{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T21:56:09Z","timestamp":1768686969876,"version":"3.49.0"},"reference-count":19,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2024,3,15]],"date-time":"2024-03-15T00:00:00Z","timestamp":1710460800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,3,15]],"date-time":"2024-03-15T00:00:00Z","timestamp":1710460800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Agencia Espa\u00f1ola de Investigaci\u00f3n","award":["TED2021-130358B-I00"],"award-info":[{"award-number":["TED2021-130358B-I00"]}]},{"name":"Fundaci\u00f3n Banco de la Rep\u00fablica","award":["Project 4649"],"award-info":[{"award-number":["Project 4649"]}]},{"name":"T\u00dcB\u0130TAK","award":["1059B192101164"],"award-info":[{"award-number":["1059B192101164"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Cryptogr. Commun."],"published-print":{"date-parts":[[2024,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper studies Abelian and consta-Abelian polyadic codes over rings defined as affine algebras over chain rings. For this purpose, we use the classical construction via splittings and multipliers of the underlying Abelian group. We also derive some results on the structure of the associated polyadic codes and the number of codes under these conditions.<\/jats:p>","DOI":"10.1007\/s12095-024-00707-0","type":"journal-article","created":{"date-parts":[[2024,3,15]],"date-time":"2024-03-15T09:02:29Z","timestamp":1710493349000},"page":"889-907","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Abelian and consta-Abelian polyadic codes over affine algebras with a finite commutative chain coefficient ring"],"prefix":"10.1007","volume":"16","author":[{"given":"G\u00fcls\u00fcm G\u00f6zde","family":"Y\u0131lmazg\u00fc\u00e7","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Javier","family":"de la Cruz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Edgar","family":"Mart\u00ednez-Moro","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,3,15]]},"reference":[{"key":"707_CR1","doi-asserted-by":"publisher","unstructured":"Brualdi, R.A., Pless, V.S.: Polyadic codes. vol. 25, pp. 3\u201317 (1989). https:\/\/doi.org\/10.1016\/0166-218X(89)90042-5. Combinatorics and complexity (Chicago, IL, 1987)","DOI":"10.1016\/0166-218X(89)90042-5"},{"key":"707_CR2","doi-asserted-by":"publisher","unstructured":"Chen, B., Dinh, H.Q., Fan, Y., Ling, S.: Polyadic constacyclic codes. IEEE Trans. Inform. Theory 61(9), 4895\u20134904 (2015). https:\/\/doi.org\/10.1109\/TIT.2015.2451656. ISSN 0018-9448","DOI":"10.1109\/TIT.2015.2451656"},{"key":"707_CR3","doi-asserted-by":"publisher","unstructured":"Dinh, H.Q., L\u00f3pez-Permouth, S.R.: Cyclic and negacyclic codes over finite chain rings. IEEE Trans. Inf. Theory 50(8), 1728\u20131744 (2004). https:\/\/doi.org\/10.1109\/TIT.2004.831789. ISSN 0018-9448","DOI":"10.1109\/TIT.2004.831789"},{"key":"707_CR4","doi-asserted-by":"publisher","unstructured":"Goyal, M., Raka, M.: Quadratic residue codes over the ring $$\\mathbb{F}_{p}[u]\/\\langle u^m-u\\rangle $$ and their Gray images. Cryptogr. Commun. 10(2), 343\u2013355 (2018). https:\/\/doi.org\/10.1007\/s12095-017-0223-z. ISSN 1936-2447,1936-2455","DOI":"10.1007\/s12095-017-0223-z"},{"key":"707_CR5","doi-asserted-by":"publisher","unstructured":"Goyal, M., Raka, M.: Duadic and triadic codes over a finite non-chain ring and their Gray images. Int. J. Inf. Coding Theory 5(1), 36\u201354 (2018). https:\/\/doi.org\/10.1504\/ijicot.2018.091834. ISSN 1753-7703,1753-7711","DOI":"10.1504\/ijicot.2018.091834"},{"key":"707_CR6","doi-asserted-by":"publisher","unstructured":"Goyal, M., Raka, M.: Polyadic constacyclic codes over a non-chain ring $$\\mathbb{F} _q[u, v]\/\\langle f(u), g(v), uv-vu\\rangle $$. J. Appl. Math. Comput. 62(1\u20132), 425\u2013447 (2020). https:\/\/doi.org\/10.1007\/s12190-019-01290-x. ISSN 1598-5865","DOI":"10.1007\/s12190-019-01290-x"},{"key":"707_CR7","doi-asserted-by":"publisher","DOI":"10.4236\/jcc.2021.95004","author":"M Goyal","year":"2021","unstructured":"Goyal, M., Raka, M.: Polyadic cyclic codes over a non-chain ring. J. Comput. Commun. (2021). https:\/\/doi.org\/10.4236\/jcc.2021.95004","journal-title":"J. Comput. Commun."},{"key":"707_CR8","doi-asserted-by":"publisher","unstructured":"Islam, H., Mart\u00ednez-Moro, E., Prakash, O.: Cyclic codes over a non-chain ring $$R_{e, q}$$ and their application to LCD codes. Discrete Math. 344(10), 11 (2021). https:\/\/doi.org\/10.1016\/j.disc.2021.112545. Paper No. 112545, ISSN 0012-365X","DOI":"10.1016\/j.disc.2021.112545"},{"key":"707_CR9","doi-asserted-by":"publisher","unstructured":"K\u00fcr\u00fcz, F., Sari, M., K\u00f6roglu, M.E.: $$m$$-adic residue codes over the ring $$\\mathbb{F}_q[v]\/(v^s-v)$$ and their applications to quantum codes. Quantum Inf. Comput. 