On some bilinear dual hyperovals

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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：Dual hyperoval ; Buratti&ndash ; Del Fra dual hyperoval ; Huybrechts dual hyperoval ; APN dual hyperoval
• 刊名：Discrete Mathematics
• 出版年：2017
• 出版时间：6 January 2017
• 年：2017
• 卷：340
• 期：1
• 页码：3154-3166
• 全文大小：512 K

文摘

It is shown in Yoshiara (2004) that, if d$d$-dimensional dual hyperovals exist in V(n,2)$V(n,2)$ (GF(2)$GF\left(2\right)$-vector space of rank n$n$), then 2d+1≤n≤(d+1)(d+2)/2+2$2d+1\le n\le (d+1)(d+2)/2+2$, and conjectured that n≤(d+1)(d+2)/2$n\le (d+1)(d+2)/2$. Known bilinear dual hyperovals in V((d+1)(d+2)/2,2)$V((d+1)(d+2)/2,2)$ are the Huybrechts dual hyperoval and the Buratti–Del Fra dual hyperoval. In this paper, we investigate on the covering map $\pi :{\mathcal{S}}_c^{\prime }(l^{\prime },GF\left(2^{r^{\prime }}\right))\to {\mathcal{S}}_c(l,GF\left(2^r\right))$, where the dual hyperovals ${\mathcal{S}}_c^{\prime }(l^{\prime },GF\left(2^{r^{\prime }}\right))$ and S_c(l,GF(2^r))${\mathcal{S}}_c(l,GF\left(2^r\right))$ are constructed in Taniguchi (2014). Using the result, we show that the Buratti–Del Fra dual hyperoval has a bilinear quotient in V(2d+1,2)$V(2d+1,2)$ if d$d$ is odd. On the other hand, we show that the Huybrechts dual hyperoval has no bilinear quotient in V(2d+1,2)$V(2d+1,2)$. We also determine the automorphism group of S_c(l,GF(2^r))${\mathcal{S}}_c(l,GF\left(2^r\right))$, and show that Aut(S_c(l₂,GF(2^rl₁)))<Aut(S_c(l,GF(2^r)))$Aut\left({\mathcal{S}}_c(l_2,GF\left(2^{r{l}_1}\right))\right)<Aut\left({\mathcal{S}}_c(l,GF\left(2^r\right))\right)$.