{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,23]],"date-time":"2025-10-23T11:10:41Z","timestamp":1761217841587,"version":"3.37.3"},"reference-count":20,"publisher":"World Scientific Pub Co Pte Ltd","issue":"01","funder":[{"DOI":"10.13039\/501100004541","name":"Ministry of Human Resource Development (IN)","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004541","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2016,3]]},"abstract":"<jats:p> In this paper, we have studied cyclic codes over the ring [Formula: see text], [Formula: see text]. We have provided the general form of the generators of a cyclic code over [Formula: see text] and obtained a minimal spanning set for such codes and determined their ranks. We have determined a necessary condition and a sufficient condition for cyclic codes over [Formula: see text] to be [Formula: see text]-free. For [Formula: see text], we have shown that [Formula: see text] is a local ring, and the complete ideal structure of [Formula: see text] is determined. Some examples are presented. <\/jats:p>","DOI":"10.1142\/s1793830916500178","type":"journal-article","created":{"date-parts":[[2016,1,4]],"date-time":"2016-01-04T03:30:53Z","timestamp":1451878253000},"page":"1650017","source":"Crossref","is-referenced-by-count":7,"title":["A note on cyclic codes over \u21244 + u\u21244"],"prefix":"10.1142","volume":"08","author":[{"given":"Rama Krishna","family":"Bandi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Maheshanand","family":"Bhaintwal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2016,2,26]]},"reference":[{"key":"S1793830916500178BIB001","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-218X(02)00432-8"},{"key":"S1793830916500178BIB002","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2003.815763"},{"key":"S1793830916500178BIB003","doi-asserted-by":"publisher","DOI":"10.1007\/s10623-006-9034-5"},{"key":"S1793830916500178BIB004","doi-asserted-by":"publisher","DOI":"10.1007\/s00200-009-0110-8"},{"key":"S1793830916500178BIB005","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-218X(02)00434-1"},{"key":"S1793830916500178BIB006","doi-asserted-by":"publisher","DOI":"10.1109\/18.761278"},{"key":"S1793830916500178BIB007","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2009.2013015"},{"key":"S1793830916500178BIB008","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2004.831789"},{"key":"S1793830916500178BIB009","doi-asserted-by":"publisher","DOI":"10.1007\/s10623-005-2773-x"},{"key":"S1793830916500178BIB010","doi-asserted-by":"publisher","DOI":"10.1109\/18.312154"},{"key":"S1793830916500178BIB011","doi-asserted-by":"publisher","DOI":"10.1006\/ffta.1997.0189"},{"key":"S1793830916500178BIB012","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2015.01.005"},{"volume-title":"The Theory of Error Correcting Codes","year":"1977","author":"MacWilliams F. 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