{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:45:34Z","timestamp":1760147134189,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,13]],"date-time":"2023-01-13T00:00:00Z","timestamp":1673568000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University","award":["PNURSP2022R231"],"award-info":[{"award-number":["PNURSP2022R231"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, we consider a semi-local ring S=Fq+uFq, where u2=u, q=ps and p is a prime number. We define a multiplication yb=\u03b2(b)y+\u03b3(b), where \u03b2 is an automorphism and \u03b3 is a \u03b2-derivation on S so that S[y;\u03b2,\u03b3] becomes a non-commutative ring which is known as skew polynomial ring. We give the characterization of S[y;\u03b2,\u03b3] and obtain the most striking results that are better than previous findings. We also determine the structural properties of 1-generator skew cyclic and skew-quasi cyclic codes. Further, We demonstrate remarkable results of the above-mentioned codes over S. Finally, we find the duality of skew cyclic and skew-quasi cyclic codes using a symmetric inner product. These codes are further generalized to double skew cyclic and skew quasi cyclic codes and a table of optimal codes is calculated by MAGMA software.<\/jats:p>","DOI":"10.3390\/sym15010225","type":"journal-article","created":{"date-parts":[[2023,1,13]],"date-time":"2023-01-13T02:05:52Z","timestamp":1673575552000},"page":"225","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["(\u03b2,\u03b3)-Skew QC Codes with Derivation over a Semi-Local Ring"],"prefix":"10.3390","volume":"15","author":[{"given":"Mohammad","family":"Ashraf","sequence":"first","affiliation":[{"name":"Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India"}]},{"given":"Amal S.","family":"Alali","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]},{"given":"Mohd","family":"Asim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7765-5668","authenticated-orcid":false,"given":"Ghulam","family":"Mohammad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1109\/18.312154","article-title":"The Z4-linearity of Kerdock, Preparata, Goethals, and related codes","volume":"40","author":"Calderbank","year":"1994","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","first-page":"10","article-title":"Skew cyclic codes of arbitrary length","volume":"2","author":"Siap","year":"2011","journal-title":"Int. J. Inform. Coding Theory"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"480","DOI":"10.2307\/1968173","article-title":"Theory of non-commutative polynomials","volume":"34","author":"Ore","year":"1933","journal-title":"Ann. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/S1385-7258(68)50022-2","article-title":"Skew polynomial rings","volume":"30","author":"Smits","year":"1968","journal-title":"Nederl. Akad. Wetensch. Proc. Ser. A 71 Indag. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2081","DOI":"10.1109\/TIT.2010.2044062","article-title":"On the construction of skew quasi cyclic codes","volume":"56","author":"Abualrub","year":"2010","journal-title":"IEEE. Trans. Inf. Theory"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1007\/s10623-011-9494-0","article-title":"Skew quasi cyclic codes over Galois rings","volume":"62","author":"Bhaintwal","year":"2012","journal-title":"Designs Codes Cryptogr."},{"key":"ref_7","first-page":"29","article-title":"Skew constacyclic codes over finite chain rings","volume":"6","author":"Jitman","year":"2012","journal-title":"Adv. Math. Commun."},{"key":"ref_8","first-page":"463","article-title":"(\u03c3,\u03b4)-codes","volume":"7","author":"Boulagouaz","year":"2013","journal-title":"Adv. Math. Commun."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1007\/s00200-007-0043-z","article-title":"Skew cyclic codes","volume":"18","author":"Boucher","year":"2007","journal-title":"Appl. Algebra Eng. Comm."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1550042","DOI":"10.1142\/S1793830915500421","article-title":"On skew cyclic codes over a semi-local ring","volume":"7","author":"Ashraf","year":"2015","journal-title":"Discrete Math. Algorithms Appl."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Ashraf, M., and Mohammad, G. (2018). Skew cyclic codes over Fq+uFq+vFq. Asian Eur. J. Math., 1850072.","DOI":"10.1142\/S1793557118500729"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"723","DOI":"10.3934\/amc.2018043","article-title":"A class of skew-cyclic codes over Z4+uZ4 with derivation","volume":"12","author":"Sharma","year":"2018","journal-title":"Adv. Math. Commun."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1007\/s12095-020-00467-7","article-title":"(\u03c3,\u03b4)-skew quasi-cyclic codes over the ring Z4+uZ4","volume":"13","author":"Fanghui","year":"2021","journal-title":"Cryptogr. Commun."},{"key":"ref_14","first-page":"218","article-title":"On skew cyclic codes over F3+vF3","volume":"2","author":"Ashraf","year":"2014","journal-title":"Int. J. Inf. Coding Theory"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1007\/s00200-013-0207-y","article-title":"Some new binary quasi-cyclic codes from codes over the ring F2+uF2+vF2+uvF2","volume":"24","author":"Aydin","year":"2013","journal-title":"Appl. Algebra Eng. Commun. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"313","DOI":"10.3934\/amc.2014.8.313","article-title":"Construction of skew cyclic codes over Fq+vFq","volume":"3","author":"Gursoy","year":"2014","journal-title":"Adv. Math. Commun."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1994","DOI":"10.1109\/18.705580","article-title":"Some new binary quasi-cyclic codes","volume":"45","author":"Heijnen","year":"1998","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_18","first-page":"2763","article-title":"(\u03b8,\u03b4\u03b8)-cyclic codes over Fq\n\t\t\t\t[u,v]\/\u2329u2-u,v2-v,uv-uv\u232a","volume":"90","author":"Prakash","year":"2022","journal-title":"Des. Codes Cryptogr."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.ffa.2016.12.002","article-title":"A generalization of quasi-twisted codes: Multi-twisted codes","volume":"45","author":"Aydin","year":"2017","journal-title":"Finite Fields Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"484","DOI":"10.2307\/1968565","article-title":"Pseudo-linear transformations","volume":"38","author":"Jacobson","year":"1937","journal-title":"Ann. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"321","DOI":"10.36045\/bbms\/1103408724","article-title":"Pseudo-linear transformation and evaluation in Ore extension","volume":"2","author":"Leroy","year":"1995","journal-title":"Bull. Belg. Math. Soc. Simon Stevin"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1007\/s10623-016-0311-7","article-title":"Some properties of skew codes over finite fields","volume":"85","author":"Cuitino","year":"2017","journal-title":"Des. Codes Cryptogr."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"35","DOI":"10.32513\/tbilisi\/1529460020","article-title":"On skew cyclic codes over Fq+vFq+v2Fq","volume":"11","author":"Ashraf","year":"2018","journal-title":"Tbilisi Math. J."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/225\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:04:52Z","timestamp":1760119492000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/225"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,13]]},"references-count":23,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["sym15010225"],"URL":"https:\/\/doi.org\/10.3390\/sym15010225","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,1,13]]}}}