### 数学代写|编码理论代写Coding theory代考|MTH 3018

statistics-lab™ 为您的留学生涯保驾护航 在代写编码理论Coding theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写编码理论Coding theory代写方面经验极为丰富，各种代写编码理论Coding theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|编码理论代写Coding theory代考|The Sphere Packing Bound

The Sphere Packing Bound, also called the Hamming Bound, is based on packing $F_{q}^{n}$ with non-overlapping spheres.

Definition 1.9.3 The sphere of radius $r$ centered at $\mathbf{u} \in F_{q}^{n}$ is the set $S_{q, n, r}(\mathbf{u})=$ $\left{\mathbf{v} \in \mathbb{F}{q}^{n} \mid \mathrm{d}{\mathbf{H}}(\mathbf{u}, \mathbf{v}) \leq r\right}$ of all vectors in $\mathbb{F}_{q}^{n}$ whose distance from $\mathbf{u}$ is at most $r$.
We need the size of a sphere, which requires use of binomial coefficients.
Definition 1.9.4 For $a, b$ integers with $0 \leq b \leq a,\left(\begin{array}{l}a \ b\end{array}\right)$ is the number of $b$-element subsets in an $a$-element set. $\left(\begin{array}{l}a \ b\end{array}\right)=\frac{a !}{b !(a-b) !}$ and is called a binomial coefficient.

The next result is the basis of the Sphere Packing Bound; part (a) is a direct count and part (b) follows from the triangle inequality of Theorem 1.6.2.
Theorem 1.9.5 The following hold.
(a) For $\mathbf{u} \in \mathbb{F}{q}^{n},\left|S{q, n, r}(\mathbf{u})\right|=\sum_{i=0}^{r}\left(\begin{array}{l}n \ i\end{array}\right)(q-1)^{i}$.
(b) If $\mathcal{C}$ is an $(n, M, d)_{q}$ code and $t=\left\lfloor\frac{d-1}{2}\right\rfloor$, then spheres of radius $t$ centered at distinct codewords are disjoint.

Theorem 1.9.6 (Sphere Packing (or Hamming) Bound) Let $d \geq 1 .$ If $t=\left\lfloor\frac{d-1}{2}\right\rfloor$, then
$$B_{q}(n, d) \leq A_{q}(n, d) \leq \frac{q^{n}}{\sum_{i=0}^{t}\left(\begin{array}{c} n \ i \end{array}\right)(q-1)^{i}}$$
Proof: Let $\mathcal{C}$ be an $(n, M, d){q}$ code. By Theorem $1.9 .5$, the spheres of radius $t$ centered at distinct codewords are disjoint, and each such sphere has $\alpha=\sum{i=0}^{t}\left(\begin{array}{l}n \ i\end{array}\right)(q-1)^{i}$ total vectors. Thus $M \alpha$ cannot exceed the number $q^{n}$ of vectors in $\mathbb{F}_{q}^{n}$. The result is now clear.

Remark 1.9.7 The Sphere Packing Bound is an upper bound on the size of a code given its length and minimum distance. Additionally the Sphere Packing Bound produces an upper bound on the minimum distance $d$ of an $(n, M){q}$ code in the following sense. Given $n, M$, and $q$, compute the smallest positive integer $s$ with $M>\frac{q^{n}}{\sum{i=0}^{s}\left(\begin{array}{c}n \ i\end{array}\right)(q-1)^{i}} ;$ for an $(n, M, d)_{q}$ code to exist, $d<2 s-1$.

## 数学代写|编码理论代写Coding theory代考|The Singleton Bound

The Singleton Bound was formulated in [1717]. As with the Sphere Packing Bound, the Singleton Bound is an upper bound on the size of a code.

Theorem 1.9.10 (Singleton Bound) For $d \leq n, A_{q}(n, d) \leq q^{n-d+1}$. Furthermore, if an $[n, k, d]{q}$ linear code exists, then $k \leq n-d+1$; i.e., $k{q}(n, d) \leq n-d+1$.

