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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si1.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=b9cd6c12cdc7de8b3440a2c5b35bb05e" title="Click to view the MathML source">c∈Sclass="mathContainer hidden">class="mathCode">$c\in S$, we can construct a bilinear dual hyperoval class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si114.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=2836667abfa544cab678be0fdd5360f0" title="Click to view the MathML source">S_c(S)class="mathContainer hidden">class="mathCode">${\mathcal{S}}_c(S)$ if c   satisfies some conditions. Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si3.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=83512e43ee40c97f8887608a40a3c3e2" title="Click to view the MathML source">c₁∈S₁class="mathContainer hidden">class="mathCode">$c_1\in S_1$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si4.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=f384c93f21c6ee5c487c1ddbd0ad036d" title="Click to view the MathML source">c₂∈S₂class="mathContainer hidden">class="mathCode">$c_2\in S_2$ for commutative presemifields class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si168.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=0ba36d835ca170d4be55a29fd399673e" title="Click to view the MathML source">S₁class="mathContainer hidden">class="mathCode">$S_1$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si169.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=fc662ff50a30bb34dd54e5c5ecb5ff10" title="Click to view the MathML source">S₂class="mathContainer hidden">class="mathCode">$S_2$, and assume class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si163.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=7a86fd95980aee222606b00414008ac5" title="Click to view the MathML source">c₁≠1class="mathContainer hidden">class="mathCode">$c_1\ne 1$ or class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si164.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=8e56251ca2505a01cca85d99727a8c02" title="Click to view the MathML source">c₂≠1class="mathContainer hidden">class="mathCode">$c_2\ne 1$. Then the dual hyperovals class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si140.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=0ad178ae40355ae205db3cf52dd09bb4" title="Click to view the MathML source">S_c1(S₁)class="mathContainer hidden">class="mathCode">${\mathcal{S}}_{c_1}(S_1)$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si10.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=63147e0d6866c9971c0401566de726aa" title="Click to view the MathML source">S_c2(S₂)class="mathContainer hidden">class="mathCode">${\mathcal{S}}_{c_2}(S_2)$ are isomorphic if and only if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si168.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=0ba36d835ca170d4be55a29fd399673e" title="Click to view the MathML source">S₁class="mathContainer hidden">class="mathCode">$S_1$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si169.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=fc662ff50a30bb34dd54e5c5ecb5ff10" title="Click to view the MathML source">S₂class="mathContainer hidden">class="mathCode">$S_2$ are isotopic with some relation between class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si11.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=d979741fc7851dec47a89c03bb256071" title="Click to view the MathML source">c₁class="mathContainer hidden">class="mathCode">$c_1$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si12.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=f636ef13a033421b28e9d001c75a7802" title="Click to view the MathML source">c₂class="mathContainer hidden">class="mathCode">$c_2$ induced by the isotopy. For the Kantor commutative presemifield class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si13.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=159f16b66024cb0195507af6f96c5445" title="Click to view the MathML source">S=(F,+,∘)class="mathContainer hidden">class="mathCode">$S=(F,+,\circ )$ with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si14.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=43afe9e4c633fbac8134d47090dce4db" title="Click to view the MathML source">c∈F_n⊂Fclass="mathContainer hidden">class="mathCode">$c\in F_n\subset F$, the dual hyperoval class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si114.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=2836667abfa544cab678be0fdd5360f0" title="Click to view the MathML source">S_c(S)class="mathContainer hidden">class="mathCode">${\mathcal{S}}_c(S)$ exists if and only if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si15.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=09a262c6c2a1643bc1aefb102868ce67" title="Click to view the MathML source">Tr(c)=1class="mathContainer hidden">class="mathCode">$Tr(c)=1$, where Tr   is the absolute trace of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si16.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=9c592aa9b7a96d3d05e10ca5ca4afaaf" title="Click to view the MathML source">F_nclass="mathContainer hidden">class="mathCode">$F_n$. The dual hyperovals class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si140.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=0ad178ae40355ae205db3cf52dd09bb4" title="Click to view the MathML source">S_c1(S₁)class="mathContainer hidden">class="mathCode">${\mathcal{S}}_{c_1}(S_1)$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si10.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=63147e0d6866c9971c0401566de726aa" title="Click to view the MathML source">S_c2(S₂)class="mathContainer hidden">class="mathCode">${\mathcal{S}}_{c_2}(S_2)$ for the Kantor commutative presemifields class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si168.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=0ba36d835ca170d4be55a29fd399673e" title="Click to view the MathML source">S₁class="mathContainer hidden">class="mathCode">$S_1$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si169.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=fc662ff50a30bb34dd54e5c5ecb5ff10" title="Click to view the MathML source">S₂class="mathContainer hidden">class="mathCode">$S_2$ are (under some conditions) isomorphic if and only if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si168.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=0ba36d835ca170d4be55a29fd399673e" title="Click to view the MathML source">S₁class="mathContainer hidden">class="mathCode">$S_1$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si169.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=fc662ff50a30bb34dd54e5c5ecb5ff10" title="Click to view the MathML source">S₂class="mathContainer hidden">class="mathCode">$S_2$ are isotopic with class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300338&_mathId=si18.gif&_user=111111111&_pii=S1071579716300338&_rdoc=1&_issn=10715797&md5=2c22d469f3c23bca790281a8e22f4a4d">class="imgLazyJSB inlineImage" height="15" width="52" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S1071579716300338-si18.gif">class="mathContainer hidden">class="mathCode">$c_1^{\sigma }=c_2$, where σ is the field automorphism of F defined by the isotopy.