<div dir="ltr"><span style="font-size:12pt"><font color="#000000"><span style="font-size:12pt"><font color="#000000"><font face="CMR12"><font face="Times New Roman"><div>Degerli liste uyeleri,</div><div><br></div><div>1-3 Subat arasinda Sabanci Universitesinde duzenlenecek olan ve asagida detaylarini bulacaginiz etkinlige katiliminiz bizi mutlu edecektir.</div><div><br></div><div>Iyi gunler dilegiyle,</div><div><br></div><div>canan kasikci</div><div><br></div><div>Ulasim icin: <div><a href="http://www.sabanciuniv.edu/en/transportation/shuttle-hours" target="_blank">http://www.sabanciuniv.edu/en/transportation/shuttle-hours</a><font color="#222222"> </font></div></div><div><font color="#222222"><br></font></div><div><strong><font size="4">Workshop on Finite Fields: Arcs, Curves and Bent Functions</font></strong></div><div><br></div><div><strong>Program:<br></strong></div><div><strong><br></strong></div><div><strong>February 1st, Monday, (FENS G032)</strong></div><div><strong>13:35-14:30: </strong>Simeon Ball, Extending small arcs to large arcs</div><div><strong>February 3rd, Wednesday, (FENS 2008)</strong></div><div><strong>11:00-12:00: </strong>Nurdagul Anbar, A new tower meeting Zink's bound with good p-rank </div><div><strong>13:30-14:30: </strong>Wilfried Meidl, <font color="#000000"><font face="Times New Roman"><font face="times new roman,serif"><font face="arial,helvetica,sans-serif"><font face="times new roman,serif"><span style="font-size:12pt">Bent, gbent and bent</span><span style="font-size:8pt"><font size="1">4</font> </span><span style="font-size:12pt">functions</span></font></font></font></font></font><br></div><div><div><span style="font-family:"F17","sans-serif";font-size:12pt"><strong><font color="#000000"></font></strong></span><br></div><div><span style="font-family:"F17","sans-serif";font-size:12pt"><strong><font color="#000000">------------------------------</font></strong><span style="font-family:"F17","sans-serif";font-size:12pt"><strong><font color="#000000">------------------------------</font></strong><span style="font-family:"F17","sans-serif";font-size:12pt"><strong><font color="#000000">------------------------------</font></strong></span></span></span></div><div><font color="#000000"><font face="arial,helvetica,sans-serif"><span style="font-family:"Times New Roman","serif""><br></span></font></font></div></div><div><strong>Abstracts:</strong></div><div><br></div><div><strong><font face="arial,helvetica,sans-serif" size="2">Extending small arcs to large arcs</font></strong></div><div><font face="arial,helvetica,sans-serif" size="2"><strong>by Simeon Ball - Universitat Polit<font face="times new roman,serif" size="3">è</font>cnica Catalunya</strong></font></div><p style="text-align:left">Let <font color="#000000"><span style="font-family:"msbm10","sans-serif";font-size:12pt">F</span><span style="font-family:cmmi8;font-size:8pt">q</span></font> denote the finite field with q elements. An arc is a set of vectors of the k-dimensional vector space over the finite field with q elements <font color="#000000"><span style="font-family:"msbm10","sans-serif";font-size:12pt">F</span><span style="font-family:cmmi8;font-size:8pt">q</span></font>, in which every subset of size k is a basis of the space, i.e. every k-subset is a set of linearly independent vectors. Alternatively (and equivalently) one can define an arc as a subset of points in the (k −1)-dimensional projective space over <font color="#000000"><span style="font-family:"msbm10","sans-serif";font-size:12pt">F</span><span style="font-family:cmmi8;font-size:8pt">q</span></font>, in which every subset of size k spans the whole space. The matrix whose columns are the vectors of an arc S generates a k-dimensional linear maximum distance separable (MDS) code over <font color="#000000"><span style="font-family:"msbm10","sans-serif";font-size:12pt">F</span><span style="font-family:cmmi8;font-size:8pt">q</span></font> of