### 数学代写|编码理论代写Coding theory代考|ELEN90030

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## 数学代写|编码理论代写Coding theory代考|Asymptotic Bounds

We now describe what happens to the bounds, excluding the Griesmer Bound, as the code length approaches infinity; these bounds are termed asymptotic bounds. We first need some terminology.

Definition 1.9.26 The information rate, or simply rate, of an $(n, M, d){q}$ code is defined to be $\frac{\log {q} M}{n}$. If the code is actually an $[n, k, d]_{q}$ linear code, its rate is $\frac{k}{n}$, measuring the number of information coordinates relative to the total number of coordinates. In either the linear or nonlinear case, the higher the rate, the higher the proportion of coordinates in a codeword that actually contain information rather than redundancy. The ratio $\frac{d}{n}$ is called the relative distance of the code; as we will see later, the relative distance is a measure of the error-correcting capability of the code relative to its length.

Each asymptotic bound will be either an upper or lower bound on the largest possible rate for a family of (possibly nonlinear) codes over $\mathbb{F}{q}$ of lengths going to infinity with relative distances approaching $\delta$. The function, called the asymptotic normalized rate function, that determines this rate is $$\alpha{q}(\delta)=\limsup {n \rightarrow \infty} \frac{\log {q} A_{q}(n, \delta n)}{n} .$$
As the exact value of $\alpha_{q}(\delta)$ is unknown, we desire upper and lower bounds on this function. An upper bound would indicate that all families with relative distances approaching $\delta$ have rates, in the limit, at most this upper bound. A lower bound indicates that there exists a family of codes of lengths approaching infinity and relative distances approaching $\delta$ whose rates are at least this bound. Three of the bounds in the next theorem involve the entropy function.
Definition 1.9.27 The entropy function is defined for $0 \leq x \leq r=1-q^{-1}$ by
$$H_{q}(x)= \begin{cases}0 & \text { if } x=0, \ x \log {q}(q-1)-x \log {q} x-(1-x) \log _{q}(1-x) & \text { if } 02. ## 数学代写|编码理论代写Coding theory代考|Hamming Codes A binary code permutation equivalent to the code of Example 1.4 .9 was discovered in 1947 by R. W. Hamming while working at Bell Telephone Laboratories. Because of patent considerations, his work was not published until 1950 ; see [895]. This Hamming code actually appeared earlier in C. E. Shannon’s seminal paper [1661]. It was also generalized to codes over fields of prime order by M. J. E. Golay [820]. Given a positive integer m, if one takes an m \times n binary matrix whose columns are nonzero and distinct, the binary code with this parity check matrix must have minimum weight at least 3 by Theorem 1.6.11. Binary Hamming codes \mathcal{H}_{m, 2} arise by choosing an m \times n parity check matrix with the maximum number of columns possible that are distinct and nonzero. Definition 1.10.1 Let m \geq 2 be an integer and