{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T14:40:37Z","timestamp":1740148837112,"version":"3.37.3"},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,3,2]],"date-time":"2021-03-02T00:00:00Z","timestamp":1614643200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,3,2]],"date-time":"2021-03-02T00:00:00Z","timestamp":1614643200000},"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":["61572132"],"award-info":[{"award-number":["61572132"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003392","name":"Natural Science Foundation of Fujian Province","doi-asserted-by":"publisher","award":["2019J01275"],"award-info":[{"award-number":["2019J01275"]}],"id":[{"id":"10.13039\/501100003392","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["U1705264"],"award-info":[{"award-number":["U1705264"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100009558","name":"University Natural Science Research Project of Anhui Province","doi-asserted-by":"publisher","award":["KJ2018A0584"],"award-info":[{"award-number":["KJ2018A0584"]}],"id":[{"id":"10.13039\/501100009558","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2022,2]]},"DOI":"10.1007\/s12190-020-01475-9","type":"journal-article","created":{"date-parts":[[2021,3,2]],"date-time":"2021-03-02T14:03:09Z","timestamp":1614693789000},"page":"83-100","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The homogeneous distance of $$(1+u^2)$$-constacyclic codes over $$\\mathbb {F}_2[u]\/\\langle u^3\\rangle $$ and its applications"],"prefix":"10.1007","volume":"68","author":[{"given":"Jian","family":"Ding","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongju","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2868-3398","authenticated-orcid":false,"given":"Changlu","family":"Lin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,3,2]]},"reference":[{"issue":"2","key":"1475_CR1","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":"1475_CR2","doi-asserted-by":"crossref","unstructured":"Shi, M., Zhu, S., Yang, S.: A class of optimal $$p$$-ary codes from one-weight codes over $$\\mathbb{F}[u]\/ <u^m>$$. J. Franlkin Inst.-Eng. Appl. Math. 350(5), 929\u2013937 (2013)","DOI":"10.1016\/j.jfranklin.2012.05.014"},{"issue":"11","key":"1475_CR3","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. Designs, Codes Cryptogr. 86(11), 2485\u20132494 (2018)","journal-title":"Designs, Codes Cryptogr."},{"issue":"9","key":"1475_CR4","doi-asserted-by":"publisher","first-page":"2091","DOI":"10.1007\/s10623-019-00609-w","volume":"87","author":"M Shi","year":"2019","unstructured":"Shi, M., Denis, K., Sol\u00e9, P.: A new distance-regular graph of diameter 3 on 1024 vertices. Designs, Codes Cryptogr 87(9), 2091\u20132101 (2019)","journal-title":"Designs , Codes Cryptogr"},{"issue":"4","key":"1475_CR5","doi-asserted-by":"publisher","first-page":"1250","DOI":"10.1109\/18.761278","volume":"45","author":"A Bonnecaze","year":"1999","unstructured":"Bonnecaze, A., Udaya, P.: Cyclic codes and self-dual codes over $$\\mathbb{F}_2+u\\mathbb{F}_2$$. IEEE Trans. Inf. Theory 45(4), 1250\u20131255 (1999)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"8","key":"1475_CR6","doi-asserted-by":"publisher","first-page":"820","DOI":"10.1016\/j.aml.2005.10.011","volume":"19","author":"JF Qian","year":"2006","unstructured":"Qian, J.F., Zhang, L.N., Zhu, S.X.: $$(1+u)$$ Constacyclic and cyclic codes over $$\\mathbb{F}_{2} + u\\mathbb{F}_{2}$$. Appl. Math. Lett. 19(8), 820\u2013823 (2006)","journal-title":"Appl. Math. Lett."},{"issue":"3","key":"1475_CR7","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1007\/s10623-006-9034-5","volume":"42","author":"T Abualrub","year":"2007","unstructured":"Abualrub, T., Saip, I.: Cyclic codes over the rings $$\\mathbb{Z}_2+u\\mathbb{Z}_2$$ and $$\\mathbb{Z}+u\\mathbb{Z}+u^2\\mathbb{Z}$$. Des. Codes Cryptogr. 