On tight Euclidean $6$-designs: an experimental result

A finite set $X\text{seq}{\text{RR}}^n$ with a weight function $w : X \longrightarrow \RR_{>0}is called \emph{Euclideant-design} in\RR^n$(supportedby$p$ concentric spheres) if the following condition holds: $$\sum _{i=1}^p\frac{w(X_i)}{\mathrm{\mid }{S}_i\mathrm{\mid }}{\int }_{S_i}f(\mathit{x})d{\sigma }_i(\mathit{x})=\sum _{\mathit{x}\in X}w(\mathit{x})f(\mathit{x}),$$ for any polynomial $f(\mathit{x})\in Pol({\text{RR}}^n)$ of degree at most $t$. Here $S_i\text{seq}{\text{RR}}^n$ is a sphere of radius $r_i\ge 0,$ $X_i=X\cap S_i,$ and ${\sigma }_i(\mathit{x})$ is an $O(n)$-invariant measure on $S_i$ such that $\mathrm{\mid }{S}_i\mathrm{\mid }=r_i^{n-1}\mathrm{\mid }{S}^{n-1}\mathrm{\mid }$, with $\mathrm{\mid }{S}_i\mathrm{\mid }$ is the surface area of $S_i$ and $\mathrm{\mid }{S}^{n-1}\mathrm{\mid }$ is a surface area of the unit sphere in ${\text{RR}}^n$. Recently, Bajnok (2006) constructed tight Euclidean $t$-designs in the plane ($n=2$) for arbitrary $t$ and $p\mathrm{.}$ 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 ${\text{RR}}^n$, for $2\le n\le 8.$