Sets with many pairs of orthogonal vectors over finite fields

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• 作者：Omran Ahmadi<sup> ; sup><sup>oahmadid@ipm.ir" class="auth_mail" title="E-mail the corresponding authorsup> ; Ali Mohammadian <sup>ali_m@ipm.ir" class="auth_mail" title="E-mail the corresponding authorsup>
• 关键词：05D05 ; 15A63 ; 11T99
• 刊名：Finite Fields and Their Applications
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：37
• 期：Complete
• 页码：179-192
• 全文大小：373 K

文摘

Let n   be a positive integer and <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S107157971500101X&_mathId=si1.gif&_user=111111111&_pii=S107157971500101X&_rdoc=1&_issn=10715797&md5=2df521debaedbe7b38c68e75c5277626" title="Click to view the MathML source">Bspan><span class="mathContainer hidden"><span class="mathCode">scroll">script">Bspan>span>span> be a non-degenerate symmetric bilinear form over <span id="mmlsi2" class="mathmlsrc">source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S107157971500101X&_mathId=si2.gif&_user=111111111&_pii=S107157971500101X&_rdoc=1&_issn=10715797&md5=87a78fe947826293d36aa064c7735b4e">ss="imgLazyJSB inlineImage" height="18" width="21" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S107157971500101X-si2.gif">script>style="vertical-align:bottom" width="21" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S107157971500101X-si2.gif">script><span class="mathContainer hidden"><span class="mathCode">scroll">subsup>struck">Fqnsubsup>span>span>span>, where q   is an odd prime power and <span id="mmlsi3" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S107157971500101X&_mathId=si3.gif&_user=111111111&_pii=S107157971500101X&_rdoc=1&_issn=10715797&md5=75e617f39e905faa30fa8f103b7e52cd" title="Click to view the MathML source">F<sub>qsub>span><span class="mathContainer hidden"><span class="mathCode">scroll">sub>struck">Fqsub>span>span>span> is the finite field with q elements. We determine the largest possible size of a subset S   of <span id="mmlsi2" class="mathmlsrc">source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S107157971500101X&_mathId=si2.gif&_user=111111111&_pii=S107157971500101X&_rdoc=1&_issn=10715797&md5=87a78fe947826293d36aa064c7735b4e">ss="imgLazyJSB inlineImage" height="18" width="21" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S107157971500101X-si2.gif">script>style="vertical-align:bottom" width="21" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S107157971500101X-si2.gif">script><span class="mathContainer hidden"><span class="mathCode">scroll">subsup>struck">Fqnsubsup>span>span>span> such that <span id="mmlsi4" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S107157971500101X&_mathId=si4.gif&_user=111111111&_pii=S107157971500101X&_rdoc=1&_issn=10715797&md5=02965a6388d3543a550060d402b570d4" title="Click to view the MathML source">|{B(<strong>xstrong>,<strong>ystrong>)|<strong>xstrong>,<strong>ystrong>&isin;S and <strong>xstrong>≠<strong>ystrong>}|=1span><span class="mathContainer hidden"><span class="mathCode">scroll">stretchy="false">|stretchy="false">{script">Bstretchy="false">(x,ystretchy="false">)stretchy="false">|x,y&isin;S and x≠ystretchy="false">}stretchy="false">|=1span>span>span>. We also pose some conjectures concerning nearly orthogonal subsets of <span id="mmlsi2" class="mathmlsrc">source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S107157971500101X&_mathId=si2.gif&_user=111111111&_pii=S107157971500101X&_rdoc=1&_issn=10715797&md5=87a78fe947826293d36aa064c7735b4e">ss="imgLazyJSB inlineImage" height="18" width="21" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S107157971500101X-si2.gif">script>style="vertical-align:bottom" width="21" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S107157971500101X-si2.gif">script><span class="mathContainer hidden"><span class="mathCode">scroll">subsup>struck">Fqnsubsup>span>span>span> where a nearly orthogonal subset T   of <span id="mmlsi2" class="mathmlsrc">source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S107157971500101X&_mathId=si2.gif&_user=111111111&_pii=S107157971500101X&_rdoc=1&_issn=10715797&md5=87a78fe947826293d36aa064c7735b4e">ss="imgLazyJSB inlineImage" height="18" width="21" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S107157971500101X-si2.gif">script>style="vertical-align:bottom" width="21" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S107157971500101X-si2.gif">script><span class="mathContainer hidden"><span class="mathCode">scroll">subsup>struck">Fqnsubsup>span>span>span> is a set of vectors in which among any three distinct vectors there are two vectors <strong class="boldFont">xstrong>, <strong class="boldFont">ystrong>   so that <span id="mmlsi5" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S107157971500101X&_mathId=si5.gif&_user=111111111&_pii=S107157971500101X&_rdoc=1&_issn=10715797&md5=57937b91253647b61935da5d417cd8be" title="Click to view the MathML source">B(<strong>xstrong>,<strong>ystrong>)=0span><span class="mathContainer hidden"><span class="mathCode">scroll">script">Bstretchy="false">(x,ystretchy="false">)=0span>span>span>.