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On Euclidean tight 4-designs

机译：关于欧几里得紧4设计

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A spherical t-design is a finite subset X in the unit sphere S~(n-1) is contained in 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 R~n as a finite set X in R~n for which ∑_(i=1)~p(w(X_i)/(|S_i|)) ∫_(S_i) f(x)dσ_i(x) = ∑_(x∈X) w(x)f(x) holds for any polynomial f(x) of deg(f) ≤ t, where {S_i, 1 ≤ i ≤ p} is the set of all the concentric spheres centered at the origin and intersect with X, X_i = X ∩S_i, and w: X → R_( >0) is a weight function of X. (The case of X is contained in 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 R~n and let 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| ≥ dim(P_e(S)) holds, where S = ∪_(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(P_e(S)) is natural and universal. In this point of view, we call a Euclidean 2e-design X with |X| = dim(P_e(S)) a tight 2e-design on p concentric spheres. Moreover if dim(P_e(S)) = dim(P_e(R~n))(= (_e~(n+e))) 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 R~n in the sense of Box and Hunter (1957) with the possible minimum size (_2~(n+2) ). We also give examples of nontrivial Euclidean tight 4-designs in 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 ≥ 4.
机译：球面t设计是R〜n中包含的单位球面S〜（n-1）中的有限子集X，它用t的平均值替换最多t个多项式的球面上的积分值。 Neumaier和Seidel（1988）概括了球形设计的概念，将R〜n中的欧几里得t-设计的概念定义为R〜n中的有限集X，为此∑_（i = 1）〜p（w（X_i）/（| S_i |））∫_（S_i）f（x）dσ_i（x）= ∑_（x∈X）w（x）f（x）成立deg（f）≤t的f（x），其中{S_i，1≤i≤p}是所有同心球的集合，所有同心球以原点为中心并与X相交，X_i = X∩S_i，并且w：X→ R_（> 0）是X的权重函数。（X的情况包含在S〜（n-1）中，并且具有恒定的权重对应于球形t设计。）Neumaier and Seidel（1988），Delsarte和Seidel（1989）证明了欧几里德2e设计基数的（Fisher型）下界。令Y为R_n的子集，令P_e（Y）为向量空间，该向量空间包含所有受限于Y且其阶数最大为e的多项式。然后从Neumaier-Seidel和Delsarte-Seidel给出的论证中，很容易看出| X |。 ≥dim（P_e（S））成立，其中S =∪_（i = 1）〜p S_i。在某些特殊情况下，由Delsarte和Seidel证明的实际下界比这更好。但是，在S上进行设计时，绑定的dim（P_e（S））是自然且通用的。从这个角度来看，我们称欧几里德2e-design X为| X |。 = dim（P_e（S））在p个同心球上进行严格的2e设计。此外，如果dim（P_e（S））= dim（P_e（R〜n））（=（_e〜（n + e）））成立，那么我们称X为欧几里得紧2e设计。我们通过在欧几里得空间上应用加法公式来研究紧密欧几里德2e设计的性质。此外，我们给出了具有恒定权重的欧几里得紧4-设计的分类。可以认为我们的主要结果是在Box和Hunter（1957）的意义上给出R〜n中2级可旋转设计的分类，并具有可能的最小尺寸（_2〜（n + 2））。我们还给出了R〜2中具有非恒定权重的非平凡欧几里得紧4e设计的示例，这与Neumaier和Seidel（1988）的推论相反，即即使对于非恒定重量情况，也没有非平凡的欧几里得紧2e设计。 2e≥4。

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