{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T14:33:49Z","timestamp":1775486029437,"version":"3.50.1"},"reference-count":27,"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;'>In this paper, we generalize the notion of self-orthogonal codes to <inline-formula><tex-math id=\"M1\">\\begin{document}$ \\sigma $\\end{document}<\/tex-math><\/inline-formula>-self-orthogonal codes over an arbitrary finite ring. Then, we study the <inline-formula><tex-math id=\"M2\">\\begin{document}$ \\sigma $\\end{document}<\/tex-math><\/inline-formula>-self-orthogonality of constacyclic codes of length <inline-formula><tex-math id=\"M3\">\\begin{document}$ p^s $\\end{document}<\/tex-math><\/inline-formula> over the finite commutative chain ring <inline-formula><tex-math id=\"M4\">\\begin{document}$ \\mathbb F_{p^m} + u \\mathbb F_{p^m} $\\end{document}<\/tex-math><\/inline-formula>, where <inline-formula><tex-math id=\"M5\">\\begin{document}$ p $\\end{document}<\/tex-math><\/inline-formula> is a prime, <inline-formula><tex-math id=\"M6\">\\begin{document}$ u^2 = 0 $\\end{document}<\/tex-math><\/inline-formula> and <inline-formula><tex-math id=\"M7\">\\begin{document}$ \\sigma $\\end{document}<\/tex-math><\/inline-formula> is an arbitrary ring automorphism of <inline-formula><tex-math id=\"M8\">\\begin{document}$ \\mathbb F_{p^m} + u \\mathbb F_{p^m} $\\end{document}<\/tex-math><\/inline-formula>. We characterize the structure of <inline-formula><tex-math id=\"M9\">\\begin{document}$ \\sigma $\\end{document}<\/tex-math><\/inline-formula>-dual code of a <inline-formula><tex-math id=\"M10\">\\begin{document}$ \\lambda $\\end{document}<\/tex-math><\/inline-formula>-constacyclic code of length <inline-formula><tex-math id=\"M11\">\\begin{document}$ p^s $\\end{document}<\/tex-math><\/inline-formula> over <inline-formula><tex-math id=\"M12\">\\begin{document}$ \\mathbb F_{p^m} + u \\mathbb F_{p^m} $\\end{document}<\/tex-math><\/inline-formula>. Further, the necessary and sufficient conditions for a <inline-formula><tex-math id=\"M13\">\\begin{document}$ \\lambda $\\end{document}<\/tex-math><\/inline-formula>-constacyclic code to be <inline-formula><tex-math id=\"M14\">\\begin{document}$ \\sigma $\\end{document}<\/tex-math><\/inline-formula>-self-orthogonal are provided. In particular, we determine all <inline-formula><tex-math id=\"M15\">\\begin{document}$ \\sigma $\\end{document}<\/tex-math><\/inline-formula>-self-dual constacyclic codes of length <inline-formula><tex-math id=\"M16\">\\begin{document}$ p^s $\\end{document}<\/tex-math><\/inline-formula> over <inline-formula><tex-math id=\"M17\">\\begin{document}$ \\mathbb F_{p^m} + u \\mathbb F_{p^m} $\\end{document}<\/tex-math><\/inline-formula>. In the end of this paper, when <inline-formula><tex-math id=\"M18\">\\begin{document}$ p $\\end{document}<\/tex-math><\/inline-formula> is an odd prime, we extend the results to constacyclic codes of length <inline-formula><tex-math id=\"M19\">\\begin{document}$ 2 p^s $\\end{document}<\/tex-math><\/inline-formula>.<\/p><\/jats:p>","DOI":"10.3934\/amc.2020127","type":"journal-article","created":{"date-parts":[[2020,12,18]],"date-time":"2020-12-18T02:04:01Z","timestamp":1608257041000},"page":"643","source":"Crossref","is-referenced-by-count":5,"title":["On $ \\sigma $-self-orthogonal constacyclic codes over $ \\mathbb F_{p^m}+u\\mathbb F_{p^m} $"],"prefix":"10.3934","volume":"16","author":[{"given":"Hongwei","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China"}]},{"given":"Jingge","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China"}]}],"member":"2321","reference":[{"key":"key-10.3934\/amc.2020127-1","doi-asserted-by":"publisher","unstructured":"E. 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