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

当某多项式函数在n维球面上积分时，如果被积多项式最多为2次，我们阐明了两个同心球面的最优欧几里得设计的条件。最优欧几里得设计的条件由(i)两个球体的半径之比，（ii）从每个球体中选择的点的数量和（iii）每个同心球体的点的配置给出。得到了一个培养公式为最优的条件，并给出了满足条件的欧几里得设计的构造。这些结果被Sankha Indian Journal of Statistics等接受为论文。此外，我们在复数域上引入了互正交t-设计（t-MOD）的概念。我们检查了t-MOD最优的一些条件。我们发现t-MOD相当于一个量子跃迁码。

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.