{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,2,6]],"date-time":"2023-02-06T00:27:54Z","timestamp":1675643274687},"reference-count":39,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2020,8,28]],"date-time":"2020-08-28T00:00:00Z","timestamp":1598572800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,8,28]],"date-time":"2020-08-28T00:00:00Z","timestamp":1598572800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100004541","name":"Ministry of Human Resource Development","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100004541","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2021,6]]},"DOI":"10.1007\/s12190-020-01425-5","type":"journal-article","created":{"date-parts":[[2020,8,28]],"date-time":"2020-08-28T16:03:46Z","timestamp":1598630626000},"page":"111-128","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A class of constacyclic codes and skew constacyclic codes over $$\\pmb {\\mathbb {Z}}_{2^s}+u\\pmb {\\mathbb {Z}}_{2^s}$$ and their gray images"],"prefix":"10.1007","volume":"66","author":[{"given":"Raj","family":"Kumar","sequence":"first","affiliation":[]},{"given":"Maheshanand","family":"Bhaintwal","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,8,28]]},"reference":[{"issue":"9","key":"1425_CR1","doi-asserted-by":"crossref","first-page":"2126","DOI":"10.1109\/TIT.2003.815763","volume":"49","author":"T Abualrub","year":"2003","unstructured":"Abualrub, T., Oehmke, R.: On the generators of $${\\mathbb{Z}}_4$$ cyclic codes of length $$2^e$$. IEEE Trans. Inf. Theory 49(9), 2126\u20132133 (2003)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"3","key":"1425_CR2","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1007\/s10623-006-9034-5","volume":"42","author":"T Abualrub","year":"2007","unstructured":"Abualrub, T., Siap, I.: Cyclic codes over the rings $${\\mathbb{Z}}_2+u{\\mathbb{Z}}_2$$ and $${\\mathbb{Z}}_2+u{\\mathbb{Z}}_2++u^2{\\mathbb{Z}}_2$$. Des. Codes Cryptogr. 42(3), 273\u2013287 (2007)","journal-title":"Des. Codes Cryptogr."},{"issue":"5","key":"1425_CR3","doi-asserted-by":"crossref","first-page":"520","DOI":"10.1016\/j.jfranklin.2009.02.001","volume":"346","author":"T Abualrub","year":"2009","unstructured":"Abualrub, T., Siap, I.: Constacyclic and cyclic codes over $${\\mathbb{F}}_2+u{\\mathbb{F}}_2$$. J. Franl. Inst. 346(5), 520\u2013529 (2009)","journal-title":"J. Franl. Inst."},{"key":"1425_CR4","unstructured":"Berlekamp, E.R.: Negacyclic codes for the Lee metric. In: Proceedings of the Conference on Combinatorial Mathematics and its Application. Chapel Hill, NC, pp. 298\u2013316 (1968)"},{"issue":"4","key":"1425_CR5","doi-asserted-by":"crossref","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"},{"key":"1425_CR6","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1007\/s00200-007-0043-z","volume":"18","author":"D Boucher","year":"2007","unstructured":"Boucher, D., Geiselmann, W., Ulmer, F.: Skew cyclic codes. Appl. Algebra Eng. Commun. Comput. 18, 379\u2013389 (2007)","journal-title":"Appl. Algebra Eng. Commun. Comput."},{"issue":"8","key":"1425_CR7","doi-asserted-by":"crossref","first-page":"1728","DOI":"10.1109\/TIT.2004.831789","volume":"50","author":"H Dinh","year":"2004","unstructured":"Dinh, H., Lopez-Permouth, S.R.: Cyclic and negacyclic codes over finite chain rings. IEEE Trans. Inf. Theory 50(8), 1728\u20131744 (2004)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"5","key":"1425_CR8","doi-asserted-by":"crossref","first-page":"940","DOI":"10.1016\/j.jalgebra.2010.05.027","volume":"324","author":"HQ Dinh","year":"2010","unstructured":"Dinh, H.Q.: Constacyclic codes of length $$p^s$$ over $$F_{p^m} +uF_{p^m}$$. J. Algebra 324(5), 940\u2013950 (2010)","journal-title":"J. Algebra"},{"key":"1425_CR9","doi-asserted-by":"crossref","first-page":"1730","DOI":"10.1109\/TIT.2009.2013015","volume":"55","author":"HQ Dinh","year":"2009","unstructured":"Dinh, H.Q.: Constacyclic codes of length $$2^s$$ over Galois extension rings of $${\\mathbb{F}}_2+u{\\mathbb{F}}_2$$. IEEE Trans. Inf. Theory 55, 1730\u20131740 (2009)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"3","key":"1425_CR10","first-page":"286","volume":"6","author":"J Gao","year":"2017","unstructured":"Gao, J., Ma, F., Fu, F.: Skew constacyclic codes over the ring $${\\mathbb{F}}_q + u{\\mathbb{F}}_q$$. Appl. Comput. Math. 6(3), 286\u2013295 (2017)","journal-title":"Appl. Comput. Math."},{"key":"1425_CR11","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1109\/18.312154","volume":"40","author":"AR Hammons Jr","year":"1994","unstructured":"Hammons Jr., A.R., Kumar, P.V., Calderbank, A.R., Sloane, N.J.A., Sol\u00e9, P.: The $$Z_4$$ linearity of Kerdock, Preparata, Goethals and related codes. IEEE Trans. Inf. Theory 40, 301\u2013319 (1994)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"8","key":"1425_CR12","doi-asserted-by":"crossref","first-page":"2237","DOI":"10.2298\/FIL1908237I","volume":"33","author":"H Islam","year":"2019","unstructured":"Islam, H., Prakasha, O.: A class of constacyclic codes over the ring $$\\frac{{\\mathbb{Z}}_4[u,~v]}{\\langle u^2, v^2, uv-vu \\rangle }$$ and their Gray images. Filomat 33(8), 2237\u20132248 (2019)","journal-title":"Filomat"},{"issue":"2","key":"1425_CR13","first-page":"101","volume":"5","author":"H Islam","year":"2018","unstructured":"Islam, H., Prakash, O.: Skew cyclic and skew $$(\\alpha _1+ u\\alpha _2 + v\\alpha _3 + uv\\alpha )$$-codes over $${\\mathbb{F}}_q + u{\\mathbb{F}}_q+v{\\mathbb{F}}_q + uv{\\mathbb{F}}_q$$. Int. J. Inf. Coding Theory 5(2), 101\u2013116 (2018)","journal-title":"Int. J. Inf. Coding Theory"},{"key":"1425_CR14","doi-asserted-by":"crossref","first-page":"39","DOI":"10.3934\/amc.2012.6.39","volume":"6","author":"S Jitman","year":"2012","unstructured":"Jitman, S., Ling, S., Udomkavanich, P.: Skew constacyclic codes over finite chain ring. Adv. Math. Commun. 6, 39\u201363 (2012)","journal-title":"Adv. Math. Commun."},{"issue":"9","key":"1425_CR15","doi-asserted-by":"crossref","first-page":"2625","DOI":"10.1016\/j.jfranklin.2011.08.005","volume":"348","author":"S Karadeniz","year":"2011","unstructured":"Karadeniz, S., Yildiz, B.: $$(1+v)$$-Constacyclic codes over $${\\mathbb{F}}_2 + u{\\mathbb{F}}_2$$. J. Frankl. Inst. 348(9), 2625\u20132632 (2011)","journal-title":"J. Frankl. Inst."},{"issue":"1","key":"1425_CR16","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1007\/s10623-011-9538-5","volume":"63","author":"HM Kiah","year":"2012","unstructured":"Kiah, H.M., Leung, K.H., Ling, S.: A note on cyclic codes over $$GR(p^2, m)$$ of length $$p^k$$. Des. Codes Cryptogr. 63(1), 105\u2013112 (2012)","journal-title":"Des. Codes Cryptogr."},{"key":"1425_CR17","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/j.dam.2017.11.020","volume":"237","author":"G Luo","year":"2018","unstructured":"Luo, G., Cao, X., Xu, G., Xu, S.: A new class of optimal linear codes with flexible parameters. Discret. Appl. Math. 237, 126\u2013131 (2018)","journal-title":"Discret. Appl. Math."},{"issue":"6","key":"1425_CR18","doi-asserted-by":"crossref","first-page":"1119","DOI":"10.1007\/s12095-017-0272-3","volume":"10","author":"G Luo","year":"2018","unstructured":"Luo, G., Cao, X.: Five classes of optimal two-weight linear codes. Cryptogr. Commun. 10(6), 1119\u20131135 (2018)","journal-title":"Cryptogr. Commun."},{"key":"1425_CR19","volume-title":"Finite Rings with Identity","author":"BR McDonald","year":"1974","unstructured":"McDonald, B.R.: Finite Rings with Identity. Marcel Dekker, New york (1974)"},{"issue":"6","key":"1425_CR20","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1007\/PL00012382","volume":"10","author":"G Norton","year":"2000","unstructured":"Norton, G., Salagean, A.: On the structure of linear and cyclic codes over a finite chain ring. Appl. Algebra Eng. Commun. Comput. 