We classify symplectic 4-dimensional semifields over \(\mathbb {F}_q\), for \(q\le 9\), thereby extending (and confirming) the previously obtained classifications for \(q\le 7\). The classification is obtained by classifying all symplectic semifield subspaces in \(\textrm{PG}(9,q)\) for \(q\le 9\) up to K-equivalence, where \(K\le \textrm{PGL}(10,q)\) 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, \(q\le 8\). For q odd, and \(q\le 9\), 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.

我们在\(\mathbb {F}_q\)上对\(q\le 9\)的辛 4 维半域进行分类，从而扩展（并确认）先前获得的\(q\le 7\)分类。通过对\(\textrm{PG}(9,q)\)中的所有辛半域子空间进行分类以获得\(q\le 9\)直到K -等价，其中\(K\le \textrm{PGL }(10,q)\)是\(\textrm{PGL}(4,q)\)在\(\textrm{PG}(3,q)\)在\(\textrm {PG}(9,q)\)二阶。我们的结果暗示不存在非关联辛 4 维半场q偶数，\(q\le 8\)。对于q odd 和\(q\le 9\)，我们的结果表明\(\mathbb {F}_q\)上的辛非关联 4 维半域的同位素类包含在 a 的 Knuth 轨道中Dickson 交换半域。