Symplectic 4-dimensional semifields of order $8^4$ and $9^4$
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Publication:6399498
DOI10.1007/S10623-023-01183-YzbMath1529.51004arXiv2205.08995MaRDI QIDQ6399498
Publication date: 18 May 2022
Abstract: We classify symplectic 4-dimensional semifields over $mathbb{F}_q$, for $qleq 9$, thereby extending (and confirming) the previously obtained classifications for $qleq 7$. The classification is obtained by classifying all symplectic semifield subspaces in $mathrm{PG}(9,q)$ for $qleq 9$ up to $K$-equivalence, where $Kleq mathrm{PGL}(10,q)$ 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, $qleq 8$. For $q$ odd, and $qleq 9$, 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.
Spreads and packing problems in finite geometry (51E23) Semifields (12K10) Translation planes and spreads in linear incidence geometry (51A40)
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