{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T12:35:41Z","timestamp":1771072541021,"version":"3.50.1"},"reference-count":53,"publisher":"American Institute of Mathematical Sciences (AIMS)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["AMC"],"published-print":{"date-parts":[[2022]]},"abstract":"<p style='text-indent:20px;'>For any odd prime <inline-formula><tex-math id=\"M3\">\\begin{document}$ p $\\end{document}<\/tex-math><\/inline-formula>, the structures and duals of <inline-formula><tex-math id=\"M4\">\\begin{document}$ \\lambda $\\end{document}<\/tex-math><\/inline-formula>-constacyclic codes of length <inline-formula><tex-math id=\"M5\">\\begin{document}$ 8p^s $\\end{document}<\/tex-math><\/inline-formula> over <inline-formula><tex-math id=\"M6\">\\begin{document}$ \\mathcal R = \\mathbb F_{p^m}+u\\mathbb F_{p^m} $\\end{document}<\/tex-math><\/inline-formula> are completely determined for all unit <inline-formula><tex-math id=\"M7\">\\begin{document}$ \\lambda $\\end{document}<\/tex-math><\/inline-formula> of the form <inline-formula><tex-math id=\"M8\">\\begin{document}$ \\lambda = \\xi^l\\in \\mathbb F_{p^m} $\\end{document}<\/tex-math><\/inline-formula>, where <inline-formula><tex-math id=\"M9\">\\begin{document}$ l $\\end{document}<\/tex-math><\/inline-formula> is even. In addition, the algebraic structures of all cyclic and negacyclic codes of length <inline-formula><tex-math id=\"M10\">\\begin{document}$ 8p^s $\\end{document}<\/tex-math><\/inline-formula> over <inline-formula><tex-math id=\"M11\">\\begin{document}$ \\mathcal R $\\end{document}<\/tex-math><\/inline-formula> are established in term of their generator polynomials. Dual codes of all cyclic and negacyclic codes of length <inline-formula><tex-math id=\"M12\">\\begin{document}$ 8p^s $\\end{document}<\/tex-math><\/inline-formula> over <inline-formula><tex-math id=\"M13\">\\begin{document}$ \\mathcal R $\\end{document}<\/tex-math><\/inline-formula> are also investigated. Furthermore, we give the number of codewords in each of those cyclic and negacyclic codes. We also obtain the number of cyclic and negacyclic codes of length <inline-formula><tex-math id=\"M14\">\\begin{document}$ 8p^s $\\end{document}<\/tex-math><\/inline-formula> over <inline-formula><tex-math id=\"M15\">\\begin{document}$ \\mathcal R $\\end{document}<\/tex-math><\/inline-formula>.<\/p><\/jats:p>","DOI":"10.3934\/amc.2020123","type":"journal-article","created":{"date-parts":[[2021,1,8]],"date-time":"2021-01-08T12:07:06Z","timestamp":1610107626000},"page":"525","source":"Crossref","is-referenced-by-count":2,"title":["Constacyclic codes of length $ 8p^s $ over $ \\mathbb F_{p^m} + u\\mathbb F_{p^m} $"],"prefix":"10.3934","volume":"16","author":[{"given":"Hai Q.","family":"Dinh","sequence":"first","affiliation":[{"name":"Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam"},{"name":"Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam"}]},{"given":"Bac T.","family":"Nguyen","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences, Thai Nguyen University of Economics and Business Administration, Thai Nguyen province, Vietnam"}]},{"given":"Paravee","family":"Maneejuk","sequence":"additional","affiliation":[{"name":"Centre of Excellence in Econometrics, Faculty of Economics, Chiang Mai University, Thailand"}]}],"member":"2321","reference":[{"key":"key-10.3934\/amc.2020123-1","doi-asserted-by":"publisher","unstructured":"T. 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