{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T14:11:35Z","timestamp":1769177495705,"version":"3.49.0"},"reference-count":20,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2020,4,6]],"date-time":"2020-04-06T00:00:00Z","timestamp":1586131200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,4,6]],"date-time":"2020-04-06T00:00:00Z","timestamp":1586131200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2020,10]]},"DOI":"10.1007\/s12190-020-01344-5","type":"journal-article","created":{"date-parts":[[2020,4,6]],"date-time":"2020-04-06T03:25:19Z","timestamp":1586143519000},"page":"71-87","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Optimal binary codes derived from $$\\mathbb {F}_{2} \\mathbb {F}_4$$-additive cyclic codes"],"prefix":"10.1007","volume":"64","author":[{"given":"Taher","family":"Abualrub","sequence":"first","affiliation":[]},{"given":"Nuh","family":"Aydin","sequence":"additional","affiliation":[]},{"given":"Ismail","family":"Aydogdu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,4,6]]},"reference":[{"issue":"3","key":"1344_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":"1344_CR2","doi-asserted-by":"publisher","first-page":"96","DOI":"10.1016\/j.ffa.2016.12.002","volume":"45","author":"N Aydin","year":"2017","unstructured":"Aydin, N., Halilovic, A.: A generalization of quasi-twisted codes: multi-twisted codes. Finite Fields Appl. 45, 96\u2013106 (2017)","journal-title":"Finite Fields Appl."},{"issue":"2","key":"1344_CR3","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 Crypt. 54(2), 167\u2013179 (2010)","journal-title":"Des. Codes Crypt."},{"issue":"2","key":"1344_CR4","doi-asserted-by":"publisher","first-page":"1098","DOI":"10.1103\/PhysRevA.54.1098","volume":"54","author":"AR Calderbank","year":"1996","unstructured":"Calderbank, A.R., Shor, P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54(2), 1098\u20131105 (1996)","journal-title":"Phys. Rev. A"},{"key":"1344_CR5","doi-asserted-by":"publisher","first-page":"1369","DOI":"10.1109\/18.681315","volume":"44","author":"AR Calderbank","year":"1998","unstructured":"Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction via codes over GF(4). IEEE Trans. Inf. Theory 44, 1369\u20131387 (1998)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1344_CR6","doi-asserted-by":"publisher","first-page":"1543","DOI":"10.1109\/18.681328","volume":"44","author":"C Carlet","year":"1998","unstructured":"Carlet, C.: $${{\\mathbb{Z}}}_{2^{k}}$$-linear codes. IEEE Trans. Inf. Theory 44, 1543\u20131547 (1998)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1344_CR7","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1109\/18.75249","volume":"37","author":"G Castagnoli","year":"1991","unstructured":"Castagnoli, G., Massey, J.L., Schoeller, P.A., Seeman, N.: On repeated-root cyclic codes. IEEE Trans. Inf. Theory 37, 337\u2013342 (1991)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1344_CR8","unstructured":"Database on Binary Quasi-Cyclic Codes. http:\/\/www.tec.hkr.se\/~chen\/research\/codes\/qc.htm. Accessed Jan 2019"},{"key":"1344_CR9","unstructured":"Grassl, M.: Table of Bounds on Linear Codes. http:\/\/www.codetables.de\/"},{"issue":"7","key":"1344_CR10","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. IEEE Trans. Inf. Theory. 45(7), 2522\u20132524 (1999)","journal-title":"IEEE Trans. Inf. Theory."},{"key":"1344_CR11","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1016\/j.ffa.2017.06.005","volume":"47","author":"C G\u00fcneri","year":"2017","unstructured":"G\u00fcneri, C., \u00d6zbudak, F., \u00d6zkaya, B., Sa\u00e7\u0131kara, E., Sepasdar, Z., Sol\u00e9, P.: Structure and performance of generalized quasi-cyclic codes. Finite Fields Appl. 47, 183\u2013202 (2017)","journal-title":"Finite Fields Appl."},{"issue":"2","key":"1344_CR12","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":"1344_CR13","doi-asserted-by":"crossref","unstructured":"Honold, T., Landjev, I.: Linear codes over finite chain rings. In: Optimal Codes and Related Topics. Sozopol, Bulgaria, pp. 116\u2013126 (1998)","DOI":"10.37236\/1489"},{"key":"1344_CR14","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1109\/18.75250","volume":"37","author":"JH Lint","year":"1991","unstructured":"Lint, J.H.: Repeated root cyclic codes. IEEE Trans. Inf. Theory 37, 343\u2013345 (1991)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1344_CR15","volume-title":"Error-Correcting Codes","author":"WW Peterson","year":"1961","unstructured":"Peterson, W.W.: Error-Correcting Codes. MIT Press, Cambridge (1961)"},{"key":"1344_CR16","volume-title":"Error-Correcting Codes","author":"WW Peterson","year":"1972","unstructured":"Peterson, W.W., Weldon Jr., E.J.: Error-Correcting Codes, 2nd edn. MIT Press, Cambridge (1972)","edition":"2"},{"issue":"6","key":"1344_CR17","first-page":"3842","volume":"65","author":"M Shi","year":"2018","unstructured":"Shi, M., Wu, R., Krotov, D.S.: On $$\\mathbb{Z}_{p}\\mathbb{Z} _{p^{k}}$$-additive codes and their duality. IEEE Trans. Inf. Theory 65(6), 3842\u20133847 (2018)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1344_CR18","doi-asserted-by":"publisher","first-page":"86","DOI":"10.1016\/j.ffa.2016.11.016","volume":"45","author":"M Shi","year":"2017","unstructured":"Shi, M., Qian, L., Sok, L., Sol\u00e9, P.: On constacyclic codes over $$\\mathbb{Z}_{4}[u]\/\\left\\langle u^{2}-1\\right\\rangle $$ and their Gray images. Finite Fields Appl. 45, 86\u201395 (2017)","journal-title":"Finite Fields Appl."},{"key":"1344_CR19","first-page":"24","volume":"5","author":"I Siap","year":"2005","unstructured":"Siap, I., Kulhan, N.: The structure of generalized quasi-cyclic codes. Appl. Math. E-Notes 5, 24\u201330 (2005)","journal-title":"Appl. Math. E-Notes"},{"issue":"1954","key":"1344_CR20","doi-asserted-by":"publisher","first-page":"2551","DOI":"10.1098\/rspa.1996.0136","volume":"452","author":"A Steane","year":"1996","unstructured":"Steane, A.: Multiple-particle interference and quantum error correction. Proc. R. Soc. Lond.Ser. A Math. Phys. Eng. Sci. 452(1954), 2551\u20132577 (1996)","journal-title":"Proc. R. Soc. Lond.Ser. A Math. Phys. Eng. Sci."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-020-01344-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s12190-020-01344-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-020-01344-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,4,6]],"date-time":"2021-04-06T00:20:19Z","timestamp":1617668419000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s12190-020-01344-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,6]]},"references-count":20,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2020,10]]}},"alternative-id":["1344"],"URL":"https:\/\/doi.org\/10.1007\/s12190-020-01344-5","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,4,6]]},"assertion":[{"value":"23 January 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 April 2020","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}