{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,16]],"date-time":"2026-02-16T16:02:23Z","timestamp":1771257743952,"version":"3.50.1"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"11","license":[{"start":{"date-parts":[[2021,8,27]],"date-time":"2021-08-27T00:00:00Z","timestamp":1630022400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,8,27]],"date-time":"2021-08-27T00:00:00Z","timestamp":1630022400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12071001"],"award-info":[{"award-number":["12071001"]}],"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":["61672036"],"award-info":[{"award-number":["61672036"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Excellent Youth Foundation of Natural Science Foundation of Anhui Province","award":["1808085J20"],"award-info":[{"award-number":["1808085J20"]}]},{"DOI":"10.13039\/501100017665","name":"Academic Fund for Outstanding Talents in Universities","doi-asserted-by":"crossref","award":["gxbjZD03"],"award-info":[{"award-number":["gxbjZD03"]}],"id":[{"id":"10.13039\/501100017665","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100003995","name":"Natural Science Foundation of Anhui Province","doi-asserted-by":"publisher","award":["2008085QA04"],"award-info":[{"award-number":["2008085QA04"]}],"id":[{"id":"10.13039\/501100003995","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":[[2022,11]]},"DOI":"10.1007\/s10623-021-00917-0","type":"journal-article","created":{"date-parts":[[2021,8,27]],"date-time":"2021-08-27T15:06:56Z","timestamp":1630076816000},"page":"2551-2562","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["On $$\\mathbb {Z}_2\\mathbb {Z}_4$$-additive polycyclic codes and their Gray images"],"prefix":"10.1007","volume":"90","author":[{"given":"Rongsheng","family":"Wu","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4990-6271","authenticated-orcid":false,"given":"Minjia","family":"Shi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,8,27]]},"reference":[{"issue":"3","key":"917_CR1","doi-asserted-by":"publisher","first-page":"1508","DOI":"10.1109\/TIT.2014.2299791","volume":"60","author":"T Abualrub","year":"2014","unstructured":"Abualrub T., Siap I., Aydin N.: $$\\mathbb{Z}_2\\mathbb{Z}_4$$-additive cyclic codes. IEEE Trans. Inf. Theory 60(3), 1508\u20131514 (2014).","journal-title":"IEEE Trans. Inf. Theory"},{"key":"917_CR2","doi-asserted-by":"crossref","unstructured":"Alahmadi A., Dougherty S. T., Leroy A., Sol\u00e9 P.: On the duality and the direction of polycyclic codes. Adv. Math. Commun. 10(4), 921\u2013929 (2016).","DOI":"10.3934\/amc.2016049"},{"issue":"6","key":"917_CR3","doi-asserted-by":"publisher","first-page":"2271","DOI":"10.12785\/amis\/070617","volume":"7","author":"I Aydogdu","year":"2013","unstructured":"Aydogdu I., Siap I.: The structure of $$\\mathbb{Z}_2\\mathbb{Z}_{2^s}$$-additive codes: bounds on the minimum distance. Appl. Math. Inf. Sci. 7(6), 2271\u20132278 (2013).","journal-title":"Appl. Math. Inf. Sci."},{"issue":"10","key":"917_CR4","doi-asserted-by":"publisher","first-page":"2089","DOI":"10.1080\/03081087.2014.952728","volume":"63","author":"I Aydogdu","year":"2015","unstructured":"Aydogdu I., Siap I.: On $$\\mathbb{Z}_{p^r}\\mathbb{Z}_{p^s}$$-additive codes. Linear Multilinear Algeb. 63(10), 2089\u20132102 (2015).","journal-title":"Linear Multilinear Algeb."},{"issue":"8","key":"917_CR5","doi-asserted-by":"publisher","first-page":"4883","DOI":"10.1109\/TIT.2016.2632163","volume":"63","author":"I Aydogdu","year":"2017","unstructured":"Aydogdu I., Abualrub T., Siap I.: $$\\mathbb{Z}_2\\mathbb{Z}_2[u]$$-cyclic and constacyclic codes. IEEE Trans. Inf. Theory 63(8), 4883\u20134893 (2017).","journal-title":"IEEE Trans. Inf. Theory"},{"key":"917_CR6","doi-asserted-by":"publisher","first-page":"101622","DOI":"10.1016\/j.ffa.2019.101622","volume":"62","author":"N Benbelkacema","year":"2020","unstructured":"Benbelkacema N., Borges J., Dougherty S.T., Fern\u00e1ndez-C\u00f3rdoba C.: On $$\\mathbb{Z}_2\\mathbb{Z}_4$$-additive complementary dual codes and related LCD codes. Finite Fields Appl. 62, 101622 (2020).","journal-title":"Finite Fields Appl."