length |S|, so arcs and linear MDS codes are equivalent objects. If a small arc can be extended to a large arc and the characteristic is odd then we prove that certain necessary conditions must be satisfied. For example, it follows from Segre’s theorem that if the characteristic is odd and an arc of size six can be extended to an arc of size q + 1 then it is necessary that the small arc is contained in a conic. We prove further results of this type. These theorems may provide new tools in the computational classification and construction of large arcs. </p><div><span style="font-family:"F17","sans-serif";font-size:12pt"><strong><font color="#000000">------------------------------</font></strong><span style="font-family:"F17","sans-serif";font-size:12pt"><strong><font color="#000000">------------------------------</font></strong><span style="font-family:"F17","sans-serif";font-size:12pt"><strong><font color="#000000">------------------------------</font></strong></span></span></span></div><div><font color="#000000"><font face="arial,helvetica,sans-serif"><span style="font-family:"Times New Roman","serif""><br></span></font></font></div></font></font></font></span></font></span><div><font face="times new roman,serif"><font color="#000000"><font face="CMR12"><font face="Times New Roman"><font color="#000000"><font face="times new roman,serif"><font face="arial,helvetica,sans-serif"><font face="arial,helvetica,sans-serif"><font face="arial,helvetica,sans-serif"><span style="font-family:"Times New Roman","serif""><div><font color="#000000"><font face="arial,helvetica,sans-serif"><strong>A new tower meeting Zink's bound with good p-rank</strong> </font></font></div></span></font></font></font></font></font></font></font></font></font><div><font face="times new roman,serif"><font face="CMR12"><font face="times new roman,serif"><font color="#000000"><font face="times new roman,serif"><font face="arial,helvetica,sans-serif"><font face="arial,helvetica,sans-serif"><font face="arial,helvetica,sans-serif"><span style="font-family:"Times New Roman","serif""><b><font face="arial,helvetica,sans-serif">by Nurdagul Anbar, Denmark Technical University </font></b></span></font></font></font></font></font></font></font></font></div></div><div><br></div><div><font color="#000000"><font face="CMMI5" size="1"><font face="CMMI5" size="1"><p><font size="2"></font></p><font face="CMMI5" size="1"><p></p><font face="CMMI5" size="1"><p><font face="Arial"></font><font color="#000000" face="Times New Roman" size="3"> </font></p><p style="margin:0in 0in 0pt;line-height:normal"><span style="font-family:"cmr10","sans-serif";font-size:10pt"><font color="#000000" face="arial,helvetica,sans-serif">In a recent work [1], Bassa, Beelen, Garcia and Stichtenoth have introduced a new recursive </font></span><font color="#000000"><span style="font-family:"cmr10","sans-serif";font-size:10pt"><font face="arial,helvetica,sans-serif">tower defined over </font></span><span style="font-family:"msbm10","sans-serif";font-size:10pt">F</span><span style="font-family:cmmi7;font-size:7pt">q</span><span style="font-family:cmmi5;font-size:5pt">n </span><span style="font-family:"cmr10","sans-serif";font-size:10pt"><font face="arial,helvetica,sans-serif">for any </font></span><span style="font-family:cmmi10;font-size:10pt">n </span><span style="font-family:"Times New Roman","serif";font-size:10pt">≥</span><span style="font-family:"cmsy10","sans-serif";font-size:10pt"> </span><span style="font-family:"cmr10","sans-serif";font-size:10pt">2, <font face="arial,helvetica,sans-serif">which results in the best known lower bound for </font></span><span style="font-size:10pt"><font face="arial,helvetica,sans-serif">Ihara's </font></span></font><font color="#000000"><font face="arial,helvetica,sans-serif"><span style="font-size:10pt">constant </span><span style="font-family:"cmr10","sans-serif";font-size:10pt"><font face="arial,helvetica,sans-serif">for any