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Codes Cryptogr."],"published-print":{"date-parts":[[2023,5]]},"abstract":"Abstract<\/jats:title>We classify symplectic 4-dimensional semifields over $$\\mathbb {F}_q$$<\/jats:tex-math>\n \n F<\/mml:mi>\n q<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, for $$q\\le 9$$<\/jats:tex-math>\n \n q<\/mml:mi>\n \u2264<\/mml:mo>\n 9<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, thereby extending (and confirming) the previously obtained classifications for $$q\\le 7$$<\/jats:tex-math>\n \n q<\/mml:mi>\n \u2264<\/mml:mo>\n 7<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The classification is obtained by classifying all symplectic semifield subspaces in $$\\textrm{PG}(9,q)$$<\/jats:tex-math>\n \n PG<\/mml:mtext>\n (<\/mml:mo>\n 9<\/mml:mn>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for $$q\\le 9$$<\/jats:tex-math>\n \n q<\/mml:mi>\n \u2264<\/mml:mo>\n 9<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> up to K<\/jats:italic>-equivalence, where $$K\\le \\textrm{PGL}(10,q)$$<\/jats:tex-math>\n \n K<\/mml:mi>\n \u2264<\/mml:mo>\n PGL<\/mml:mtext>\n (<\/mml:mo>\n 10<\/mml:mn>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is the lift of $$\\textrm{PGL}(4,q)$$<\/jats:tex-math>\n \n PGL<\/mml:mtext>\n (<\/mml:mo>\n 4<\/mml:mn>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> under the Veronese embedding of $$\\textrm{PG}(3,q)$$<\/jats:tex-math>\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><\/jats:alternatives><\/jats:inline-formula> in $$\\textrm{PG}(9,q)$$<\/jats:tex-math>\n \n PG<\/mml:mtext>\n (<\/mml:mo>\n 9<\/mml:mn>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of degree two. Our results imply the non-existence of non-associative symplectic 4-dimensional semifields for q<\/jats:italic> even, $$q\\le 8$$<\/jats:tex-math>\n \n q<\/mml:mi>\n \u2264<\/mml:mo>\n 8<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. For q<\/jats:italic> odd, and $$q\\le 9$$<\/jats:tex-math>\n \n q<\/mml:mi>\n \u2264<\/mml:mo>\n 9<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, our results imply that the isotopism class of a symplectic non-associative 4-dimensional semifield over $$\\mathbb {F}_q$$<\/jats:tex-math>\n \n F<\/mml:mi>\n q<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is contained in the Knuth orbit of a Dickson commutative semifield.<\/jats:p>","DOI":"10.1007\/s10623-023-01183-y","type":"journal-article","created":{"date-parts":[[2023,2,5]],"date-time":"2023-02-05T15:03:10Z","timestamp":1675609390000},"page":"1935-1949","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Symplectic 4-dimensional semifields of order $$8^4$$ and $$9^4$$"],"prefix":"10.1007","volume":"91","author":[{"given":"Michel","family":"Lavrauw","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8590-0301","authenticated-orcid":false,"given":"John","family":"Sheekey","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,2,5]]},"reference":[{"key":"1183_CR1","unstructured":"Bamberg J., Betten A., Cara P., De Beule J., Lavrauw M., Neunhoeffer M.: FinInG\u2014a GAP package for Finite Incidence Geometry, 1.01 (2014). https:\/\/gap-packages.github.io\/FinInG\/."},{"key":"1183_CR2","unstructured":"Betten A., Braun M., Fripertinger H., Kerber A., Kohnert A., Wassermann A.: Error-Correcting Linear Codes. 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