},{"key":"917_CR7","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1016\/j.ffa.2013.09.004","volume":"25","author":"TP Berger","year":"2014","unstructured":"Berger T.P., Amrani N.E.: Codes over finite quotients of polynomial ring. Finite Fields Appl. 25, 165\u2013181 (2014).","journal-title":"Finite Fields Appl."},{"issue":"1","key":"917_CR8","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1007\/s10623-010-9437-1","volume":"61","author":"M Bilal","year":"2011","unstructured":"Bilal M., Borges J., Dougherty S.T., Fern\u00e1ndez-C\u00e1rdoba C.: Maximum distance separable codes over $$\\mathbb{Z}_4$$ and $$\\mathbb{Z}_2\\times \\mathbb{Z}_4$$. Des. Codes Cryptogr. 61(1), 31\u201340 (2011).","journal-title":"Des. Codes Cryptogr."},{"issue":"7","key":"917_CR9","doi-asserted-by":"publisher","first-page":"1377","DOI":"10.1007\/s10623-017-0401-1","volume":"86","author":"J Borges","year":"2018","unstructured":"Borges J., Fern\u00e1ndez-C\u00f3rdoba C.: A characterization of $$\\mathbb{Z}_2\\mathbb{Z}_2[u]$$-linear codes. Des. Codes Cryptogr. 86(7), 1377\u20131389 (2018).","journal-title":"Des. Codes Cryptogr."},{"issue":"5","key":"917_CR10","doi-asserted-by":"publisher","first-page":"1688","DOI":"10.1109\/18.771247","volume":"45","author":"J Borges","year":"1999","unstructured":"Borges J., Rif\u00e0 J.: A characterization of 1-perfect additive codes. IEEE Trans. Inf. Theory 45(5), 1688\u20131697 (1999).","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"2","key":"917_CR11","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1007\/s10623-009-9316-9","volume":"54","author":"J Borges","year":"2010","unstructured":"Borges J., Fern\u00e1ndez-C\u00f3rdoba C., Pujol J., Rif\u00e0 J., Villanueva M.: $$\\mathbb{Z}_2\\mathbb{Z}_4$$-linear codes: generator matrices and duality. Des. Codes Cryptogr. 54(2), 167\u2013179 (2010).","journal-title":"Des. Codes Cryptogr."},{"issue":"11","key":"917_CR12","doi-asserted-by":"publisher","first-page":"6348","DOI":"10.1109\/TIT.2016.2611528","volume":"62","author":"J Borges","year":"2016","unstructured":"Borges J., Fern\u00e1ndez-C\u00f3rdoba C., Ten-Valls R.: $$\\mathbb{Z}_2\\mathbb{Z}_4$$-additive cyclic codes, generator polynomials, and dual codes. IEEE Trans. Inf. Theory 62(11), 6348\u20136354 (2016).","journal-title":"IEEE Trans. Inf. Theory"},{"key":"917_CR13","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. Symbolic Comput. 24, 235\u2013265 (1997).","journal-title":"J. Symbolic Comput."},{"key":"917_CR14","volume-title":"An Algebraic Approach to the Association Schemes of Coding Theory, (Philips Research Reports)","author":"P Delsarte","year":"1973","unstructured":"Delsarte P.: An Algebraic Approach to the Association Schemes of Coding Theory, (Philips Research Reports). Historical Jrl, Ann Arbor (1973)."},{"issue":"2","key":"917_CR15","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1007\/s00200-015-0273-4","volume":"27","author":"ST Dougherty","year":"2016","unstructured":"Dougherty S.T., Liu H.W., Yu L.: One weight $$\\mathbb{Z}_2\\mathbb{Z}_4$$-additive codes. AAECC 27(2), 123\u2013138 (2016).","journal-title":"AAECC"},{"key":"917_CR16","unstructured":"Fan Y., Liu H.L.: $$\\mathbb{Z}_2\\mathbb{Z}_4$$-additive cyclic codes are asymptotically good, (2019). arXiv:1911.09350."},{"issue":"3","key":"917_CR17","doi-asserted-by":"crossref","first-page":"455","DOI":"10.3934\/amc.2020028","volume":"14","author":"A Fotue-Tabue","year":"2020","unstructured":"Fotue-Tabue A., Mart\u00edez-Moro E., Blackford J.T.: On polycyclic codes over a finite chain ring. Adv. Math. Commun. 14(3), 455\u2013466 (2020).","journal-title":"Adv. Math. Commun."},{"key":"917_CR18","unstructured":"Grassl M.: Table of bounds on linear codes. (1995). [Online]. http:\/\/www.codetables.de\/."},{"issue":"2","key":"917_CR19","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. Inf. Theory 40(2), 301\u2013319 (1994).","journal-title":"IEEE Trans. Inf. Theory"},{"key":"917_CR20","doi-asserted-by":"crossref","unstructured":"Hou X.T., Gao J.: $$\\mathbb{Z}_p\\mathbb{Z}_p[v]$$-additive cyclic codes are asymptotically good. J. Appl. Math. Comput. 