non-prime finite fields. </font></span></font></font></p><p style="margin:0in 0in 0pt;line-height:normal"><font color="#000000"><font face="arial,helvetica,sans-serif"><span style="font-family:"cmr10","sans-serif""><font face="arial,helvetica,sans-serif"><br></font></span></font></font></p><p style="margin:0in 0in 0pt;line-height:normal"><font color="#000000"><font face="arial,helvetica,sans-serif"><span style="font-family:"cmr10","sans-serif""><font face="arial,helvetica,sans-serif"><font size="2">In this work, we have investigated a tower </font><span style="font-family:"cmsy10","sans-serif"">F</span><span style="font-family:cmmi10">=<span style="font-family:"msbm10","sans-serif";font-size:10pt">F</span><span style="font-family:cmmi7;font-size:7pt">q</span><span style="font-family:"cmr5","sans-serif";font-size:5pt">3</span></span><span style="font-family:"cmr5","sans-serif""> </span><span style="font-family:"cmr10","sans-serif"">=(</span><span style="font-family:cmmi10">F</span><span style="font-family:"cmr7","sans-serif"">1</span><span style="font-family:"cmsy10","sans-serif""> <span style="color:black;line-height:115%;font-family:"Cambria Math","serif"">⊂ </span></span><span style="font-family:cmmi10">F</span><span style="font-family:"cmr7","sans-serif"">2 <span style="color:black;line-height:115%;font-family:"Cambria Math","serif"">⊂</span><span style="font-family:"cmsy10","sans-serif""><font face="Arial">.......</font></span></span><span style="font-family:"cmr10","sans-serif"">)</span></font></span></font></font><font color="#000000"><font size="2"><span style="font-family:"cmr10","sans-serif""><font face="arial,helvetica,sans-serif">arising from the Drinfeld modular interpretation of the tower in [1]. More precisely, we have shown that the limit of </font></span><span style="font-family:"cmsy10","sans-serif"">F</span><span style="font-family:cmmi10">=</span><span style="font-family:"msbm10","sans-serif"">F</span></font><span style="font-family:cmmi7;font-size:7pt">q</span><span style="font-family:"cmr5","sans-serif";font-size:5pt">3<font face="arial,helvetica,sans-serif"> </font></span><font face="arial,helvetica,sans-serif"><span style="font-family:"cmr10","sans-serif";font-size:10pt"><font face="arial,helvetica,sans-serif">attains Zink's bound and that the</font> </span><span style="font-size:10pt">p</span></font><span style="font-family:"cmr10","sans-serif";font-size:10pt"><font face="arial,helvetica,sans-serif">-torsion limit of the tower</font> </span><span style="font-family:"cmsy10","sans-serif";font-size:10pt">F</span><span style="font-family:cmmi10;font-size:10pt">=</span><span style="font-family:"msbm10","sans-serif";font-size:10pt">F</span><span style="font-family:cmmi7;font-size:7pt">q</span><span style="font-family:"cmr5","sans-serif";font-size:5pt">3 </span><span style="font-family:"cmr10","sans-serif";font-size:10pt"><font face="arial,helvetica,sans-serif">satisfies</font></span></font></p><p style="margin:0in 0in 0pt;line-height:normal"><font color="#000000" face="Arial"><span style="font-family:"cmr10","sans-serif";font-size:10pt"></span></font><br></p><p style="margin:0in 0in 0pt;line-height:normal"><font color="#000000" face="Arial"><span style="font-family:"cmr10","sans-serif";font-size:10pt"><img width="439" height="53" alt="Inline image 2" src="cid:ii_1528523421e148b5"></span></font></p><p style="margin:0in 0in 0pt;line-height:normal"><font color="#000000" face="Arial"><span style="font-family:"cmr10","sans-serif";font-size:10pt"></span></font><br></p><p style="margin:0in 0in 0pt;line-height:normal"><font color="#000000"><span style="font-family:"cmr10","sans-serif";font-size:10pt"><font face="arial,helvetica,sans-serif"> </font></span></font></p><p style="margin:0in 0in 0pt;line-height:normal"><font color="#000000"><span style="font-family:"cmr10","sans-serif";font-size:10pt"><font face="arial,helvetica,sans-serif">where </font></span><span style="font-family:"Cambria