22(5\u20136), 427\u2013439 (2022). https:\/\/doi.org\/10.26421\/qic22.5-6-4. ISSN 1533-7146","DOI":"10.26421\/qic22.5-6-4"},{"key":"707_CR10","doi-asserted-by":"publisher","unstructured":"Lim, C.J.: Consta-abelian polyadic codes. IEEE Trans. Inform. Theory 51(6), 2198\u20132206 (2005). https:\/\/doi.org\/10.1109\/TIT.2005.847734. ISSN 0018-9448","DOI":"10.1109\/TIT.2005.847734"},{"key":"707_CR11","doi-asserted-by":"publisher","unstructured":"Lim, C.J.: Consta-abelian polyadic codes. IEEE Trans. Inform. Theory 51(6), 2198\u20132206 (2005). https:\/\/doi.org\/10.1109\/TIT.2005.847734. ISSN 0018-9448","DOI":"10.1109\/TIT.2005.847734"},{"key":"707_CR12","doi-asserted-by":"publisher","unstructured":"Ling, S., Xing, C.: Polyadic codes revisited. IEEE Trans. Inform. Theory, 50 (1): 200\u2013207, 2004. https:\/\/doi.org\/10.1109\/TIT.2003.821986. ISSN 0018-9448","DOI":"10.1109\/TIT.2003.821986"},{"key":"707_CR13","doi-asserted-by":"publisher","unstructured":"Mart\u00ednez-Moro, E., R\u00faa, I.F.: Multivariable codes over finite chain rings: serial codes. SIAM J. Discrete Math. 20(4), 947\u2013959 (2006). https:\/\/doi.org\/10.1137\/050632208. ISSN 0895-4801","DOI":"10.1137\/050632208"},{"key":"707_CR14","doi-asserted-by":"publisher","unstructured":"Mart\u00ednez-Moro, E., R\u00faa, I.F.: On repeated-root multivariable codes over a finite chain ring. Des. Codes Cryptogr. 45(2), 219\u2013227 (2007). https:\/\/doi.org\/10.1007\/s10623-007-9114-1. ISSN 0925-1022","DOI":"10.1007\/s10623-007-9114-1"},{"key":"707_CR15","doi-asserted-by":"publisher","unstructured":"Mart\u00ednez-Moro, E., Pi\u00f1era Nicol\u00e1s, A., R\u00faa, I.F.: Multivariable codes in principal ideal polynomial quotient rings with applications to additive modular bivariate codes over $$\\mathbb{F}_4$$. J. Pure Appl. Algebra 222(2), 359\u2013367 (2018). https:\/\/doi.org\/10.1016\/j.jpaa.2017.04.007. ISSN 0022-4049","DOI":"10.1016\/j.jpaa.2017.04.007"},{"key":"707_CR16","doi-asserted-by":"publisher","unstructured":"Mart\u00ednez-Moro, E., Pi\u00f1era Nicol\u00e1s, A., R\u00faa, I.F.: Codes over affine algebras with a finite commutative chain coefficient ring. Finite Fields Appl. 49, 94\u2013107 (2018). https:\/\/doi.org\/10.1016\/j.ffa.2017.09.008. ISSN 1071-5797","DOI":"10.1016\/j.ffa.2017.09.008"},{"key":"707_CR17","unstructured":"McDonald, B.R.: Finite rings with identity. Pure and Applied Mathematics, vol. 28. Marcel Dekker, Inc., New York (1974)"},{"key":"707_CR18","doi-asserted-by":"publisher","unstructured":"Norton, G.H., S\u0103l\u0103gean, A.: On the structure of linear and cyclic codes over a finite chain ring. Appl. Algebra Engrg. Comm. Comput. 10(6), 489\u2013506 (2000). https:\/\/doi.org\/10.1007\/PL00012382. ISSN 0938-1279","DOI":"10.1007\/PL00012382"},{"key":"707_CR19","doi-asserted-by":"publisher","unstructured":"Puninski, G.: Serial rings. Kluwer Academic Publishers, Dordrecht (2001). https:\/\/doi.org\/10.1007\/978-94-010-0652-1. isbn:0-7923-7187-9","DOI":"10.1007\/978-94-010-0652-1"}],"container-title":["Cryptography and Communications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-024-00707-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12095-024-00707-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-024-00707-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,18]],"date-time":"2024-06-18T09:45:52Z","timestamp":1718703952000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12095-024-00707-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,15]]},"references-count":19,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2024,7]]}},"alternative-id":["707"],"URL":"https:\/\/doi.org\/10.1007\/s12095-024-00707-0","relation":{},"ISSN":["1936-2447","1936-2455"],"issn-type":[{"value":"1936-2447","type":"print"},{"value":"1936-2455","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,3,15]]},"assertion":[{"value":"10 October 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 March 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 March 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declaration"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"The authors declare no competing interests.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}