66(1-2), 871\u2013884 (2021).","DOI":"10.1007\/s12190-020-01466-w"},{"key":"917_CR21","unstructured":"Hu P., Liu X.S.: Binary LCD codes from $$\\mathbb{Z}_2\\mathbb{Z}_2[u]$$ (2019). arXiv:1903.11380v1."},{"issue":"3","key":"917_CR22","doi-asserted-by":"publisher","first-page":"227","DOI":"10.3934\/amc.2009.3.227","volume":"3","author":"SR L\u00f3pez-Permouth","year":"2009","unstructured":"L\u00f3pez-Permouth S.R., Parra-Avila B.R., Szabo S.: Dual generalizations of the concept of cyclicity of codes. Adv. Math. Commun. 3(3), 227\u2013234 (2009).","journal-title":"Adv. Math. Commun."},{"issue":"1","key":"917_CR23","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1016\/j.ffa.2012.10.002","volume":"19","author":"SR L\u00f3pez-Permouth","year":"2013","unstructured":"L\u00f3pez-Permouth S.R., \u00d6zadam H., \u00d6zbudak F., Szabo S.: Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes. Finite Fields Appl. 19(1), 16\u201338 (2013).","journal-title":"Finite Fields Appl."},{"key":"917_CR24","doi-asserted-by":"publisher","first-page":"332","DOI":"10.1016\/j.ffa.2018.12.003","volume":"56","author":"S Mahmoudi","year":"2019","unstructured":"Mahmoudi S., Samei K.: Additive codes over Galois rings. Finite Fields Appl. 56, 332\u2013350 (2019).","journal-title":"Finite Fields Appl."},{"issue":"4","key":"917_CR25","doi-asserted-by":"publisher","first-page":"947","DOI":"10.1137\/050632208","volume":"20","author":"E Mart\u00edez-Moro","year":"2006","unstructured":"Mart\u00edez-Moro E., R\u00faa I.F.: Multivariable codes over finite chain rings: serial codes. SIAM J. Discrete Math. 20(4), 947\u2013959 (2006).","journal-title":"SIAM J. Discrete Math."},{"issue":"2","key":"917_CR26","doi-asserted-by":"publisher","first-page":"219","DOI":"10.1007\/s10623-007-9114-1","volume":"45","author":"E Mart\u00edez-Moro","year":"2007","unstructured":"Mart\u00edez-Moro E., R\u00faa I.F.: On repeated-root multivariable codes over a finite chain rings. Des. Codes Cryptogr. 45(2), 219\u2013227 (2007).","journal-title":"Des. Codes Cryptogr."},{"key":"917_CR27","doi-asserted-by":"publisher","first-page":"94","DOI":"10.1016\/j.ffa.2017.09.008","volume":"49","author":"E Mart\u00edez-Moro","year":"2018","unstructured":"Mart\u00edez-Moro E., Pi\u00f1era-Nicol\u00e1s A., R\u00faa I.F.: Codes over affine algebras with a finite communicative chain coefficient ring. Finite Fields Appl. 49, 94\u2013107 (2018).","journal-title":"Finite Fields Appl."},{"issue":"107","key":"917_CR28","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1016\/0012-365X(92)90563-U","volume":"106","author":"JL Massey","year":"1992","unstructured":"Massey J.L.: Linear codes with complementary dual. Discrete Math. 106(107), 337\u2013342 (1992).","journal-title":"Discrete Math."},{"key":"917_CR29","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1016\/j.ffa.2017.01.003","volume":"45","author":"A Nikseresht","year":"2017","unstructured":"Nikseresht A.: Dual of codes over finite quotients of polynomial rings. Finite Fields Appl. 45, 323\u2013340 (2017).","journal-title":"Finite Fields Appl."},{"issue":"3","key":"917_CR30","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1017\/S0004972700019973","volume":"64","author":"GH Norton","year":"2001","unstructured":"Norton G.H., S\u01cel\u01cegean A.: Strong Gr\u00f6bner basis for polynomials over a principal ideal ring. Bull. Austral. Math. Soc. 64(3), 505\u2013528 (2001).","journal-title":"Bull. Austral. Math. Soc."},{"issue":"2","key":"917_CR31","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1016\/S1071-5797(03)00003-0","volume":"9","author":"GH Norton","year":"2003","unstructured":"Norton G.H., S\u01cel\u01cegean A.: Cyclic codes and minimal strong Gr\u00f6bner basis over a principal ideal ring. Finite Fields Appl. 9(2), 237\u2013249 (2003).","journal-title":"Finite Fields Appl."