n=2^{m}-1. Let H_{m, 2} be an m \times n matrix whose columns are all 2^{m}-1 distinct nonzero binary m-tuples. A code with this parity check matrix is called a binary Hamming code. Changing the column order of H_{m, 2} produces a set of pairwise permutation equivalent codes. Any code in this list is denoted \mathcal{H}{m, 2} and is a \left[2^{m}-1,2^{m}-1-m, 3\right]{2} code. The code \mathcal{H}{3,2} of Example 1.4 .9 is indeed a binary Hamming code. These codes are generalized to Hamming codes \mathcal{H}{m, q} over \mathbb{F}_{q}, all with minimum weight 3 again from Theorem 1.6.11. Definition 1.10.2 Let m \geq 2 be an integer and n=\left(q^{m}-1\right) /(q-1). There are a total of n 1-dimensional subspaces of \mathbb{F}{q}^{m}. Let H{m, q} be an m \times n matrix whose columns are all nonzero m-tuples with one column from each of the distinct 1-dimensional subspaces of F_{q}^{m}. A code with this parity check matrix is called a Hamming code over \mathbb{F}{q}. Re-scaling columns and/or changing column order of H{m, q} produces a set of pairwise monomially equivalent codes. Any code in this list is denoted \mathcal{H}{m, q} and is a \left[\left(q^{m}-1\right) /(q-1),\left(q^{m}-1\right) /(q-1)-m, 3\right]{q} code. The code \mathcal{H}{m, q}^{\perp} is called a simplex code. Example 1.10.3 The parity check matrix of the code in Example 1.9.14 is$$ \left[\begin{array}{cc|cc} -1 & -1 & 1 & 0 \ -1 & 1 & 0 & 1 \end{array}\right] . $$This code satisfies the definition of a Hamming [4,2,3]{3} code, and so \mathcal{H}_{2,3} is the appropriate labeling of this code. ## 数学代写|编码理论代写Coding theory代考|Reed-Muller Codes In 1954 the binary Reed-Muller codes were first constructed and examined by D. E. Muller [1409], and a majority logic decoding algorithm for them was described by I. S. Reed [1581]. The non-binary Reed-Muller codes, called generalized Reed-Muller codes, were developed in [1089,1887]; see also Example 16.4.11 and Section 2.8. We define binary Reed-Muller codes recursively based on the (\mathbf{u} \mid \mathbf{u}+\mathbf{v}) construction; see [1323]. Other constructions of Reed-Muller codes can be found in Chapters 2,16 , and 20 . Definition 1.11.1 For i \in{1,2}, let \mathcal{C}{i} be linear codes both of length n over \mathbb{F}{q}. The (\mathbf{u} \mid \mathbf{u}+\mathbf{v}) construction produces the linear code \mathcal{C} of length 2 n given by \mathcal{C}={(\mathbf{u}, \mathbf{u}+\mathbf{v}) \mid \left.\mathbf{u} \in \mathcal{C}{1}, \mathbf{v} \in \mathcal{C}{2}\right} Remark 1.11.2 Let \mathcal{C}{i}, for i \in{1,2}, be \left[n, k{i}, d_{i}\right]{q} codes with generator and parity check matrices G{i} and H_{i}, respectively. \mathcal{C} obtained by the (\mathbf{u} \mid \mathbf{u}+\mathbf{v}) construction is a \left[2 n, k_{1}+\right. \left.k_{2}, \min \left{2 d_{1}, d_{2}\right}\right]{q} code with generator and parity check matrices$$ G=\left[\begin{array}{c|c} G{1} & G_{1} \
\hline \mathbf{0}{k{2} \times n} & G_{2}
