-017-0258-1.","journal-title":"Cryptogr. Commun."},{"key":"1834_CR29","unstructured":"Gleason A.M.: Weight polynomials of codes and the MacWilliams identities. \u201cActes Congr. Int. Math.\u201d, Nice, France 3, 211\u2013215 (1970)."},{"key":"1834_CR30","unstructured":"Grassl M. Code tables: bounds on the parameters of various types of codes. https:\/\/www.codetables.de\/."},{"key":"1834_CR31","unstructured":"Guan C., Li R., Ma Z.: On Euclidean, Hermitian and symplectic quasi-cyclic complementary dual codes (2017). arXiv:2301.00945."},{"issue":"2","key":"1834_CR32","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1109\/18.312154","volume":"40","author":"AR Hammons","year":"1994","unstructured":"Hammons A.R., Kumar P.V., Calderbank A.R., Sloane N.J.A., Sol\u00e9 P.: The $$\\mathbb{Z} _4$$-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inform. Theory 40(2), 301\u2013319 (1994). https:\/\/doi.org\/10.1109\/18.312154.","journal-title":"IEEE Trans. Inform. Theory"},{"key":"1834_CR33","doi-asserted-by":"publisher","first-page":"677","DOI":"10.1007\/s12095-018-0319-0","volume":"11","author":"M Harada","year":"2018","unstructured":"Harada M., Saito K.: Binary linear complementary dual codes. Cryptogr. Commun. 11, 677\u2013696 (2018). https:\/\/doi.org\/10.1007\/s12095-018-0319-0.","journal-title":"Cryptogr. Commun."},{"key":"1834_CR34","unstructured":"Pless V., Huffman W.C. (eds.): Handbook of Coding Theory, volume 1 (Part 1: Algebraic Coding), Volume 2 (Part 2: Connections, Part 3: Applications). Elsevier\/North-Holland, Amsterdam (1998)."},{"issue":"4","key":"1834_CR35","doi-asserted-by":"publisher","first-page":"779","DOI":"10.1007\/s12095-024-00699-x","volume":"16","author":"S Li","year":"2024","unstructured":"Li S., Shi M., Liu H.: Several constructions of optimal LCD codes over small finite fields. Cryptogr. Commun. 16(4), 779\u2013800 (2024). https:\/\/doi.org\/10.1007\/s12095-024-00699-x.","journal-title":"Cryptogr. Commun."},{"key":"1834_CR36","doi-asserted-by":"publisher","DOI":"10.1016\/j.heliyon.2024.e38238","author":"Y Liu","year":"2024","unstructured":"Liu Y., Li R.: On the largest minimum distances of $$[n,6]$$ LCD codes. Heliyon (2024). https:\/\/doi.org\/10.1016\/j.heliyon.2024.e38238.","journal-title":"Heliyon"},{"issue":"7","key":"1834_CR37","doi-asserted-by":"publisher","first-page":"19137","DOI":"10.3934\/math.2024933","volume":"9","author":"Y Liu","year":"2024","unstructured":"Liu Y., Li R., Fu Q., Song H.: On the minimum distances of binary optimal LCD codes with dimension 5. AIMS Math. 9(7), 19137\u201319153 (2024). https:\/\/doi.org\/10.3934\/math.2024933.","journal-title":"AIMS Math."},{"key":"1834_CR38","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1016\/0012-365X(92)90563-U","volume":"106\u2013107","author":"JL Massey","year":"1992","unstructured":"Massey J.L.: Linear codes with complementary duals. Discret. Math. 106\u2013107, 337\u2013342 (1992). https:\/\/doi.org\/10.1016\/0012-365X(92)90563-U.","journal-title":"Discret. Math."},{"key":"1834_CR39","doi-asserted-by":"publisher","first-page":"778","DOI":"10.1016\/j.procs.2017.03.166","volume":"107","author":"Y Rao","year":"2017","unstructured":"Rao Y., Li R., Lv L., Chen G., Zuo F.: On binary LCD cyclic codes. Procedia Comput. Sci. 107, 778\u2013783 (2017). https:\/\/doi.org\/10.1016\/j.procs.2017.03.166.","journal-title":"Procedia Comput. Sci."},{"key":"1834_CR40","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2024.102381","author":"G Wang","year":"2024","unstructured":"Wang G., Liu S., Liu H.: New constructions of optimal binary LCD codes. Finite Fields Appl. (2024). https:\/\/doi.org\/10.1016\/j.ffa.2024.102381.","journal-title":"Finite Fields Appl."},{"issue":"3","key":"1834_CR41","doi-asserted-by":"publisher","first-page":"555","DOI":"10.1353\/ajm.1999.0024","volume":"121","author":"JA Wood","year":"1999","unstructured":"Wood J.A.: Duality for modules over finite rings and applications to coding theory. Am. J. Math. 121(3), 555\u2013575 (1999). https:\/\/doi.org\/10.1353\/ajm.1999.0024.","journal-title":"Am. J. Math."}],"container-title":["Designs, Codes and Cryptography"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-026-01834-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10623-026-01834-w","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-026-01834-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,29]],"date-time":"2026-05-29T06:19:46Z","timestamp":1780035586000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10623-026-01834-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,20]]},"references-count":41,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2026,5]]}},"alternative-id":["1834"],"URL":"https:\/\/doi.org\/10.1007\/s10623-026-01834-w","relation":{},"ISSN":["0925-1022","1573-7586"],"issn-type":[{"value":"0925-1022","type":"print"},{"value":"1573-7586","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,20]]},"assertion":[{"value":"26 July 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 January 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 March 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 April 2026","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 conflict of interest to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"96"}}