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

Abstract

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