# Tight Gaussian 4-Designs

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DOI: 10.1007/s10801-005-2505-3

- Cite this article as:
- Bannai, E. & Bannai, E. J Algebr Comb (2005) 22: 39. doi:10.1007/s10801-005-2505-3

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## Abstract

A Gaussian *t*-design is defined as a finite set *X* in the Euclidean space ℝ^{n} satisfying the condition: \(\frac{1}{V({\mathbb R}^n)}\int_{{\mathbb R}^n} f(x)e^{-\alpha^2||x||^2}dx=\sum_{u\in X}\omega(u)f(u)\) for any polynomial *f*(*x*) in *n* variables of degree at most *t*, here α is a constant real number and ω is a positive weight function on *X*. It is easy to see that if *X* is a Gaussian 2*e*-design in ℝ^{n}, then \(|X|\geq {n+e\choose e}\). We call *X* a tight Gaussian 2*e*-design in ℝ^{n} if \(|X|={n+e\choose e}\) holds. In this paper we study tight Gaussian 2*e*-designs in ℝ^{n}. In particular, we classify tight Gaussian 4-designs in ℝ^{n} with constant weight \(\omega=\frac{1}{|X|}\) or with weight \(\omega(u)=\frac{e^{-\alpha^2||u||^2}} {\sum_{x\in X}e^{-\alpha^2||x||^2}}\). Moreover we classify tight Gaussian 4-designs in ℝ^{n} on 2 concentric spheres (with arbitrary weight functions).