{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T00:53:33Z","timestamp":1761785613843,"version":"build-2065373602"},"reference-count":23,"publisher":"Springer Science and Business Media LLC","issue":"11","license":[{"start":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T00:00:00Z","timestamp":1753833600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T00:00:00Z","timestamp":1753833600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100007465","name":"UiT The Arctic University of Norway","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100007465","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Des. Codes Cryptogr."],"published-print":{"date-parts":[[2025,11]]},"abstract":"Abstract<\/jats:title>\n \n We consider the cyclic presentation of\n \n \n $$\\textrm{PG}(3, q )$$<\/jats:tex-math>\n \n \n PG<\/mml:mtext>\n (<\/mml:mo>\n 3<\/mml:mn>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n whose points are in the finite field\n \n \n $$\\mathbb {F}_{q^4}$$<\/jats:tex-math>\n \n \n F<\/mml:mi>\n \n q<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msup>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and describe the known ovoids therein. We revisit the set\n \n \n $$\\mathcal {O}$$<\/jats:tex-math>\n \n O<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , consisting of\n \n \n $$(q^2+1)$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n \n q<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n +<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n th roots of unity in\n \n \n $$\\mathbb {F}_{q^4}$$<\/jats:tex-math>\n \n \n F<\/mml:mi>\n \n q<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msup>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , and prove that it forms an elliptic quadric within the cyclic presentation of\n \n \n $$\\textrm{PG}(3, q )$$<\/jats:tex-math>\n \n \n PG<\/mml:mtext>\n (<\/mml:mo>\n 3<\/mml:mn>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Additionally, following the work of Glauberman on Suzuki groups, we offer a new description of Suzuki\u2013Tits ovoids in the cyclic presentation of\n \n \n $$\\textrm{PG}(3, q )$$<\/jats:tex-math>\n \n \n PG<\/mml:mtext>\n (<\/mml:mo>\n 3<\/mml:mn>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , characterizing them as the zeroes of a polynomial over\n \n \n $$\\mathbb {F}_{q^4}$$<\/jats:tex-math>\n \n \n F<\/mml:mi>\n \n q<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msup>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n .\n <\/jats:p>","DOI":"10.1007\/s10623-025-01695-9","type":"journal-article","created":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:53:42Z","timestamp":1753890822000},"page":"4765-4778","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Ovoids in the cyclic presentation of $$\\textrm{PG}(3, q )$$"],"prefix":"10.1007","volume":"93","author":[{"given":"Kanat","family":"Abdukhalikov","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Simeon","family":"Ball","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Duy","family":"Ho","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tabriz","family":"Popatia","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,7,30]]},"reference":[{"key":"1695_CR1","doi-asserted-by":"publisher","first-page":"2283","DOI":"10.1007\/s10623-021-00915-2","volume":"89","author":"K Abdukhalikov","year":"2021","unstructured":"Abdukhalikov K., Ho D.: Extended cyclic codes, maximal arcs and ovoids. 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