Symplectic 4-dimensional semifields of order $$8^4$$ and $$9^4$$

Abstract

AbstractWe classify symplectic 4-dimensional semifields over $$\mathbb {F}_q$$ F q , for $$q\le 9$$ q ≤ 9 , thereby extending (and confirming) the previously obtained classifications for $$q\le 7$$ q ≤ 7 . The classification is obtained by classifying all symplectic semifield subspaces in $$\textrm{PG}(9,q)$$ PG ( 9 , q ) for $$q\le 9$$ q ≤ 9 up to K-equivalence, where $$K\le \textrm{PGL}(10,q)$$ K ≤ PGL ( 10 , q ) is the lift of $$\textrm{PGL}(4,q)$$ PGL ( 4 , q ) under the Veronese embedding of $$\textrm{PG}(3,q)$$ PG ( 3 , q ) in $$\textrm{PG}(9,q)$$ PG ( 9 , q ) of degree two. Our results imply the non-existence of non-associative symplectic 4-dimensional semifields for q even, $$q\le 8$$ q ≤ 8 . For q odd, and $$q\le 9$$ q ≤ 9 , our results imply that the isotopism class of a symplectic non-associative 4-dimensional semifield over $$\mathbb {F}_q$$ F q is contained in the Knuth orbit of a Dickson commutative semifield.