10(6), 489\u2013506 (2000)","journal-title":"Appl. Algebra Eng. Commun. Comput."},{"key":"1425_CR21","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.ffa.2015.12.003","volume":"38","author":"M Ozen","year":"2016","unstructured":"Ozen, M., Uzekmek, F.Z., Aydin, N., Ozzaim, N.T.: Cyclic and some constacyclic codes over the ring $$\\frac{{\\mathbb{Z}}_4[u]}{\\langle u^2 - 1 \\rangle }$$. Finite Fields Appl. 38, 27\u201339 (2016)","journal-title":"Finite Fields Appl."},{"issue":"4","key":"1425_CR22","first-page":"289","volume":"4","author":"A Sharma","year":"2017","unstructured":"Sharma, A., Bhaintwal, M.: A class of skew constacyclic codes over $${\\mathbb{Z}}_4+u{\\mathbb{Z}}_4$$. Int. J. Inf. Coding Theory 4(4), 289\u2013303 (2017)","journal-title":"Int. J. Inf. Coding Theory"},{"issue":"4","key":"1425_CR23","doi-asserted-by":"crossref","first-page":"723","DOI":"10.3934\/amc.2018043","volume":"12","author":"A Sharma","year":"2018","unstructured":"Sharma, A., Bhaintwal, M.: A class of skew-cyclic codes over $${\\mathbb{Z}}_4+u{\\mathbb{Z}}_4$$ with derivation. Adv. Math. Commun. 12(4), 723\u2013739 (2018)","journal-title":"Adv. Math. Commun."},{"issue":"10","key":"1425_CR24","doi-asserted-by":"crossref","first-page":"6240","DOI":"10.1109\/TIT.2017.2742499","volume":"63","author":"M Shi","year":"2017","unstructured":"Shi, M., Guan, Y., Sole, P.: Two new families of two-weight codes. IEEE Trans. Inf. Theory 63(10), 6240\u20136246 (2017)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1425_CR25","doi-asserted-by":"crossref","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., Aydin, N., Sole, P.: On constacyclic codes over $$\\frac{{\\mathbb{Z}}_4[u]}{\\langle u^2 - 1 \\rangle }$$ and their Gray images. Finite Fields Appl. 45, 86\u201395 (2017)","journal-title":"Finite Fields Appl."},{"issue":"12","key":"1425_CR26","doi-asserted-by":"crossref","first-page":"2346","DOI":"10.1109\/LCOMM.2016.2614934","volume":"20","author":"M Shi","year":"2016","unstructured":"Shi, M., Liu, Y., Sol\u00e9, P.: Optimal two-weight codes from trace codes over $${\\mathbb{F}}_2+u{\\mathbb{F}}_2$$. IEEE Commun. Lett. 20(12), 2346\u20132349 (2016)","journal-title":"IEEE Commun. Lett."},{"key":"1425_CR27","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1016\/j.dam.2016.09.050","volume":"219","author":"M Shi","year":"2017","unstructured":"Shi, M., Liu, Y., Sol\u00e9, P.: Optimal binary codes from trace codes over a non-chain ring. Discret. Appl. Math. 219, 176\u2013181 (2017)","journal-title":"Discret. Appl. Math."},{"key":"1425_CR28","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1007\/s00200-017-0345-8","volume":"29","author":"M Shi","year":"2018","unstructured":"Shi, M., Qian, L., Sol\u00e9, P.: New few weight codes from trace codes over a local Ring. Appl. Algebra Eng. Commun. Comput. 29, 335\u2013350 (2018)","journal-title":"Appl. Algebra Eng. Commun. Comput."},{"issue":"5","key":"1425_CR29","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1007\/s12095-016-0206-5","volume":"9","author":"M Shi","year":"2017","unstructured":"Shi, M., Wu, R., Liu, Y., Sol\u00e9, P.: Two and three weight codes over $${\\mathbb{F}}_p+u{\\mathbb{F}}_p$$. Cryptogr. Commun. 9(5), 637\u2013646 (2017)","journal-title":"Cryptogr. Commun."},{"key":"1425_CR30","doi-asserted-by":"publisher","DOI":"10.1007\/s40840-017-0553-1","author":"M Shi","year":"2019","unstructured":"Shi, M., Wu, R., Qian, L., Sok, L., Sol\u00e9, P.: New classes of $$p$$-ary few weight codes. Bull. Malays. Math. Sci. Soc. (2019). https:\/\/doi.org\/10.1007\/s40840-017-0553-1","journal-title":"Bull. Malays. Math. Sci. Soc."