DOI Number : 10.5614/itbj.sci.2011.43.1.3
Hits : 38

On Tight Euclidean 6-Designs: An Experimental Result

Djoko Suprijanto

Combinatorial Mathematics Research Group
Faculty of Mathematics and Natural Sciences
Institut Teknologi Bandung 40132, INDONESIA.

Abstract. A finite set $X \seq \RR^n$ with a weight function $w : X \longrightarrow \RR_{>0}$ is called \emph{Euclidean $t$-design} in $\RR^n$ (supported by $p$ concentric spheres) if the following condition holds: \[ \sum_{i="1"}^p \frac{w(X_i)}{|S_i|}\int_{S_i} f(\boldsymbol x)d\sigma_i(\boldsymbol x) =\sum_{\boldsymbol x \in X}w(\boldsymbol x) f(\boldsymbol x), \] for any polynomial $f(\boldsymbol x) \in \mbox{Pol}(\RR^n)$ of degree at most $t$. Here $S_i \seq \RR^n$ is a sphere of radius $r_i \geq 0,$ $X_i=X \cap S_i,$ and $\sigma_i(\boldsymbol x)$ is an $O(n)$-invariant measure on $S_i$ such that $|S_i|=r_i^{n-1}|S^{n-1}|$, with $|S_i|$ is the surface area of $S_i$ and $|S^{n-1}|$ is a surface area of the unit sphere in $\RR^n$. Recently, Bajnok (2006) constructed tight Euclidean $t$-designs in the plane ($n="2"$) for arbitrary $t$ and $p.$ In this paper we show that for case $t="6"$ and $p="2",$ tight Euclidean $6$-designs constructed by Bajnok is the unique configuration in $\RR^n$, for $2 \leq n \leq 8.$

Keywords: Euclidean designs; spherical designs; tight designs; distance sets; association schemes

Download Article
Bahasa Indonesia | English


Begin on 10 October 2014 this website is no longer activated for article process in Journal of Mathematical and Fundamental Sciences, Journal of Engineering and Technological Sciences, Journal of ICT Research and Applications and Journal of Visual Art and Design. The next process will be proceeded under new website at

For detail information please contact us to:

       ITB Journal Visitor Number #15801157       
       Jl. Tamansari 64, Bandung 40116, Indonesia Visitor IP Address #       
       Tel : +62-22-250 1759 ext. 121 2011 Institut Teknologi Bandung       
       Fax : +62-22-250 4010, +62-22-251 1215 XHTML + CSS + RSS       
       E-mail : or Developed by AVE