On some bilinear dual hyperovals

详细信息    查看全文

• 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si1.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=687fe321ae4845dff9bb032121e41e88" title="Click to view the MathML source">dclass="mathContainer hidden">class="mathCode">$d$-dimensional dual hyperovals exist in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si2.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=0a29779ee1747cd881f97a29f97ddfaf" title="Click to view the MathML source">V(n,2)class="mathContainer hidden">class="mathCode">$V(n,2)$ (class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si3.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=b10b0dc58a4898bafb619f37e4e3d0f3" title="Click to view the MathML source">GF(2)class="mathContainer hidden">class="mathCode">$GF\left(2\right)$-vector space of rank class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si4.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=ab1d1dffc44cdf1e158e52ed78437bec" title="Click to view the MathML source">nclass="mathContainer hidden">class="mathCode">$n$), then class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si5.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=4be7c5f9c6b29bcaab75975a001ebfb7" title="Click to view the MathML source">2d+1≤n≤(d+1)(d+2)/2+2class="mathContainer hidden">class="mathCode">$2d+1\le n\le (d+1)(d+2)/2+2$, and conjectured that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si6.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=8ae76d592c14d3110c907f389a8ab28f" title="Click to view the MathML source">n≤(d+1)(d+2)/2class="mathContainer hidden">class="mathCode">$n\le (d+1)(d+2)/2$. Known bilinear dual hyperovals in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si7.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=4ce7d68b3bc049e382e75a5e0846a6d7" title="Click to view the MathML source">V((d+1)(d+2)/2,2)class="mathContainer hidden">class="mathCode">$V((d+1)(d+2)/2,2)$ are the Huybrechts dual hyperoval and the Buratti–Del Fra dual hyperoval. In this paper, we investigate on the covering map class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si8.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=4015b4508e4663bfbd68105409b7864a">class="imgLazyJSB inlineImage" height="21" width="228" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0012365X16302230-si8.gif">class="mathContainer hidden">class="mathCode">$\pi :{\mathcal{S}}_c^{\prime }(l^{\prime },GF\left(2^{r^{\prime }}\right))\to {\mathcal{S}}_c(l,GF\left(2^r\right))$, where the dual hyperovals class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si9.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=0917a484dabe48e41d51234d4cb58d26">class="imgLazyJSB inlineImage" height="21" width="92" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0012365X16302230-si9.gif">class="mathContainer hidden">class="mathCode">${\mathcal{S}}_c^{\prime }(l^{\prime },GF\left(2^{r^{\prime }}\right))$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si10.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=67f8c3dcb50cc3daa6b8dc1fdb3a3a7c" title="Click to view the MathML source">S_c(l,GF(2^r))class="mathContainer hidden">class="mathCode">${\mathcal{S}}_c(l,GF\left(2^r\right))$ are constructed in Taniguchi (2014). Using the result, we show that the Buratti–Del Fra dual hyperoval has a bilinear quotient in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si11.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=f9ed7f106d4206deb81a4274af57f214" title="Click to view the MathML source">V(2d+1,2)class="mathContainer hidden">class="mathCode">$V(2d+1,2)$ if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si1.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=687fe321ae4845dff9bb032121e41e88" title="Click to view the MathML source">dclass="mathContainer hidden">class="mathCode">$d$ is odd. On the other hand, we show that the Huybrechts dual hyperoval has no bilinear quotient in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si11.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=f9ed7f106d4206deb81a4274af57f214" title="Click to view the MathML source">V(2d+1,2)class="mathContainer hidden">class="mathCode">$V(2d+1,2)$. We also determine the automorphism group of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si10.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=67f8c3dcb50cc3daa6b8dc1fdb3a3a7c" title="Click to view the MathML source">S_c(l,GF(2^r))class="mathContainer hidden">class="mathCode">${\mathcal{S}}_c(l,GF\left(2^r\right))$, and show that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si15.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=6f7f4805193c4fda037b5db965767da9" title="Click to view the MathML source">Aut(S_c(l₂,GF(2^rl₁)))<Aut(S_c(l,GF(2^r)))class="mathContainer hidden">class="mathCode">$Aut\left({\mathcal{S}}_c(l_2,GF\left(2^{r{l}_1}\right))\right)<Aut\left({\mathcal{S}}_c(l,GF\left(2^r\right))\right)$.