Remark 1.9.11 In addition to providing an upper bound on code size, the Singleton Bound yields the upper bound $d \leq n-\log {q}(M)+1$ on the minimum distance of an $(n, M, d){q}$ code.
Definition 1.9.12 A code for which equality holds in the Singleton Bound is called maximum distance separable (MDS). No code of length $n$ and minimum distance $d$ has more codewords than an MDS code with parameters $n$ and $d$; equivalently, no code of length $n$ with $M$ codewords has a larger minimum distance than an MDS code with parameters $n$ and $M$. MDS codes are discussed in Chapters $3,6,8,14$, and 33 .
The following theorem is proved using Theorem 1.6.11.
Theorem $1.9 .13 \mathcal{C}$ is an $[n, k, n-k+1]{q} M D S$ code if and only if $\mathcal{C}^{\perp}$ is an $[n, n-k, k+1]{q}$ MDS code.
Example 1.9.14 Let $\mathcal{H}{2,3}$ be the $[4,2]{3}$ ternary linear code with generator matrix
$$G_{2,3}=\left[\begin{array}{cc|cc} 1 & 0 & 1 & 1 \ 0 & 1 & 1 & -1 \end{array}\right] .$$
Examining inner products of the rows of $G_{2,3}$, we see that $\mathcal{H}{2,3}$ is self-orthogonal of dimension half its length; so it is self-dual. Using Theorem $1.6 .2(\mathrm{~h}), A{0}\left(\mathcal{H}{2,3}\right)=1, A{3}\left(\mathcal{H}{2,3}\right)=8$, and $A{i}\left(\mathcal{H}{2,3}\right)=0$ otherwise. In particular $\mathcal{H}{2,3}$ is a $[4,2,3]_{3}$ code and hence is MDS.

## 数学代写|编码理论代写Coding theory代考|The Griesmer Bound

The Griesmer Bound $[855]$ is a lower bound on the length of a linear code given its dimension and minimum weight.

Theorem 1.9.18 (Griesmer Bound) Let $\mathcal{C}$ be an $[n, k, d]{q}$ linear code with $k \geq 1$. Then $$n \geq \sum{i=0}^{k-1}\left[\frac{d}{q^{i}}\right] .$$
Remark 1.9.19 One can interpret the Griesmer Bound as an upper bound on the code size given its length and minimum weight. Specifically, $B_{q}(n, d) \leq q^{k}$ where $k$ is the largest positive integer such that $n \geq \sum_{i=0}^{k-1}\left\lceil\frac{d}{q^{2}}\right\rceil$. This bound can also be interpreted as a lower bound on the length of a linear code of given dimension and minimum weight; that is, $n_{q}(k, d) \geq \sum_{i=0}^{k-1}\left\lceil\frac{d}{q^{2}}\right\rceil$. Finally, the Griesmer Bound can be understood as an upper bound on the minimum weight given the code length and dimension; given $n$ and $k, d_{q}(n, k)$ is at most the largest $d$ for which the bound holds.

Example 1.9.20 Suppose we wish to find the smallest code length $n$ such that an $[n, 4,3]{2}$ code can exist. By the Griesmer Bound $n \geq\left\lceil\frac{3}{1}\right\rceil+\left\lceil\frac{3}{2}\right\rceil+\left\lceil\frac{3}{4}\right\rceil+\left\lceil\frac{3}{8}\right\rceil=3+2+1+1=7$. Note that equality in this bound is attained by the $[7,4,3]{2}$ code $\mathcal{H}_{3,2}$ of Examples 1.4.9 and 1.6.10.

## 数学代写|编码理论代写Coding theory代考|The Sphere Packing Bound

Sphere Packing Bound，也称为 Hamming Bound，是基于 PackingFqn具有不重叠的球体。

(a) 为在∈Fqn,|小号q,n,r(在)|=∑一世=0r(n 一世)(q−1)一世.
(b) 如果C是一个(n,米,d)q代码和吨=⌊d−12⌋, 然后是半径球吨以不同码字为中心是不相交的。

## 数学代写|编码理论代写Coding theory代考|The Singleton Bound

Singleton Bound 是在 [1717] 中制定的。与 Sphere Packing Bound 一样，Singleton Bound 是代码大小的上限。

G2,3=[1011 011−1].

## 数学代写|编码理论代写Coding theory代考|The Griesmer Bound

Griesmer 绑定[855]是给定其尺寸和最小权重的线性代码长度的下限。

n≥∑一世=0ķ−1[dq一世].

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。