Math","serif";font-size:10pt">γ</span><span style="font-family:"cmr10","sans-serif";font-size:10pt"> (</span><span style="font-family:cmmi10;font-size:10pt">F</span><span style="font-family:cmmi7;font-size:7pt">n</span><span style="font-family:"cmr10","sans-serif";font-size:10pt">) and </span><span style="font-family:cmmi10;font-size:10pt">g</span><span style="font-family:"cmr10","sans-serif";font-size:10pt">(</span><span style="font-family:cmmi10;font-size:10pt">F</span><span style="font-family:cmmi7;font-size:7pt">n</span><span style="font-family:"cmr10","sans-serif";font-size:10pt">) is <font face="arial,helvetica,sans-serif">the </font></span><font face="arial,helvetica,sans-serif"><span style="font-size:10pt">p</span><span style="font-family:"cmr10","sans-serif";font-size:10pt"><font face="arial,helvetica,sans-serif">-rank and the genus of</font> </span></font><span style="font-family:cmmi10;font-size:10pt">F</span><span style="font-family:cmmi7;font-size:7pt">n</span><span style="font-family:"cmr10","sans-serif";font-size:10pt">, <font face="arial,helvetica,sans-serif">respectively. Moreover, we have shown that the equality holds in (1) in the case of prime </font></span><font face="arial,helvetica,sans-serif"><span style="font-size:10pt">q</span><span style="font-family:"cmr10","sans-serif";font-size:10pt">. <font face="arial,helvetica,sans-serif">This is a joint work with Peter Beelen and Nhut Nguyen.</font></span></font></font></p><p><font face="Arial" size="2"><font color="#000000" face="times new roman,serif"><strong>References:</strong></font></font><font face="Arial" size="2"><font color="#000000"><font face="times new roman,serif"><span style="font-family:"cmr10","sans-serif";font-size:11pt">[1] <font face="arial,helvetica,sans-serif" size="1">A. Bassa, P. Beelen, A. Garcia, H. Stichtenoth, Towers of Function Fields over Non-prime Finite</font></span></font><font size="1"><font face="arial,helvetica,sans-serif"> </font><span style="font-family:"cmr10","sans-serif""><font face="arial,helvetica,sans-serif">Fields, Moscow Mathematical Journal </font>15 (1) (2015) 1{29</span></font></font><font size="1">.<br></font></font></p></font><p><font face="Arial"></font></p></font><p><font face="Arial"></font></p></font></font></font></div><font face="CMR10"><font face="CMR10"></font></font><font face="times new roman,serif"><font face="CMR10" size="3"><font face="CMR10" size="3"><font face="CMR10" size="2"><font face="CMMI5" size="1"><font face="CMMI5" size="1"><font size="2"><font face="arial,helvetica,sans-serif" size="2"><font face="CMR10" size="2"><font face="arial,helvetica,sans-serif" size="2"><font face="arial,helvetica,sans-serif"><font size="2"><font size="2"><p style="margin:0in 0in 0pt;line-height:normal"><font color="#000000"><strong><span style="font-family:"F17","sans-serif";font-size:12pt">-----------------------------------------------------------------------------------------</span></strong></font></p><p style="margin:0in 0in 0pt;line-height:normal"><font color="#000000"><span style="font-family:"F17","sans-serif";font-size:12pt"><br></span></font></p><span style="font-family:"F16","sans-serif""><font color="#000000" face="arial,helvetica,sans-serif"><p style="margin:0in 0in 0pt;line-height:normal"><font color="#000000"><strong><font face="times new roman,serif"><span style="font-size:12pt">Bent, gbent and bent</span><span style="font-size:8pt">4 </span><span style="font-size:12pt">functions</span></font></strong></font></p><font color="#000000" size="3"><strong></strong></font><p style="margin:0in 0in 0pt;line-height:normal"><span style="font-family:"F16","sans-serif";font-size:12pt"><font color="#000000" face="times new roman,serif"><strong>by Wilfried Meidl, RICAM Linz</strong></font></span></p><font color="#000000" face="Times New Roman" size="3"></font><p style="margin:0in 0in 0pt;line-height:normal"><span style="font-family:"F16","sans-serif";font-size:12pt"><font