On Euclidean tight 4-designs

Abstract

A spherical $t$-design is a finite subset $X$ in the unit sphere $S^{n-1}\subset {\mathbf{R}}^n$which replaces the value of the integral on the sphere of any polynomial of degree at most $t$ by the average of the values of the polynomial on the finite subset $X$. Generalizing the concept of spherical designs, Neumaier and Seidel (1988) defined the concept of Euclidean $t$-design in ${\mathbf{R}}^n$ as a finite set $X$ in ${\mathbf{R}}^n$ for which ${\sum }_{i=1}^p\left(w\right(X_i)/(\left|S_i\right|\left)\right){\int }_{S_i}f\left(x\right)d{\sigma }_i\left(x\right)={\sum }_{x\in X}w\left(x\right)f\left(x\right)$ holds for any polynomial $f\left(x\right)$ of $\mathrm{deg}\left(f\right)\le t$, where $\{S_i,1\le i\le p\}$ is the set of all the concentric spheres centered at the origin and intersect with $X$, $X_i=X\cap S_i$, and $w:X\to {\mathbf{R}}_{>0}$ is a weight function of $X$. (The case of $X\subset S^{n-1}$ and with a constant weight corresponds to a spherical $t$-design.) Neumaier and Seidel (1988), Delsarte and Seidel (1989) proved the (Fisher type) lower bound for the cardinality of a Euclidean $2e$-design. Let $Y$ be a subset of ${\mathbf{R}}^n$ and let ${𝒫}_e\left(Y\right)$ be the vector space consisting of all the polynomials restricted to $Y$ whose degrees are at most $e$. Then from the arguments given by Neumaier-Seidel and Delsarte-Seidel, it is easy to see that $\left|X\right|\ge \dim \left({𝒫}_e\right(S\left)\right)$ holds, where $S={\cup }_{i=1}^p{S}_i$. The actual lower bounds proved by Delsarte and Seidel are better than this in some special cases. However as designs on $S$, the bound $\dim \left({𝒫}_e\right(S\left)\right)$ is natural and universal. In this point of view, we call a Euclidean $2e$-design $X$ with $\left|X\right|=\dim \left({𝒫}_e\right(S\left)\right)$ a tight $2e$-design on $p$ concentric spheres. Moreover if $\dim \left({𝒫}_e\right(S\left)\right)=\dim \left({𝒫}_e\right({\mathbf{R}}^n\left)\right)(=\left(\begin{array}{c}n+e\\ e\end{array}\right))$ holds, then we call $X$ a Euclidean tight $2e$-design. We study the properties of tight Euclidean $2e$-designs by applying the addition formula on the Euclidean space. Furthermore, we give the classification of Euclidean tight 4-designs with constant weight. It is possible to regard our main result as giving the classification of rotatable designs of degree 2 in ${\mathbf{R}}^n$in the sense of Box and Hunter (1957) with the possible minimum size $\left(\begin{array}{c}n+2\\ 2\end{array}\right)$. We also give examples of nontrivial Euclidean tight 4-designs in ${\mathbf{R}}^2$ with nonconstant weight,which give a counterexample to the conjecture of Neumaier and Seidel (1988) that there are no nontrivial Euclidean tight $2e$-designs even for the nonconstant weight case for $2e\ge 4$.