},{"issue":"2","key":"1425_CR31","first-page":"175","volume":"17","author":"M Shi","year":"2018","unstructured":"Shi, M., Zhu, H., Sol\u00e9, P.: Optimal three-weight cubic codes. Appl. Comput. Math. 17(2), 175\u2013184 (2018)","journal-title":"Appl. Comput. Math."},{"issue":"10","key":"1425_CR32","first-page":"6240","volume":"63","author":"M Shi","year":"2017","unstructured":"Shi, M., Guan, Y., Sol\u00e9, P.: Two new families of two-weight codes. IEEE Commun. Lett. 63(10), 6240\u20136246 (2017)","journal-title":"IEEE Commun. Lett."},{"key":"1425_CR33","doi-asserted-by":"crossref","first-page":"2527","DOI":"10.1109\/18.796397","volume":"45","author":"J Wolfmann","year":"1999","unstructured":"Wolfmann, J.: Negacyclic and cyclic codes over $${\\mathbb{Z}}_4$$. IEEE Trans. Inf. Theory 45, 2527\u20132532 (1999)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1425_CR34","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1007\/s10623-009-9309-8","volume":"54","author":"B Yildiz","year":"2010","unstructured":"Yildiz, B., Karadenniz, S.: Linear codes over $${\\mathbb{F}}_2 + u{\\mathbb{F}}_2 + v{\\mathbb{F}}_2 + uv{\\mathbb{F}}_2$$. Des. Codes Cryptogr. 54, 61\u201381 (2010)","journal-title":"Des. Codes Cryptogr."},{"issue":"3","key":"1425_CR35","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1007\/s10623-010-9399-3","volume":"58","author":"B Yildiz","year":"2011","unstructured":"Yildiz, B., Karadeniz, S.: Cyclic codes over $${\\mathbb{F}}_2 + u{\\mathbb{F}}_2 + v{\\mathbb{F}}_2 + uv{\\mathbb{F}}_2$$. Des. Codes Cryptogr. 58(3), 221\u2013234 (2011)","journal-title":"Des. Codes Cryptogr."},{"key":"1425_CR36","doi-asserted-by":"crossref","first-page":"1888","DOI":"10.1016\/j.jfranklin.2010.10.007","volume":"347","author":"B Yildiz","year":"2010","unstructured":"Yildiz, B., Karadenniz, S.: Self-dual codes over $${\\mathbb{F}}_2 + u{\\mathbb{F}}_2 + v{\\mathbb{F}}_2 + uv{\\mathbb{F}}_2$$. J. Franklin Inst. 347, 1888\u20131894 (2010)","journal-title":"J. Franklin Inst."},{"issue":"5","key":"1425_CR37","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1007\/s11424-014-3241-3","volume":"27","author":"H Yu","year":"2014","unstructured":"Yu, H., Zhu, S., Kai, X.: $$(1-uv)$$-constacyclic codes over $${\\mathbb{F}}_p + u{\\mathbb{F}}_p + v{\\mathbb{F}}_ + uv{\\mathbb{F}}_p$$. J. Syst. Sci. Complex. 27(5), 811\u2013816 (2014)","journal-title":"J. Syst. Sci. Complex."},{"key":"1425_CR38","first-page":"226","volume":"2","author":"B Yildiz","year":"2014","unstructured":"Yildiz, B., Aydin, N.: On cyclic codes over $${\\mathbb{Z}}_4+u{\\mathbb{Z}}_4$$ and thier $${\\mathbb{Z}}_4$$-images. Int. J. Inf. Coding Theory 2, 226\u2013237 (2014)","journal-title":"Int. J. Inf. Coding Theory"},{"key":"1425_CR39","unstructured":"http:\/\/magma.maths.usyd.edu.au\/magma\/"}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-020-01425-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-020-01425-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-020-01425-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,8,28]],"date-time":"2021-08-28T00:12:24Z","timestamp":1630109544000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-020-01425-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,28]]},"references-count":39,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2021,6]]}},"alternative-id":["1425"],"URL":"https:\/\/doi.org\/10.1007\/s12190-020-01425-5","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,8,28]]},"assertion":[{"value":"6 April 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 August 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 August 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 August 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}