42(3), 273\u2013287 (2007)","journal-title":"Des. Codes Cryptogr."},{"issue":"4","key":"1475_CR8","first-page":"737","volume":"35","author":"M Al-Ashker","year":"2011","unstructured":"Al-Ashker, M., Hamoudeh, M.: Cyclic codes over $$\\mathbb{Z}_2+u\\mathbb{Z}_2+\\cdots +u^{k-1}\\mathbb{Z}_2$$. Turk. J. Math. 35(4), 737\u2013749 (2011)","journal-title":"Turk. J. Math."},{"issue":"1","key":"1475_CR9","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10623-013-9843-2","volume":"74","author":"AK Singh","year":"2015","unstructured":"Singh, A.K., Kewat, P.K.: On cyclic codes over the ring $$\\mathbb{Z}_p[u]\/\\langle u^k\\rangle $$. Desi. Codes Cryptogr. 74(1), 1\u201313 (2015)","journal-title":"Desi. Codes Cryptogr."},{"issue":"5","key":"1475_CR10","doi-asserted-by":"publisher","first-page":"520","DOI":"10.1016\/j.jfranklin.2009.02.001","volume":"346","author":"T Abualrub","year":"2009","unstructured":"Abualrub, T., Saip, I.: Constacyclic codes over $$\\mathbb{F}_{2} + u\\mathbb{F}_{2}$$. J. Frankl. Inst. 346(5), 520\u2013529 (2009)","journal-title":"J. Frankl. Inst."},{"issue":"5","key":"1475_CR11","doi-asserted-by":"publisher","first-page":"751","DOI":"10.1016\/j.jfranklin.2010.02.003","volume":"347","author":"X Kai","year":"2010","unstructured":"Kai, X., Zhu, S.X., Li, P.: $$(1+\\lambda u)$$-Constacyclic codes over $$\\mathbb{F}_{p}[u]\/\\langle u^k\\rangle $$. J. Frankl. Inst. 347(5), 751\u2013762 (2010)","journal-title":"J. Frankl. Inst."},{"key":"1475_CR12","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1016\/j.ffa.2015.01.008","volume":"34","author":"R Sobhani","year":"2015","unstructured":"Sobhani, R.: Complete classification of $$(\\delta +\\alpha u^2)$$-constacyclic codes of length $$p^k$$ over $$\\mathbb{F}_{p^m}+u\\mathbb{F}_{p^m}+u^2\\mathbb{F}_{p^m}$$. Finite Fields Their Appl. 34, 123\u2013138 (2015)","journal-title":"Finite Fields Their Appl."},{"key":"1475_CR13","unstructured":"Sobhani, R., Sun, Z., Wang, L., Zhu, S. X.: A note on complete classification of $$(\\delta +\\alpha u^2)$$-constacyclic codes of length $$p^k$$ over $${\\mathbb{F}}_{p^m}+u{\\mathbb{F}}_{p^m}+u^2{\\mathbb{F}}_{p^m}$$, arXiv:1702.04694V1"},{"issue":"12","key":"1475_CR14","doi-asserted-by":"publisher","first-page":"2840","DOI":"10.1016\/j.disc.2017.08.002","volume":"340","author":"Y Cao","year":"2017","unstructured":"Cao, Y., Cao, Y.L., Ma, F.: Complete classification of $$(\\delta +\\alpha u^2)$$-constacyclic codes over $$\\mathbb{F}_{2^m}[u]\/\\langle u^4\\rangle $$ of oddly even length. Discrete Math. 340(12), 2840\u20132852 (2017)","journal-title":"Discrete Math."},{"issue":"11","key":"1475_CR15","doi-asserted-by":"publisher","first-page":"3106","DOI":"10.1016\/j.disc.2018.07.024","volume":"341","author":"HR Mahmoodi","year":"2018","unstructured":"Mahmoodi, H.R., Sobhani, R.: On some constacyclic codes over the ring $$\\mathbb{F}_{p^m}[u]\/\\langle u^4\\rangle $$. Discrete Math. 341(11), 3106\u20133122 (2018)","journal-title":"Discrete Math."},{"issue":"1\u20132","key":"1475_CR16","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1007\/s12190-018-1183-y","volume":"59","author":"J Ding","year":"2019","unstructured":"Ding, J., Li, H., Liang, J.: $$(1+\\lambda u^2)$$-constacyclic code of arbitrary length over $$\\mathbb{F}_{p^m}[u]\/\\langle u^3\\rangle $$. J. Appl. Math. Comput. 59(1\u20132), 343\u2013359 (2019)","journal-title":"J. Appl. Math. Comput."