### 数学代写|matlab代写|Hamming Codes with Maple

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## 数学代写|matlab代写|Hamming Codes with Maple

In this section, we will show how Maple can be used to construct and correct errors in Hamming codes. We will consider the $[15,11]$ Hamming code.
Because some of the functions that we will use are in the Maple ListTools, LinearAlgebra, and Modular packages, we will begin by including these packages. In addition, we will enter the following interface command to cause Maple to display all matrices of size $50 \times 50$ and smaller throughout the remainder of this Maple session.

with(ListTools):
$>$ with(LinearAl gebra):
$>$ with (Modular):
$>$ interface $(r$ tablesize=50) :
Next we will construct the parity check matrix $H$ for the code. We first enter the length $m=4$ of the vectors that form the columns of $H$.
$>m:=4:$
Recall that the columns of $H$ are binary expressions of the integers $1,2, \ldots, 2^{m}-1$. We can obtain the binary expression of an integer in Maple by using the Maple convert function. For example, we can obtain the binary expression of the integer 4 by entering the following command.
$>c b:=$ convert $(4$, base 2$)$;
$c b:=[0,0,1]$
The entries in the preceding result for $c b$ are the coefficients in the expression $0 \cdot\left(2^{0}\right)+0 \cdot\left(2^{1}\right)+1 \cdot\left(2^{2}\right)$ of the integer 4 . Note that $c b$ contains only three positions, whereas for the columns of $H$ we want binary vectors of length $m=4$ positions. That is, to be placed as the fourth column of $H$, we would want the number 4 to be converted to binary vector $[0,0,1,0]$. Furthermore, the binary digits in $c b$ are the reverse of how they should be expressed in the fourth column of $H$. To be directly placed as the fourth column of $H$, the number 4 should be converted to the binary vector $[0,1,0,0]$. We can use the following command to take care of these two problems.

## 数学代写|matlab代写|Hamming Codes with MATLAB

In this section, we will show how MATLAB can be used to construct and correct errors in Hamming codes. We will consider the [15, 11] Hamming code.

We will begin by constructing the parity check matrix $H$ for the code. We first enter the length $m=4$ of the vectors that form the columns of $H$.
$$\mathrm{m}=4$$
Recall that the columns of $H$ are binary expressions of the integers $1,2, \ldots, 2^{m}-1$. We can obtain the binary expression of an integer in MATLAB by using the MATLAB dec2bin function. For example, we can obtain the binary expression of the integer 4 by entering the following command.
The first parameter in the preceding command is the normal base 10 expression of an integer, while the second parameter is the number of bits that are to be used in the binary expression of the integer. The output displayed for $c b$ is a string containing the coefficients in the expression $4=0 \cdot\left(2^{3}\right)+1 \cdot\left(2^{2}\right)+0 \cdot\left(2^{1}\right)+0 \cdot\left(2^{0}\right)$ as characters. However, rather than having these coefficients stored as characters in a string, we want these coefficients to be stored as positions in a vector. We can take care of this by entering the following for loop.
$>$ for $i=1:$ length $(\mathrm{cb})$
$b v(i)=\operatorname{str} 2 \operatorname{num}(c b(i))$;
end
The MATLAB length function is designed to count the number of characters in a string. Thus, the preceding for loop takes each of the binary digits in the string $c b$, converts them from characters into numbers using the MATLAB str2num function, and stores the resulting integers as positions in the vector $b v$. To see the contents of $b v$, we can enter the following command.
$\gg \mathrm{bv}$
$$b v=$$

## 数学代写|matlab代写|Computer Exercises

1. Use a Hadamard matrix to construct the codewords in a $(31,16)$ code with 32 codewords. What is the maximum number of bit errors that are guaranteed to be uniquely correctable in this code? Correct the vector (0011110000101100001011000011110) to a codeword in this code.
2. As we mentioned in Section $3.3$, the $(32,16)$ Reed-Muller code was used in the Mariner 9 space probe when it transmitted photographs of Mars back to Earth.
(a) Construct the codewords in the $(32,16)$ Reed-Muller code. What is the maximum number of bit errors that are guaranteed to be uniquely correctable in this code?
(b) Correct the vector (11100101011010011110101101101001) to a codeword in the $(32,16)$ Reed-Muller code.
3. Find a parity check matrix for the code in Example 3.4.Find a parity check matrix for the code for which you constructed a generator matrix in Exercise 7 .
4. Let $C$ be the $[31,26]$ Hamming code.
5. (a) Construct the parity check matrix $H$ and a generator matrix $G$ for $C$.
6. (b) Find the number of codewords in $C$.
7. (c) Construct the codeword $w G$ in $C$ that results from the vector $\mathbf{w}=(10110101110110111110111000)$
8. (d) Correct the vector (1101011100110110110101011110111) to a codeword in $C$.
9. (e) Correct the vector (1101010110111010110011000011101) to a codeword in $C$.
10. Let $C$ be the $[63,57]$ Hamming code.
11. (a) Construct the parity check matrix and a generator matrix for $C$.
12. (b) Find the number of codewords in $C$.
13. (c) Construct two of the codewords in $C$.
14. Consider the Maple command on page 83 in which we used the convert function to convert the syndrome of $r$ from the binary expression of an integer into the normal base 10 expression of the integer, thereby revealing the position in $r$ that contained an error. Recall that we could have also identified the position in $r$ that contained an error by finding the number of the column in $H$ that matched the syndrome of $r$. Write a routine or sequence of commands in Maple to replace the convert command on page 83 in which you find the position in $r$ that contains an error by finding the number of the column in $H$ that matches the syndrome of $r$.
15. Consider the MATLAB command on page 88 in which we used the bin2dec function to convert the syndrome of $r$ from the binary expression of an integer into the normal base 10 expression of the integer, thereby revealing the position in $r$ that contained an error. Recall that we could have also identified the position in $r$ that contained an error by finding the number of the column in $H$ that matched the syndrome of $r$. Write a routine or sequence of commands in MATLAB to replace the bin2dec command on page 88 in which you find the position in $r$ that contains an error by finding the number of the column in $H$ that matches the syndrome of $r$.

