A variation of the dual hyperoval ${\mathcal {S}}_c$ using presemifields

Abstract

In Discret Math 337:65–75, 2014, we construct a bilinear dual hyperoval called ${\mathcal {S}}_c(l,GF(2^r))$, or simply ${\mathcal {S}}_c$, for $rl\ge 4$ and $c\in GF(2^r)$ with $Tr(c)=1$. In this note, we modify the bilinear mapping of ${\mathcal {S}}_c$ for $l \ge 2$ using multiplications of presemifields, and have a dual hyperoval ${\mathcal {S}}_{c}^{'}$ from this bilinear mapping. We also investigate on the isomorphism problems of these dual hyperovals under the conditions that $c\ne 1$ and the presemifields are not isotopic to commutative presemifields (see Theorem 2 for precise statement), and see that, under these conditions, ${\mathcal {S}}_{c}^{'}$ is not isomorphic to the dual hyperovals in Taniguchi (Discret Math 337:65–75, 2014).