{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,27]],"date-time":"2025-06-27T22:40:02Z","timestamp":1751064002323,"version":"3.41.0"},"reference-count":0,"publisher":"Combinatorial Press","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ars Comb."],"published-print":{"date-parts":[[2025,6,28]]},"abstract":"<jats:p>&lt;p&gt;We consider a ring &lt;span class=\"math inline\"&gt;\\(R_{u^3} = \\mathbb{F}_2+u\\mathbb{F}_2+u^2\\mathbb{F}_2+u^3\\mathbb{F}_2, u^4=0\\)&lt;\/span&gt;. We discuss the structure of self-dual codes, Type I codes and Type II codes over the ring &lt;span class=\"math inline\"&gt;\\(R_{u^3}\\)&lt;\/span&gt; in terms of the structure of their Torsion and Residue codes. We construct Type I and Type II optimal codes over the ring &lt;span class=\"math inline\"&gt;\\(R_{u^3}\\)&lt;\/span&gt; for different lengths.&lt;\/p&gt;<\/jats:p>","DOI":"10.61091\/ars163-08","type":"journal-article","created":{"date-parts":[[2025,6,27]],"date-time":"2025-06-27T22:01:43Z","timestamp":1751061703000},"page":"113-121","source":"Crossref","is-referenced-by-count":0,"title":["Type I and type II codes over the ring \\({\\mathbb F}_2+u{\\mathbb F}_2+u^2{\\mathbb F}_2+u^3{\\mathbb F}_2\\)"],"prefix":"10.61091","volume":"163","author":[{"name":"Department of Mathematics, VIT-AP University Amaravati, A.P. India 2Department of Mathematics (SoT) PDEU Gandhinagar, Gujarat, India","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ankur","family":"Singh","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"R.","family":"Kumar","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"name":"Department of Mathematics, Amity University, Gwalior, Madhya Pradesh, India","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"39747","published-online":{"date-parts":[[2025,6,28]]},"container-title":["Ars Combinatoria"],"original-title":[],"deposited":{"date-parts":[[2025,6,27]],"date-time":"2025-06-27T22:01:44Z","timestamp":1751061704000},"score":1,"resource":{"primary":{"URL":"https:\/\/combinatorialpress.com\/ars-articles\/volume-163\/type-i-and-type-ii-codes-over-the-ring-mathbb-f_2umathbb-f_2u2mathbb-f_2u3mathbb-f_2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,28]]},"references-count":0,"URL":"https:\/\/doi.org\/10.61091\/ars163-08","relation":{},"ISSN":["0381-7032","2817-5204"],"issn-type":[{"value":"0381-7032","type":"print"},{"value":"2817-5204","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,28]]}}}