论文标题
Symplectic 4维半场订单$ 8^4 $和$ 9^4 $
Symplectic 4-dimensional semifields of order $8^4$ and $9^4$
论文作者
论文摘要
我们将Sympletic 4维半场对$ \ Mathbb {f} _Q $，$ Q \ leq 9 $进行分类，从而扩展（并确认）以前获得的$ Q \ leq 7 $的分类。分类是通过对$ \ mathrm {pg}（9，q）$中的所有符合性半场子空间进行分类来获得的，以$ q \ leq 9 $最高$ k $ - 等价性，其中$ k \ leq \ leq \ leq \ mathrm {pgl}（pgl}（10，q）$是$ \ mathrm的提升， $ \ mathrm {pg}（3，q）$ in $ \ mathrm {pg}（9，q）二维的$。我们的结果意味着非缔合4维半场不存在$ q $，$ q \ leq 8 $。对于$ q $奇数和$ q \ leq 9 $，我们的结果表明，在$ \ mathbb {f} _q $上的同位素非相关性4维半场的同位素类别包含在迪克森通勤半场的Knuth Orbit中。
We classify symplectic 4-dimensional semifields over $\mathbb{F}_q$, for $q\leq 9$, thereby extending (and confirming) the previously obtained classifications for $q\leq 7$. The classification is obtained by classifying all symplectic semifield subspaces in $\mathrm{PG}(9,q)$ for $q\leq 9$ up to $K$-equivalence, where $K\leq \mathrm{PGL}(10,q)$ is the lift of $\mathrm{PGL}(4,q)$ under the Veronese embedding of $\mathrm{PG}(3,q)$ in $\mathrm{PG}(9,q)$ of degree two. Our results imply the non-existence of non-associative symplectic 4-dimensional semifields for $q$ even, $q\leq 8$. For $q$ odd, and $q\leq 9$, 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.
