Algebro-combinatorial generalizations of the Victoir method for constructing weighted designs

Abstract

A weighted $t$-design in $\mathbb{R}^d$ is a finite weighted set that exactly integrates all polynomials of degree at most $t$ with respect to a given probability measure. A fundamental problem is to construct weighted $t$-designs with as few points as possible. Victoir (2004) proposed a method to reduce the size of weighted $t$-designs while preserving the $t$-design property by using combinatorial objects such as combinatorial designs or orthogonal arrays with two levels. In this paper, we give an algebro-combinatorial generalization of both Victoir's method and its variant by the present authors (2014) in the framework of Euclidean polynomial spaces, enabling us to reduce the size of weighted designs obtained from the classical product rule. Our generalization allows the use of orthogonal arrays with arbitrary levels, whereas Victoir only treated the case of two levels. As an application, we present a construction of equi-weighted $5$-designs with $O(d^4)$ points for product measures such as Gaussian measure $π^{-d/2} e^{-\sum_{i=1}^d x_i^2} dx_1 \cdots dx_d$ on $\mathbb{R}^d$ or equilibrium measure $π^{-d} \prod_{i=1}^d (1-x_i^2)^{-1/2} dx_1 \cdots dx_d$ on $(-1,1)^d$, where $d$ is any integer at least 5. The construction is explicit and does not rely on numerical approximations. Moreover, we establish an existence theorem of Gaussian $t$-designs with $N$ points for any $t \geq 2$, where $N< q^{t}d^{t-1}=O(d^{t-1})$ for fixed sufficiently large prime power $q$. As a corollary of this result, we give an improvement of a famous theorem by Milman (1988) on isometric embeddings of the classical finite-dimensional Banach spaces.