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 an id="mmlsi1" class="mathmlsrc">an 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">dan>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">dath>an>an>an>-dimensional dual hyperovals exist in an id="mmlsi2" class="mathmlsrc">an 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)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si2.gif" overflow="scroll">Vrow>(n,2)row>ath>an>an>an> (an id="mmlsi3" class="mathmlsrc">an 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)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si3.gif" overflow="scroll">GFrow>(2)row>ath>an>an>an>-vector space of rank an id="mmlsi4" class="mathmlsrc">an 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">nan>an class="mathContainer hidden">an class="mathCode">ath altimg="si4.gif" overflow="scroll">nath>an>an>an>), then an id="mmlsi5" class="mathmlsrc">an 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+2an>an class="mathContainer hidden">an class="mathCode">ath altimg="si5.gif" overflow="scroll">2d+1≤n≤row>(d+1)row>row>(d+2)row>/2+2ath>an>an>an>, and conjectured that an id="mmlsi6" class="mathmlsrc">an 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)/2an>an class="mathContainer hidden">an class="mathCode">ath altimg="si6.gif" overflow="scroll">n≤row>(d+1)row>row>(d+2)row>/2ath>an>an>an>. Known bilinear dual hyperovals in an id="mmlsi7" class="mathmlsrc">an 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)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si7.gif" overflow="scroll">Vrow>(row>(d+1)row>row>(d+2)row>/2,2)row>ath>an>an>an> are the Huybrechts dual hyperoval and the Buratti&ndash;Del Fra dual hyperoval. In this paper, we investigate on the covering map an id="mmlsi8" class="mathmlsrc"><a 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">ass="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">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si8.gif" overflow="scroll">π:row>athvariant="script">Srow>row>crow>row>′row>row>(row>lrow>row>′row>,GFrow>(row>2row>row>row>rrow>row>′row>row>)row>)row>&rarr;row>athvariant="script">Srow>row>crow>row>(l,GFrow>(row>2row>row>rrow>)row>)row>ath>an>an>an>, where the dual hyperovals an id="mmlsi9" class="mathmlsrc"><a 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">ass="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">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si9.gif" overflow="scroll">row>athvariant="script">Srow>row>crow>row>′row>row>(row>lrow>row>′row>,GFrow>(row>2row>row>row>rrow>row>′row>row>)row>)row>ath>an>an>an> and an id="mmlsi10" class="mathmlsrc">an 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))an>an class="mathContainer hidden">an class="mathCode">ath altimg="si10.gif" overflow="scroll">row>athvariant="script">Srow>row>crow>row>(l,GFrow>(row>2row>row>rrow>)row>)row>ath>an>an>an> are constructed in Taniguchi (2014). Using the result, we show that the Buratti&ndash;Del Fra dual hyperoval has a bilinear quotient in an id="mmlsi11" class="mathmlsrc">an 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)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si11.gif" overflow="scroll">Vrow>(2d+1,2)row>ath>an>an>an> if an id="mmlsi1" class="mathmlsrc">an 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">dan>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">dath>an>an>an> is odd. On the other hand, we show that the Huybrechts dual hyperoval has no bilinear quotient in an id="mmlsi11" class="mathmlsrc">an 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)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si11.gif" overflow="scroll">Vrow>(2d+1,2)row>ath>an>an>an>. We also determine the automorphism group of an id="mmlsi10" class="mathmlsrc">an 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))an>an class="mathContainer hidden">an class="mathCode">ath altimg="si10.gif" overflow="scroll">row>athvariant="script">Srow>row>crow>row>(l,GFrow>(row>2row>row>rrow>)row>)row>ath>an>an>an>, and show that an id="mmlsi15" class="mathmlsrc">an 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)))an>an class="mathContainer hidden">an class="mathCode">ath altimg="si15.gif" overflow="scroll">Autrow>(row>athvariant="script">Srow>row>crow>row>(row>lrow>row>2row>,GFrow>(row>2row>row>rrow>lrow>row>1row>row>)row>)row>)row><Autrow>(row>athvariant="script">Srow>row>crow>row>(l,GFrow>(row>2row>row>rrow>)row>)row>)row>ath>an>an>an>.