Bilinear dual hyperovals from binary commutative presemifields

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• 作者：Hiroaki Taniguchi ^{taniguchi@t.kagawa-nct.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：05B25 ; 51A45 ; 51E20 ; 51E21
• 刊名：Finite Fields and Their Applications
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：42
• 期：Complete
• 页码：93-101
• 全文大小：305 K

文摘

For a binary commutative presemifield S   with an element c∈S$c\in S$, we can construct a bilinear dual hyperoval S_c(S)${\mathcal{S}}_c(S)$ if c   satisfies some conditions. Let c₁∈S₁$c_1\in S_1$ and c₂∈S₂$c_2\in S_2$ for commutative presemifields S₁$S_1$ and S₂$S_2$, and assume c₁≠1$c_1\ne 1$ or c₂≠1$c_2\ne 1$. Then the dual hyperovals S_c1(S₁)${\mathcal{S}}_{c_1}(S_1)$ and S_c2(S₂)${\mathcal{S}}_{c_2}(S_2)$ are isomorphic if and only if S₁$S_1$ and S₂$S_2$ are isotopic with some relation between c₁$c_1$ and c₂$c_2$ induced by the isotopy. For the Kantor commutative presemifield S=(F,+,∘)$S=(F,+,\circ )$ with c∈F_n⊂F$c\in F_n\subset F$, the dual hyperoval S_c(S)${\mathcal{S}}_c(S)$ exists if and only if Tr(c)=1$Tr(c)=1$, where Tr   is the absolute trace of F_n$F_n$. The dual hyperovals S_c1(S₁)${\mathcal{S}}_{c_1}(S_1)$ and S_c2(S₂)${\mathcal{S}}_{c_2}(S_2)$ for the Kantor commutative presemifields S₁$S_1$ and S₂$S_2$ are (under some conditions) isomorphic if and only if S₁$S_1$ and S₂$S_2$ are isotopic with $c_1^{\sigma }=c_2$, where σ is the field automorphism of F defined by the isotopy.