{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T02:30:09Z","timestamp":1747189809936,"version":"3.40.5"},"reference-count":29,"publisher":"World Scientific Pub Co Pte Ltd","issue":"03","funder":[{"DOI":"10.13039\/100010718","name":"Absa Bank Limited","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100010718","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2021,6]]},"abstract":"<jats:p>[Formula: see text]-Direct codes are an extension to the class of linear codes having complementary duals (LCD codes). Defined over a finite field [Formula: see text], it is comprised of [Formula: see text] linear codes [Formula: see text], [Formula: see text] with [Formula: see text], where [Formula: see text] is the dual with respect to [Formula: see text]. A 2-Direct code [Formula: see text] with respect to [Formula: see text] is comprised of only LCD codes: [Formula: see text]. On the contrary, two LCD codes do not set up a [Formula: see text]-Direct code in general. This paper presents some important and generalized results on [Formula: see text]-Direct codes, in that it attempts to construct [Formula: see text]-Direct codes from LCD codes. The class of [Formula: see text]-cyclic maximum rank distance (MRD) codes as having complementary duals over [Formula: see text] are generalized. Dual bases such as self-dual basis and self-dual normal basis play a crucial role in constructions. Further, construction implausibility of [Formula: see text]-Direct codes from almost self-dual bases is also dealt. Results obtained are demonstrated through examples.<\/jats:p>","DOI":"10.1142\/s1793830921500671","type":"journal-article","created":{"date-parts":[[2020,12,2]],"date-time":"2020-12-02T09:03:03Z","timestamp":1606899783000},"page":"2150067","source":"Crossref","is-referenced-by-count":2,"title":["Some important results on \ud835\udcaf-Direct codes"],"prefix":"10.1142","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6656-1087","authenticated-orcid":false,"given":"Meenakshi","family":"Devi","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Sol Plaatje University, Private Bag X5008, Kimberley, South Africa"}]},{"given":"R. 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