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## 计算机代写|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 $\mathbb{F}_q^n$ with non-overlapping spheres.

Definition 1.9.3 The sphere of radius $r$ centered at $\mathbf{u} \in \mathbb{F}q^n$ 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 $\mathbf{u} \in \mathbb{F}q^n,\left|S{q, n, r}(\mathbf{u})\right|=\sum_{i=0}^r\left(\begin{array}{c}n \ i\end{array}\right)(q-1)^i$.
(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 $\alpha=\sum{i=0}^t\left(\begin{array}{c}n \ i\end{array}\right)(q-1)^i$ total vectors. Thus $M \alpha$ cannot exceeed the number $q^n$ of vectors in $\mathbb{F}_q^n$. The résult is now clearr.

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, $d<2 s-1$.

## 计算机代写|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 $\mathcal{C}$ in an $[n, k, n \quad k \mid 1]_q M D S$ eode if and only if $\mathcal{C}^{\perp}$ in an $\left[\begin{array}{lll}n, n & k, k \mid 1]_q\end{array}\right.$ 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.

# 编码理论代考

## 计算机代写|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 Packing $\mathbb{F}q^n$ 具有不重㕷的球体。 定义 1.9.3 半径球体 $r$ 以 $\mathbf{u} \in \mathbb{F} q^n$ 是集合 $S q, n, r(\mathbf{u})=\backslash 1 \mathrm{eft}$ 的分隔符缺失或无法识别 中的所有向量 $\mathbb{F}_q^n$ 准的距离 $\mathbf{u}{\text {最多是 } r .}$

(a) 为 $\mathbf{u} \in \mathbb{F} q^n,|S q, n, r(\mathbf{u})|=\sum_{i=0}^r(n i)(q-1)^i$.
(b) 如果 $\mathcal{C}$ 是一个 $(n, M, d)q$ 代码和 $t=\left\lfloor\frac{d-1}{2}\right\rfloor$, 然后是半径球 定理 $1.9 .6$ (球形包装 (或汉明) 约束) 让 $d \geq 1$. 如果 $t=\left\lfloor\frac{d-1}{2}\right\rfloor$ ，然后 $$B_q(n, d) \leq A_q(n, d) \leq \frac{q^n}{\sum{i=0}^t(n i)(q-1)^i} .$$

## 计算机代写|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。

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## 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。

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