Lower bounds for designs in symmetric spaces

00:00 08-12-2010
<p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"><span style="FONT-FAMILY: 'Arial', 'sans-serif'; FONT-SIZE: 14px"><font face="Arial, Helvetica, sans-serif"></font></span> </p> <p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"><span style="FONT-FAMILY: 'Arial', 'sans-serif'; FONT-SIZE: 14px"><font face="Arial, Helvetica, sans-serif"></font></span> </p> <p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"><span style="FONT-FAMILY: 'Arial', 'sans-serif'; FONT-SIZE: 14px"><font face="Arial, Helvetica, sans-serif"> A design is a (finite) subset of points in space on which simple functions </font></span></p> <p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"><span style="FONT-FAMILY: 'Arial', 'sans-serif'; FONT-SIZE: 14px"><font face="Arial, Helvetica, sans-serif">("low-degree polynomials") average to their total average.<br /><br />We will discuss what 'simple' means in various nice spaces ('nice' means graphs and Riemannian manifolds </font></span></p> <p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"><span style="FONT-FAMILY: 'Arial', 'sans-serif'; FONT-SIZE: 14px"><font face="Arial, Helvetica, sans-serif">with many symmetries) and give easy proofs to some known (and possibly some new)  bounds on the cardinality of designs. <br /><br />In particular, the following geometric claim holds: a design is large, </font></span></p> <p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"><span style="FONT-FAMILY: 'Arial', 'sans-serif'; FONT-SIZE: 14px"><font face="Arial, Helvetica, sans-serif">because a union of "spheres" around its points "covers" the whole space.<br /><br />As a corollary, we will sketch simple proofs of (essentially) the best known bounds for </font></span></p> <p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"><span style="FONT-FAMILY: 'Arial', 'sans-serif'; FONT-SIZE: 14px"><font face="Arial, Helvetica, sans-serif">binary linear codes and lattice packings in the Euclidean space.</font></span></p> <p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"><span style="FONT-FAMILY: 'Arial', 'sans-serif'; FONT-SIZE: 14px"><font face=""> </font></span></p>