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**On Euclidean tight 4-designs.**
*(English)*
Zbl 1104.05016

A spherical \(t\)-design is a finite subset \(X\) of the unit sphere \(\mathbb{S}^{n-1}\subset\mathbb{R}^n\), which replaces the value of the integral on the sphere of any polynomial of degeree at most \(t\) by average of the values of the polynomial on the finite subset \(X\). Generalizing the concept of spherical designs, A. Neumaier and J. J. Seidel [(*) Indag. Math. 50, No. 3, 321–334 (1988; Zbl 0657.10033)] defined the concept of Euclidean \(t\)-design in \(\mathbb{R}^n\) as a finite subset \(X\) of \(\mathbb{R}^n\) for which
\[
\sum^p_{i=1} (w(X_i)/|S_i|) \int_{S_i} f(x)\,d\sigma_i(x)= \sum_{x\in X} w(x) f(x)
\]
holds for any polynomial \(f(x)\) of \(\deg f\leq t\), where \(\{S_i\mid 1\leq i\leq p\}\) is the set of all concentric spheres centered at the origin and intersecting with \(X\) in \(X_i\) and \(w: X\to R_{> 0}\) is a weight function of \(X\). (The case of \(X\subset\mathbb{S}^{n-1}\) and with a constant weight corresponds to a spherical \(t\)-design.) A. Neumaier and J. J. Seidel (*) and P. Delsarte and J. J. Seidel [(**) Linear Algebra Appl. 114/115, 213–230 (1989; Zbl 0671.05014)] proved (Fisher type) lower bounds for the cardinality of a Euclidean \(2e\)-design.

Let \(Y\) be a subset of \(\mathbb{R}^n\) and let \({\mathcal P}_e(Y)\) be the vector space consisting of all the polynomials restricted to \(Y\) whose degrees are at most \(e\). Then from the arguments given by A. Neumaier and J. J. Seidel (*) and P. Delsarte and J. J. Seidel (**), it is easy to see that \(|X|\geq \dim({\mathcal P}_e(S))\) holds, where \(S=\bigcup^p_{i=1} S_i\). The actual lower bounds proved by Delsarte and Seidel are better than this in some special cases. However as design on \(S\), the bound \(\dim({\mathcal P}_e(S))\) is natural and universal.

In this point of view the authors called a Euclidean \(2e\)-design \(X\) with \(|X|\geq \dim({\mathcal P}_s(S))\) a tight \(2e\)-design on \(p\) concentric spheres. Moreover if \(\dim({\mathcal P}_e(S))= \dim({\mathcal P}_e(\mathbb{R}^n))\) \((={n+e\over e})\) holds, then \(X\) is called an Euclidean tight \(2e\)-design.

In this paper the authors studied the properties of tight Euclidean \(2e\)-designs by applying the addition formula on the Euclidean space. Furthermore, they gave a classification of Euclidean tight 4-designs with constant weight.

The main result of the paper can be regarded as giving the classification of rotatable designs of degree 2 in \(\mathbb{R}^n\) in the sense of G. E. P. Box and J. S. Hunter [Ann. Math. Stat. 28, 195–241 (1957; Zbl 0080.35901)] with the possible minimum size \({n+2\over 2}\).

The authors also gave examples of nontrivial Euclidean tight 4-designs in \(\mathbb{R}^2\) with nonconstant weight, which give a counterexample to the conjecture of A. Neumaier and J. J. Seidel (*) that there are no nontrivial Euclidean tight \(2e\)-designs (even for constant weight \(2e\geq 4)\).

Let \(Y\) be a subset of \(\mathbb{R}^n\) and let \({\mathcal P}_e(Y)\) be the vector space consisting of all the polynomials restricted to \(Y\) whose degrees are at most \(e\). Then from the arguments given by A. Neumaier and J. J. Seidel (*) and P. Delsarte and J. J. Seidel (**), it is easy to see that \(|X|\geq \dim({\mathcal P}_e(S))\) holds, where \(S=\bigcup^p_{i=1} S_i\). The actual lower bounds proved by Delsarte and Seidel are better than this in some special cases. However as design on \(S\), the bound \(\dim({\mathcal P}_e(S))\) is natural and universal.

In this point of view the authors called a Euclidean \(2e\)-design \(X\) with \(|X|\geq \dim({\mathcal P}_s(S))\) a tight \(2e\)-design on \(p\) concentric spheres. Moreover if \(\dim({\mathcal P}_e(S))= \dim({\mathcal P}_e(\mathbb{R}^n))\) \((={n+e\over e})\) holds, then \(X\) is called an Euclidean tight \(2e\)-design.

In this paper the authors studied the properties of tight Euclidean \(2e\)-designs by applying the addition formula on the Euclidean space. Furthermore, they gave a classification of Euclidean tight 4-designs with constant weight.

The main result of the paper can be regarded as giving the classification of rotatable designs of degree 2 in \(\mathbb{R}^n\) in the sense of G. E. P. Box and J. S. Hunter [Ann. Math. Stat. 28, 195–241 (1957; Zbl 0080.35901)] with the possible minimum size \({n+2\over 2}\).

The authors also gave examples of nontrivial Euclidean tight 4-designs in \(\mathbb{R}^2\) with nonconstant weight, which give a counterexample to the conjecture of A. Neumaier and J. J. Seidel (*) that there are no nontrivial Euclidean tight \(2e\)-designs (even for constant weight \(2e\geq 4)\).

Reviewer: Ratnakaram Nava Mohan (Nankai)

### MSC:

05B30 | Other designs, configurations |

05E30 | Association schemes, strongly regular graphs |

51M99 | Real and complex geometry |

62K99 | Design of statistical experiments |

### Keywords:

experimental design; rotatable design; tight design; spherical design; 2-distance set; Euclidean space; addition formula; polynomial
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\textit{E. Bannai} and \textit{E. Bannai}, J. Math. Soc. Japan 58, No. 3, 775--804 (2006; Zbl 1104.05016)

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