color="#000000"><br></font></span></p></font><p style="margin:0in 0in 0pt;line-height:normal"><br></p></span><div style="margin:0in 0in 0pt;text-align:left;line-height:normal"><font face="times new roman,serif"><span style="font-family:"F16","sans-serif";font-size:12pt"><font color="#000000" face="times new roman,serif">After recalling bent functions, vectorial bent functions and their rela</font></span><span style="font-size:12pt"><font color="#000000">tions to difference sets, two types of generalizations of bent functions are</font></span><font color="#000000" size="3"> </font></font><span style="font-size:12pt"><font color="#000000" face="times new roman,serif">discussed. The first one are gbent functions (generalized bent functions), which are functions from <font color="#000000"><span style="line-height:115%;font-family:"msbm10","sans-serif";font-size:12pt">F</span>_{<span style="line-height:115%;font-family:"cmr8","sans-serif";font-size:8pt">2</span>}_{<span style="line-height:115%;font-family:"Arial","sans-serif";font-size:8pt"> </span>}<span style="line-height:115%;font-family:"Arial","sans-serif";font-size:8pt">ⁿ<font size="3"><font face="Times New Roman">to </font><font face="msbm10">Z</font></font><font face="CMR8" size="1"><font face="CMR8" size="1">2</font></font><font face="CMMI6" size="1"><font face="CMMI6" size="1">k  <span style="font-size:12pt"><font face="Times New Roman">with a flat spectrum with respect <span style="font-size:12pt"><font face="Times New Roman">to a <span style="font-size:12pt"><font color="#000000" face="times new roman,serif">generalized Walsh transform. Recent results on their structure, particularly on their relation with (vectorial) bent and semibent functions will be presented. </font></span></font></span></font></span></font></font></span></font></font></span></div><div style="margin:0in 0in 0pt;text-align:left;line-height:normal"><span style="font-size:12pt"><font color="#000000" face="times new roman,serif"><font color="#000000"><span style="line-height:115%;font-family:"Arial","sans-serif";font-size:8pt"><font face="CMMI6" size="1"><font face="CMMI6" size="1"><span style="font-size:12pt"><font face="Times New Roman"><span style="font-size:12pt"><font face="Times New Roman"><span style="font-size:12pt"><font color="#000000" face="times new roman,serif">The second are bent<font size="1">4</font> </font></span></font></span></font></span></font></font></span></font></font></span><span style="font-size:12pt"><font color="#000000" face="times new roman,serif"><font color="#000000"><span style="line-height:115%;font-family:"Arial","sans-serif";font-size:8pt"><font face="CMMI6" size="1"><font face="CMMI6" size="1"><span style="font-size:12pt"><font face="Times New Roman"><span style="font-size:12pt"><font face="Times New Roman"><span style="font-size:12pt"><font color="#000000" face="times new roman,serif"><font size="1"><span style="font-size:12pt"><font face="times new roman,serif">functions which appear as component functions </font></span><span style="font-size:12pt"><font color="#000000"><font face="times new roman,serif">of the recently discovered modified planar functions, i.e., functions describing <span style="font-size:12pt"><font color="#000000"><font face="cmr12">(</font><span style="line-height:115%;font-family:"cmr12","sans-serif";font-size:12pt">2ⁿ</span><font face="CMMI12">; <span style="line-height:115%;font-family:"cmr12","sans-serif";font-size:12pt">2ⁿ</span></font><font face="CMMI12">; <span style="line-height:115%;font-family:"cmr12","sans-serif";font-size:12pt">2ⁿ</span></font><font face="CMMI12">; </font><font face="CMR12">1)-relative difference sets and give rise to projective planes.</font></font></span></font></font></span></font></font></span></font></span></font></span></font></font></span></font></font></span></div></font></font></font></font></font></font></font></font></font></font></font></font></font></div>