},{"key":"1475_CR17","unstructured":"Constaninescu, I.: Lineare Codes \u00fcber Restklassenringen ganzer Zahlen und ihre Automorphismen bez\u00fcglich einer verallgemeinerten Hamming-Metrik (Ph.D. dissertation), Technische University, M\u00fcnchen, Germany, (1995)"},{"issue":"7","key":"1475_CR18","doi-asserted-by":"publisher","first-page":"2522","DOI":"10.1109\/18.796395","volume":"45","author":"M Greferath","year":"1999","unstructured":"Greferath, M., Schmidt, S.E.: Gray isometries for finite chain rings and a nonlinear Ternary $$(36,3^{12},15)$$ code. IEEE Trans. Inf. Theory 45(7), 2522\u20132524 (1999)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1475_CR19","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1006\/jcta.1999.3033","volume":"92","author":"M Greferath","year":"2000","unstructured":"Greferath, M., Schmidt, S.E.: Finite ring combinatorics and MacWilliams\u2019s equivalence theorem. J. Comb. Theory Ser. A 92, 17\u201328 (2000)","journal-title":"J. Comb. Theory Ser. A"},{"key":"1475_CR20","doi-asserted-by":"crossref","unstructured":"Qian, J. F., Zhang, L.N., Zhu, S.X.: Constacyclic and cyclic codes over $${\\mathbb{F}}_{2}+u{\\mathbb{F}}_{2}+u^2{\\mathbb{F}}_{2}$$. IEICE Trans. Fundam. E89-A(6), 1863-1865 (2006)","DOI":"10.1093\/ietfec\/e89-a.6.1863"},{"issue":"12","key":"1475_CR21","doi-asserted-by":"publisher","first-page":"2837","DOI":"10.1007\/s10623-018-0479-0","volume":"86","author":"J Li","year":"2018","unstructured":"Li, J., Zhang, A., Feng, K.: Linear codes over $$\\mathbb{F}_q[x]\/\\langle x^2\\rangle $$ and GR$$(p^2, m)$$ reaching the Griesmer bound. Des. Codes Cryptogr. 86(12), 2837\u20132855 (2018)","journal-title":"Des. Codes Cryptogr."},{"issue":"6","key":"1475_CR22","doi-asserted-by":"publisher","first-page":"1706","DOI":"10.1016\/j.disc.2016.01.020","volume":"339","author":"HQ Dinh","year":"2016","unstructured":"Dinh, H.Q., Dhompongsa, S., Sriboonchitta, S.: Repeated-root constacyclic codes of prime power length over $$\\mathbb{F}_{p^m}[u]\/\\langle u^a\\rangle $$ and their duals. Discrete Math. 339(6), 1706\u20131715 (2016)","journal-title":"Discrete Math."},{"issue":"4","key":"1475_CR23","doi-asserted-by":"publisher","first-page":"926","DOI":"10.1016\/j.ins.2010.10.021","volume":"181","author":"M Han","year":"2011","unstructured":"Han, M., Ye, Y.P., Zhu, S.X., Xu, C., Dou, B.: Cyclic codes over $$R=\\mathbb{F}_{p^m}+u\\mathbb{F}_{p^m}+\\cdots +u^{k-1}\\mathbb{F}_{p^m}$$ with length $$p^sn$$. Inf. Sci. 181(4), 926\u2013934 (2011)","journal-title":"Inf. Sci."},{"issue":"2","key":"1475_CR24","first-page":"171","volume":"1","author":"ST Dougherty","year":"2010","unstructured":"Dougherty, S.T., Kim, J., Liu, H.: Constructions of self-dual codes over finite commutative chain rings. Int. J. Inf. Coding Theory 1(2), 171\u2013190 (2010)","journal-title":"Int. J. Inf. Coding Theory"},{"key":"1475_CR25","unstructured":"Grassl, M.: Bound on the minimum distance of linear codes, http:\/\/www.codetables.de. Accessed 25 May (2020)"}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-020-01475-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-020-01475-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-020-01475-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,1,25]],"date-time":"2022-01-25T15:24:02Z","timestamp":1643124242000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-020-01475-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,2]]},"references-count":25,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,2]]}},"alternative-id":["1475"],"URL":"https:\/\/doi.org\/10.1007\/s12190-020-01475-9","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"type":"print","value":"1598-5865"},{"type":"electronic","value":"1865-2085"}],"subject":[],"published":{"date-parts":[[2021,3,2]]},"assertion":[{"value":"7 October 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 December 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 December 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 March 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}