Journal of the Mathematical Society of Japan

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 $\sum_{i=1}^p(w(X_i)/(|S_i|)) \int_{S_i}f(x)d\sigma_i(x) = \sum_{x\in X}w(x)f(x)$ holds for any polynomial $f(x)$ of $\deg (f)\leq t$, where $\{S_i, 1\leq i \leq p\}$ is the set of all the concentric spheres centered at the origin and intersect with $X$, $X_i=X\cap S_i$, and $w:X\rightarrow \mathbf{R}_{>0}$ 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 $\mathscr{P}_e(Y)$ 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 $|X|\geq \dim(\mathscr{P}_e(S))$ holds, where $S=\cup_{i=1}^pS_i$. The actual lower bounds proved by Delsarte and Seidel are better than this in some special cases. However as designs on $S$, the bound $\dim(\mathscr{P}_e(S))$ is natural and universal. In this point of view, we call a Euclidean $2e$-design $X$ with $|X| =\dim(\mathscr{P}_e(S))$ a tight $2e$-design on $p$ concentric spheres. Moreover if $\dim(\mathscr{P}_e(S)) = \dim(\mathscr{P}_e(\mathbf{R}^n)) (=\left(\begin{array}{c}n+e\\e\end{array}\right))$ 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 $2e\geq 4$.

Article information

Source
J. Math. Soc. Japan, Volume 58, Number 3 (2006), 775-804.

Dates
First available in Project Euclid: 23 August 2006

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1156342038

Digital Object Identifier
doi:10.2969/jmsj/1156342038

Mathematical Reviews number (MathSciNet)
MR2254411

Zentralblatt MATH identifier
1104.05016

Citation

BANNAI, Eiichi; BANNAI, Etsuko. On Euclidean tight 4-designs. J. Math. Soc. Japan 58 (2006), no. 3, 775--804. doi:10.2969/jmsj/1156342038. https://projecteuclid.org/euclid.jmsj/1156342038