On Euclidean $t$-designs

A Euclidean $t$-design, as introduced by Neumaier and Seidel (1988), is a finite set ${\cal X} \subset \mathbb{R}^n$ with a weight function $w: {\cal X} \rightarrow \mathbb{R}^+$ for which $$\sum_{r \in R} W_r \overline{f}_{S_{r}} = \sum_{{\bf x} \in {\cal X}} w({\bf x}) f({\bf x})$$ holds for every polynomial $f$ of total degree at most $t$; here $R$ is the set of norms of the points in ${\cal X}$, $W_r$ is the total weight of all elements of ${\cal X}$ with norm $r$, $S_r$ is the $n$-dimensional sphere of radius $r$ centered at the origin, and $\overline{f}_{S_{r}}$ is the average of $f$ over $S_{r}$. Neumaier and Seidel (1988), as well as Delsarte and Seidel (1989), also proved a Fisher-type inequality $|{\cal X}| \geq N(n,|R|,t)$ (assuming that the design is antipodal if $t$ is odd). For fixed $n$ and $|R|$ we have $N(n,|R|,t)=O(t^{n-1})$. In Part I of this paper we provide a recursive construction for Euclidean $t$-designs in $\mathbb{R}^n$. Namely, we show how to use certain Gauss--Jacobi quadrature formulae to "lift" a Euclidean $t$-design in $\mathbb{R}^{n-1}$ to a Euclidean $t$-design in $\mathbb{R}^{n}$, preserving both the norm spectrum $R$ and the weight sum $W_r$ for each $r \in R$. A Euclidean design with exactly $N(n,|R|,t)$ points is called tight. In Part II of this paper we construct tight Euclidean designs for $n=2$ and every $t$ and $|R|$ with $|R| \leq \frac{t+5}{4}$. We also provide examples for tight Euclidean designs with $(n,|R|,t) \in \{(3,2,5),(3,3,7),(4,2,7)\}$.