{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T09:58:37Z","timestamp":1772013517117,"version":"3.50.1"},"reference-count":25,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2026,2,22]],"date-time":"2026-02-22T00:00:00Z","timestamp":1771718400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ongoing Research Funding program","award":["ORF-2026-871"],"award-info":[{"award-number":["ORF-2026-871"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Let p be a prime with p\u2209{2,5} and let q=pm. This paper studies cyclic and self-orthogonal linear codes of length n over the finite local non-Frobenius ring Rp,u,v=Fq+uFq+vFq,\u00a0u2=v2=uv=vu=0. We define an Fq-linear Gray map \u03c0n:Rp,u,vn\u2192Fq6n and investigate the structural properties of Gray images of cyclic codes under this map. It is shown that \u03c0n preserves self-orthogonality and, when gcd(n,p)=1, transforms any cyclic code over Rp,u,v into a quasi-cyclic code over Fq of length 6n with index dividing 6. Moreover, we completely characterize the possible quasi-cyclic indices of the Gray images, proving that only the values l\u2208{1,3,6} can occur, and we establish necessary and sufficient conditions for each case in terms of the generators of the associated cyclic code. Several explicit examples are provided to illustrate the theoretical results and the resulting quasi-cyclic structures.<\/jats:p>","DOI":"10.3390\/e28020250","type":"journal-article","created":{"date-parts":[[2026,2,23]],"date-time":"2026-02-23T10:00:52Z","timestamp":1771840852000},"page":"250","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Gray Images of Cyclic and Self-Orthogonal Codes over Fq+uFq+vFq"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6824-6985","authenticated-orcid":false,"given":"Sami H.","family":"Saif","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-2520-2699","authenticated-orcid":false,"given":"Alhanouf Ali","family":"Alhomaidhi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,2,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1016\/S0019-9958(72)90223-9","article-title":"Codes over certain rings","volume":"20","author":"Blake","year":"1972","journal-title":"Inf. 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