Symplectic 4-dimensional semifields of order 8(4) and 9(4)

We classify symplectic 4-dimensional semifields over Fq, 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 Fq is contained in the Knuth orbit of a Dickson commutative semifield