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# 数学代写|组合数学代写Combinatorial Mathematics代考|MA1510 Finite subsets on spheres

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## 数学代写|组合数学代写Combinatorial Mathematics代考|Finite subsets on spheres

In this section, we first explain that similar methods to the study of finite subsets of association schemes are used for the study of finite subsets of spheres. The origin of this study can be found in Delsarte, Goethals, and Seidel [163]. This will be described as Delsarte’s theory on spheres. See also the survey article [38].

The main purpose of the study that we call algebraic combinatorics on a sphere is to study “good” finite subsets of the sphere. What “good” means can be regarded as a part of the problem. Actually, what “good” is not unique and there are various viewpoints. Among them, roughly speaking, we will first treat the following two viewpoints: the viewpoint from coding theory and the viewpoint from design theory.

## 数学代写|组合数学代写Combinatorial Mathematics代考|Study of ffnite sets on the sphere from the viewpoint of coding theory

We divide this subsection into parts (a) to (e) as follows.
(a) Let $N$ be a natural number. Among all the $N$-element subsets of the sphere, find subsets with the property that the minimum value of the non-zero distances (i. e., the minimum distance) is the largest. Then classify such sets. (Such sets are called “optimal” codes.)

This problem is also called the “Tammes problem” [447]. This problem, in botany, originated from the problem to study the locations of the pollen grain on a pistil of flower. The classification of optimal codes for the 2-dimensional sphere $S^2$ (in the 3-dimensional Euclidean space) had been known for $N \leq 12$ and $N=24$ until relatively recently (for the details, see Ericson and Zinoviev [180]). This problem for $N=13$ was solved by Musin and Tarasov (2012) [363] and for $N=14$ also by Musin and Tarasov (2015) [364].
The following problem (b) is in a similar direction to problem (a).
(b) Suppose that a positive real number is given. Among all the subsets of the sphere, what is the largest size of them having the property that the minimum distance is greater than or equal to that given positive real number? Then also classify those subsets with this largest size.

In the unit sphere $S^{n-1}$ in the real Euclidean space $\mathbb{R}^n$, the problem of finding a subset whose Euclidean distance between the distinct points in it are at least 1 (or equivalently the geodesic distance on the sphere is at least $\pi / 3$, or the central angle is at least 60 degrees, or the minimum inner product is at most $1 / 2$ ) is well known as the problem of finding the kissing number $k(n)$.

## 数学代写|组合数学代写Combinatorial Mathematics代考|Study of ffnite sets on the sphere from the viewpoint of coding theory

(a) 让 $N$ 是 个自然数。在所有的 $N$ – 球体的元拜子集，找到具有非雬距离 (即最小距离) 的最小值最大的属性的子集。然后对这 些集合进行分宩。（这样的集合称为“最优”代码。）

(b) 假设给定一个正实数。在球体的所有子集中，具有最小距离大于或等于给定正实数的性质的最大尺寸是多少? 然后也对具有最 大尺寸的那些子焦进行分类。

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