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

Lavrauw, Michel and Sheekey, John (2023) Symplectic 4-dimensional semifields of order 8(4) and 9(4). Designs, Codes, and Cryptography, 91 (5). pp. 1935-1949. ISSN 0925-1022 (Print) 1573-7586 (Online)

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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.
Item Type: Article
Uncontrolled Keywords: Commutative; Semifield; Symplectic; Veronese variety
Divisions: Faculty of Engineering and Natural Sciences
Depositing User: Michel Lavrauw
Date Deposited: 07 Sep 2023 16:42
Last Modified: 07 Sep 2023 16:42

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