},{"key":"917_CR32","volume-title":"Error-Correcting Codes","author":"WW Peterson","year":"1972","unstructured":"Peterson W.W., Weldon E.J.: Error-Correcting Codes, 2nd edn. MIT Press, Cambridge (1972).","edition":"2"},{"issue":"2","key":"917_CR33","doi-asserted-by":"publisher","first-page":"923","DOI":"10.1109\/TIT.2019.2927034","volume":"66","author":"LQ Qian","year":"2020","unstructured":"Qian L.Q., Cao X.W.: Bounds and optimal $$q$$-ary codes derived from the $$\\mathbb{Z}_qR$$-cyclic codes. IEEE Trans. Inf. Theory 66(2), 923\u2013935 (2020).","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"2","key":"917_CR34","doi-asserted-by":"publisher","first-page":"590","DOI":"10.1109\/18.556115","volume":"43","author":"J Rif\u00e0","year":"1997","unstructured":"Rif\u00e0 J., Pujol J.: Translation invariant propeliniear codes. IEEE Trans. Inf. Theory 43(2), 590\u2013598 (1997).","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"2","key":"917_CR35","doi-asserted-by":"publisher","first-page":"413","DOI":"10.1016\/j.dam.2005.03.016","volume":"154","author":"A S\u01cel\u01cegean","year":"2006","unstructured":"S\u01cel\u01cegean A.: Repeated-root cyclic and nagacyclic codes over a finite chain ring. Discrete Appl. Math. 154(2), 413\u2013419 (2006).","journal-title":"Discrete Appl. Math."},{"issue":"6","key":"917_CR36","doi-asserted-by":"publisher","first-page":"3841","DOI":"10.1109\/TIT.2018.2883759","volume":"65","author":"MJ Shi","year":"2019","unstructured":"Shi M.J., Wu R.S., Krotov D.S.: On $$\\mathbb{Z}_p\\mathbb{Z}_{p^k}$$-additive codes and their dality. IEEE Trans. Inf. Theory 65(6), 3841\u20133847 (2019).","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"3","key":"917_CR37","doi-asserted-by":"publisher","first-page":"443","DOI":"10.1007\/s12095-019-00391-5","volume":"12","author":"MJ Shi","year":"2020","unstructured":"Shi M.J., Wang C.C., Wu R.S., Hu Y., Chang Y.Q.: One-weight and two-weight $$\\mathbb{Z}_2\\mathbb{Z}_2[u, v]$$-additive codes. Cryptogr. Commun. 12(3), 443\u2013454 (2020).","journal-title":"Cryptogr. Commun."},{"issue":"3","key":"917_CR38","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).","journal-title":"Am. J. Math."},{"key":"917_CR39","unstructured":"Wood J.A.: Lecture notes on the MacWilliams identities and extension theorem, Lectures for the Simat International School and Conference on Coding Theory, November 30\u2013December 2, (2008)."},{"issue":"2","key":"917_CR40","doi-asserted-by":"publisher","first-page":"253","DOI":"10.1007\/s12095-019-00397-z","volume":"12","author":"T Yao","year":"2020","unstructured":"Yao T., Zhu S.X.: $$\\mathbb{Z}_p\\mathbb{Z}_{p^s}$$-additive cyclic codes are asymptotically good. Cryptogr. Commun. 12(2), 253\u2013264 (2020).","journal-title":"Cryptogr. Commun."},{"key":"917_CR41","doi-asserted-by":"publisher","first-page":"101633","DOI":"10.1016\/j.ffa.2020.101633","volume":"63","author":"T Yao","year":"2020","unstructured":"Yao T., Zhu S.X., Kai X.S.: Asymptotically good $$\\mathbb{Z}_{p^r}\\mathbb{Z}_{p^s}$$-additive cyclic codes. Finite Fields Appl. 63, 101633 (2020).","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\/s10623-021-00917-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10623-021-00917-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-021-00917-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,7]],"date-time":"2024-09-07T09:43:47Z","timestamp":1725702227000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10623-021-00917-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,27]]},"references-count":41,"journal-issue":{"issue":"11","published-print":{"date-parts":[[2022,11]]}},"alternative-id":["917"],"URL":"https:\/\/doi.org\/10.1007\/s10623-021-00917-0","relation":{},"ISSN":["0925-1022","1573-7586"],"issn-type":[{"value":"0925-1022","type":"print"},{"value":"1573-7586","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,8,27]]},"assertion":[{"value":"17 December 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 May 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 July 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 August 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}