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

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