\end{array}\right] \text { and } H=\left[\begin{array}{c|c}
H_{1} & \mathbf{0}{\left(n-k{1}\right) \times n} \
\hline-H_{2} & H_{2}
\end{array}\right]
$$We now define the binary Reed-Muller codes. Definition 1.11.3 Let r and m be integers with 0 \leq r \leq m and 1 \leq m. The r^{\text {th }} order binary Reed-Muller (RM) code of length 2^{m}, denoted \mathcal{R} \mathcal{M}(r, m), is defined recursively. The code \mathcal{R} \mathcal{M}(0, m)={\mathbf{0}, 1}, the \left[2^{m}, 1,2^{m}\right]{2} binary repetition code, and \mathcal{R} \mathcal{M}(m, m)=\mathbb{F}{q}^{2^{m}}, a \left[2^{m}, 2^{m}, 1\right]_{2} code. For 1 \leq r<m, define$$
\mathcal{R} \mathcal{M}(r, m)={(\mathbf{u}, \mathbf{u}+\mathbf{v}) \mid \mathbf{u} \in \mathcal{R} \mathcal{M}(r, m-1), \mathbf{v} \in \mathcal{R} \mathcal{M}(r-1, m-1)}
H_{q}(x)= \begin{cases}0 & \text { if } x=0, \ x \log {q}(q-1)-x \log {q} x-(1 -x) \log _{q}(1-x) & \text { 如果 } 02$。 ## 数学代写|编码理论代写Coding theory代考|Hamming Codes 等效于示例代码的二进制代码排列1.4.9由 RW Hamming 于 1947 年在贝尔电话实验室工作时发现。由于专利的考虑，他的作品直到 1950 年才出版；见[895]。这个汉明码实际上早先出现在 CE Shannon 的开创性论文 [1661] 中。它也被 MJE Golay [820] 推广到素数域上的代码。 给定一个正整数米, 如果一个人拿米×n根据定理 1.6.11，其列非零且不同的二进制矩阵，具有该奇偶校验矩阵的二进制码必须具有至少 3 的最小权重。二进制汉明码H米,2通过选择一个米×n具有最大可能不同且非零列数的奇偶校验矩阵。 定义 1.10.1 让米≥2是一个整数并且n=2米−1. 让H米,2豆米×n列全部为的矩阵2米−1不同的非零二进制米-元组。具有这种奇偶校验矩阵的代码称为二进制汉明码。更改的列顺序H米,2产生一组成对置换等效码。此列表中的任何代码都表示H米,2并且是一个[2米−1,2米−1−米,3]2代码。 编码H3,2示例1.4.9确实是二进制汉明码。这些代码被推广到汉明码H米,q超过Fq，从定理 1.6.11 再次具有最小权重 3。 定义 1.10.2 让米≥2是一个整数并且n=(q米−1)/(q−1). 共有n的一维子空间Fq米. 让H米,q豆米×n列全部为非零的矩阵米- 元组中每个不同的一维子空间中的一列Fq米. 具有这种奇偶校验矩阵的代码称为汉明码Fq. 重新缩放列和/或更改列顺序H米,q产生一组成对的单项式等价码。此列表中的任何代码都表示H米,q并且是一个[(q米−1)/(q−1),(q米−1)/(q−1)−米,3]q代码。编码H米,q⊥称为单纯形码。例 1.10.3 例 1.9.14 中代码的奇偶校验矩阵为 [−1−110 −1101].该代码满足汉明的定义[4,2,3]3代码，等等H2,3是此代码的适当标记。 ## 数学代写|编码理论代写Coding theory代考|Reed-Muller Codes 1954 年，DE Muller [1409] 首次构建和检查了二进制 Reed-Muller 码，IS Reed [1581] 描述了它们的多数逻辑解码算法。非二进制 Reed-Muller 码，称为广义 Reed-Muller 码，是在[1089,1887]; 另见示例 16.4.11 和第 2.8 节。我们基于递归定义二进制 Reed-Muller 码(在∣在+在)建造; 见[1323]。Reed-Muller 码的其他结构可以在第 2,16 和 20 章中找到。 定义 1.11.1 对于一世∈1,2， 让C一世都是长度的线性码n超过Fq. 这(在∣在+在)构造产生线性代码C长度2n由\mathcal{C}={(\mathbf{u}, \mathbf{u}+\mathbf{v}) \mid$ $\left.\mathbf{u} \in \mathcal{C}{1}, \ mathbf{v} \in \mathcal{C}{2}\right}\mathcal{C}={(\mathbf{u}, \mathbf{u}+\mathbf{v}) \mid$ \$\left.\mathbf{u} \in \mathcal{C}{1}, \ mathbf{v} \in \mathcal{C}{2}\right}

G=\left[\begin{array}{c|c} G{1} & G_{1} \ \hline \mathbf{0}{k{2} \times n} & G_{2} \end{array }\right] \text { 和 } H=\left[\begin{array}{c|c} H_{1} & \mathbf{0}{\left(nk{1}\right) \times n} \ \hline-H_{2} & H_{2} \end{array}\right]G=\left[\begin{array}{c|c} G{1} & G_{1} \ \hline \mathbf{0}{k{2} \times n} & G_{2} \end{array }\right] \text { 和 } H=\left[\begin{array}{c|c} H_{1} & \mathbf{0}{\left(nk{1}\right) \times n} \ \hline-H_{2} & H_{2} \end{array}\right]

R米(r,米)=(在,在+在)∣在∈R米(r,米−1),在∈R米(r−1,米−1)

## 有限元方法代写

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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。