{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,8]],"date-time":"2026-03-08T17:16:53Z","timestamp":1772990213562,"version":"3.50.1"},"reference-count":7,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2025,4,21]],"date-time":"2025-04-21T00:00:00Z","timestamp":1745193600000},"content-version":"vor","delay-in-days":110,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2025,1]]},"abstract":"<jats:p>\n                    <jats:italic>t<\/jats:italic>\n                    \u2010(\n                    <jats:italic>v<\/jats:italic>\n                    ,\n                    <jats:italic>k<\/jats:italic>\n                    , \u039b) designs are well\u2010known objects of study. Here, we define\n                    <jats:italic>t<\/jats:italic>\n                    \u2010group divisible designs (\n                    <jats:italic>t<\/jats:italic>\n                    \u2010GDDs) and specifically study\n                    <jats:italic>t<\/jats:italic>\n                    \u2010GDDs, where the number of groups are less than the block size. After a short survey of\n                    <jats:italic>t<\/jats:italic>\n                    \u2010GDDs, we will present some examples of 4\u2010GDDs with block size 5 and two groups and prove that the necessary conditions are sufficient for group sizes 3, 4, 5, 6, 7, 9, and 10. Though the necessary conditions for the existence of\n                    <jats:italic>t<\/jats:italic>\n                    \u2010designs are well known, the actual spectrums for 4\u2010(\n                    <jats:italic>v<\/jats:italic>\n                    , 5, \u039b) and for 4\u2010GDD (\n                    <jats:italic>n<\/jats:italic>\n                    , 2, 5; \u039b\n                    <jats:sub>1<\/jats:sub>\n                    , \u039b\n                    <jats:sub>2<\/jats:sub>\n                    ) are obtained as a ready source for researchers interested in the constructions of such designs.\n                  <\/jats:p>","DOI":"10.1155\/ijmm\/8832205","type":"journal-article","created":{"date-parts":[[2025,4,21]],"date-time":"2025-04-21T05:53:01Z","timestamp":1745214781000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["New Constructions and Necessary Conditions for 4\u2010GDDs With Block Size 5"],"prefix":"10.1155","volume":"2025","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7865-7430","authenticated-orcid":false,"given":"Dinkayehu M.","family":"Woldemariam","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3238-0389","authenticated-orcid":false,"given":"D. G.","family":"Sarvate","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2025,4,21]]},"reference":[{"key":"e_1_2_9_1_2","doi-asserted-by":"publisher","DOI":"10.1006\/jcta.1998.2868"},{"key":"e_1_2_9_2_2","doi-asserted-by":"publisher","DOI":"10.4153\/cjm-1963-069-5"},{"key":"e_1_2_9_3_2","first-page":"79","volume-title":"t-Designs With t\u2009\u2265\u20093, Handbook of Combinatorial Design Theory","author":"Khosrovshahi G. B.","year":"2007"},{"key":"e_1_2_9_4_2","first-page":"50","article-title":"3-GDDs with Block Size 4","volume":"82","author":"Sarvate D. G.","year":"2018","journal-title":"Bulletin of the ICA"},{"key":"e_1_2_9_5_2","first-page":"329","article-title":"A Short Note on 3-GDDs With Three Groups and Block Size 4","volume":"114","author":"Sarvate D. G.","year":"2020","journal-title":"Journal of Combinatorial Mathematics and Combinatorial Computing"},{"key":"e_1_2_9_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(75)90006-0"},{"key":"e_1_2_9_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/s0304-0208(08)73309-7"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/ijmm\/8832205","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1155\/ijmm\/8832205","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/ijmm\/8832205","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,8]],"date-time":"2026-03-08T12:17:29Z","timestamp":1772972249000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/ijmm\/8832205"}},"subtitle":[],"editor":[{"given":"Shikha","family":"Binwal","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2025,1]]},"references-count":7,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,1]]}},"alternative-id":["10.1155\/ijmm\/8832205"],"URL":"https:\/\/doi.org\/10.1155\/ijmm\/8832205","archive":["Portico"],"relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"value":"0161-1712","type":"print"},{"value":"1687-0425","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1]]},"assertion":[{"value":"2023-11-09","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-04-07","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-04-21","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"8832205"}}