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

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