Posted on Categories:Combinatorics, 数学代写, 组合学, 组合数学

# 数学代写|组合数学代写Combinatorial Mathematics代考|MA1510 Association schemes obtained from tight spherical t-designs

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|组合数学代写Combinatorial Mathematics代考|Association schemes obtained from tight spherical t-designs

The contents here are due to Delsarte, Goethals, and Seidel [163]. We faithfully follow [163] in the discussion below (see also [32, Section 7.2]).

Let $A$ be a subset of $[-1,1)$ and let $X$ be an $A$-code on $S^{n-1}$. For $\alpha, \beta \in A^{\prime}(=A \cup{1})$ and for $x, y \in X$, we define
\begin{aligned} & v_\alpha(x)=|{z \in X \mid x \cdot z=\alpha}|, \ & p_{\alpha, \beta}(x, y)=|{z \in X \mid x \cdot z=\alpha, z \cdot y=\beta}| . \end{aligned}
By definition, we have $p_{\alpha, \alpha}(x, x)=v_\alpha(x)$ and $v_1(x)=1$. If for each $\alpha \in A^{\prime}, v_\alpha(x)$ is a constant independent of the choice of $x \in X$, then the $A$-code $X$ is said to be distanceinvariant. Moreover, for any $\alpha, \beta, \gamma \in A^{\prime}$, if $p_{\alpha, \beta}(x, y)$ is a constant independent of the choice of $x, y \in X$ such that $x \cdot y=y$, an association scheme is attached to the $A$-code $X$. To be more precise, let $A^{\prime}=\left{\alpha_0(=1), \alpha_1, \ldots, \alpha_s\right}$, and let $R_i=\left{(x, y) \in X \times X \mid x \cdot y=\alpha_i\right}$. Then
$$X \times X=R_0 \cup R_1 \cup \cdots \cup R_{\mathrm{S}}$$
holds, and $\left(X,\left{R_i\right}_{0 \leq i \leq s}\right)$ becomes an association scheme. The reference [163] gives a condition when an $A$-code $X$ on $S^{n-1}$ has this good property that an association scheme is attached to it (see also [32]). Now, let us define some more notation. Let
$$x^i=\sum_{\ell=0}^i f_{i, \ell} G_{\ell}^{(n)}(x)$$
be the Gegenbauer expansion of the monomial $x^i$. Also, let
$$F_{i, j}(x)=\sum_{\ell=0}^{\min {i, j}} f_{i, \ell} f_{j, \ell} G_{\ell}^{(n)}(x) .$$

## 数学代写|组合数学代写Combinatorial Mathematics代考|Connections of spherical designs with group theory, number theory, modular forms

(a) $t$-Designs obtained as orbits of finite groups
Let us consider what kinds of spherical $t$-designs there are. The most natural way of the construction is to consider orbits of a finite group $G$ in the orthogonal group $\mathrm{O}(n)$. Namely, for $x \in S^{n-1}$ we consider the orbit $x^G$ of $x$ by $G$ as follows:
$$X=x^G=\left{x^g \mid g \in G\right} \subset S^{n-1} .$$
There are many possibilities for $G$. We expect that if we take larger finite subgroups $G$ in $O(n)$, then we may get better designs. This topic was already treated in [32, Chapter 6] in a detailed way, so here we just mention the points that we think important, leaving the details to the book [32]. The most important finite groups are: real reflection groups (including Weyl groups and more generally Coxeter groups), the Conway group Co.0 in the 24-dimensional space and their various subgroups, and Clifford groups. The research in this direction was started by Sobolev in the 1960s (cf. [412]). See also Sidelnikov [428, 429]. There are some works on properties of finite groups such that orbits become $t$-designs (for details, see [32]). Here we just mention some important facts that we believe interesting.

For each $n \geq 3$, among spherical $t$-designs that are obtained as an orbit of a finite group $G$ of $O(n)$, those with large $t$ are not yet found. We remark the following.

## 数学代写|组合数学代写Combinatorial Mathematics代考|Association schemes obtained from tight spherical t-designs

$$v_\alpha(x)=|z \in X| x \cdot z=\alpha\left|, \quad p_{\alpha, \beta}(x, y)=\right| z \in X|x \cdot z=\alpha, z \cdot y=\beta| .$$

\left 缺少或无法识别的分隔符 $\quad$.然后
$$X \times X=R_0 \cup R_1 \cup \cdots \cup R_{\mathrm{S}}$$

$$x^i=\sum_{\ell=0}^i f_{i, \ell} G_{\ell}^{(n)}(x)$$

$$F_{i, j}(x)=\sum_{\ell=0}^{\min i, j} f_{i, \ell} f_{j, \ell} G_{\ell}^{(n)}(x) .$$

## 数学代写|组合数学代写Combinatorial Mathematics代考|Connections of spherical designs with group theory, number theory, modular forms

(一种) $t$-作为有限群轨道获得的设计

\left 缺少或无法识别的分隔符

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。