Duality of statistical optimality and physical optimality for spherical designs and their constructions

在n维球面上积分多项式函数时，如果被积多项式至多为2次，我们阐明了两个同心球面的最优欧氏设计的条件。最优欧几里得设计的条件是：(I)两个球体的半径之比，(Ii)从每个球体中选择的点数，(Iii)每个同心球体的点的配置。我们得到了容积式是最优的条件，并给出了满足该条件的欧几里得设计的构造。这些结果被接受为Sankha印度统计杂志等的论文。此外，我们在复数域上引入了相互正交t-设计(t-mod)的概念。我们研究了t-mod是最优的一些条件。我们发现t-mod等价于量子跳转码。

When we integrate some polynomial function on a n-dimensional sphere, if the integrand polynomial is at most degree 2, we clarified the condition on an optimal Euclidean design with two concentric spheres. Conditions on an optimal Euclidean design is given by (i) the ratio the radius of two spheres, (ii) the number of points chosen from each sphere and (iii) the configuration of points for each concentric sphere. We obtained such conditions that a cubature formula is optimal and gave constructions of Euclidean designs which satisfy the conditions. These results were accepted as papers for Sankha Indian Journal of Statistics, etc.Moreover, we introduced a notion of mutually orthogonal t-designs (t-MOD) overthe complex number field. We examined some conditions for a t-MOD to be optimal. And we find that t-MOD is equivalent to a quantum jump code.