## 数学代写|matlab代写|Hamming Codes with Maple

>与（线性代数）：
>与（模块化）：
>界面(rtablesize=50) ：

>米:=4:

>Cb:=兑换(4, 基数 2);
Cb:=[0,0,1]

## 数学代写|matlab代写|Hamming Codes with MATLAB

$$\数学{m} = 4 R和C一个ll吨H一个吨吨H和C这l在米ns这FH一个r和b一世n一个r是和Xpr和ss一世这ns这F吨H和一世n吨和G和rs1,2,…,2米−1.在和C一个n这b吨一个一世n吨H和b一世n一个r是和Xpr和ss一世这n这F一个n一世n吨和G和r一世n米一个吨大号一个乙b是在s一世nG吨H和米一个吨大号一个乙d和C2b一世nF在nC吨一世这n.F这r和X一个米pl和,在和C一个n这b吨一个一世n吨H和b一世n一个r是和Xpr和ss一世这n这F吨H和一世n吨和G和r4b是和n吨和r一世nG吨H和F这ll这在一世nGC这米米一个nd.吨H和F一世rs吨p一个r一个米和吨和r一世n吨H和pr和C和d一世nGC这米米一个nd一世s吨H和n这r米一个lb一个s和10和Xpr和ss一世这n这F一个n一世n吨和G和r,在H一世l和吨H和s和C这ndp一个r一个米和吨和r一世s吨H和n在米b和r这Fb一世吨s吨H一个吨一个r和吨这b和在s和d一世n吨H和b一世n一个r是和Xpr和ss一世这n这F吨H和一世n吨和G和r.吨H和这在吨p在吨d一世spl一个是和dF这rCb一世s一个s吨r一世nGC这n吨一个一世n一世nG吨H和C这和FF一世C一世和n吨s一世n吨H和和Xpr和ss一世这n4=0⋅(23)+1⋅(22)+0⋅(21)+0⋅(20)一个sCH一个r一个C吨和rs.H这在和在和r,r一个吨H和r吨H一个nH一个在一世nG吨H和s和C这和FF一世C一世和n吨ss吨这r和d一个sCH一个r一个C吨和rs一世n一个s吨r一世nG,在和在一个n吨吨H和s和C这和FF一世C一世和n吨s吨这b和s吨这r和d一个sp这s一世吨一世这ns一世n一个在和C吨这r.在和C一个n吨一个ķ和C一个r和这F吨H一世sb是和n吨和r一世nG吨H和F这ll这在一世nGF这rl这这p.>F这r一世=1:l和nG吨H(Cb)$$b在(一世)=字符串⁡2在一个⁡(Cb(一世))$;和nd吨H和米一个吨大号一个乙l和nG吨HF在nC吨一世这n一世sd和s一世Gn和d吨这C这在n吨吨H和n在米b和r这FCH一个r一个C吨和rs一世n一个s吨r一世nG.吨H在s,吨H和pr和C和d一世nGF这rl这这p吨一个ķ和s和一个CH这F吨H和b一世n一个r是d一世G一世吨s一世n吨H和s吨r一世nG$Cb$,C这n在和r吨s吨H和米Fr这米CH一个r一个C吨和rs一世n吨这n在米b和rs在s一世nG吨H和米一个吨大号一个乙s吨r2n在米F在nC吨一世这n,一个nds吨这r和s吨H和r和s在l吨一世nG一世n吨和G和rs一个sp这s一世吨一世这ns一世n吨H和在和C吨这r$b在$.吨这s和和吨H和C这n吨和n吨s这F$b在$,在和C一个n和n吨和r吨H和F这ll这在一世nGC这米米一个nd.$≫b在$bv=$\$

## 数学代写|matlab代写|Computer Exercises

1. 使用 Hadamard 矩阵构造码字(31,16)32 个码字的代码。在此代码中保证唯一可纠正的最大误码数是多少？将向量 (0011110000101100001011000011110) 更正为此代码中的一个码字。
2. 正如我们在章节中提到的3.3， 这(32,16)水手 9 号太空探测器将火星照片传回地球时使用了 Reed-Muller 代码。
(a) 在(32,16)里德-穆勒码。在此代码中保证唯一可纠正的最大误码数是多少？
(b) 将向量 (11100101011010011110101101101001) 修正为(32,16)里德-穆勒码。
3. 为示例 3.4 中的代码找到一个奇偶校验矩阵。为您在练习 7 中构造生成器矩阵的代码找到一个奇偶校验矩阵。
4. 让C成为[31,26]汉明码。
5. (a) 构造奇偶校验矩阵H和一个生成矩阵G为了C.
6. (b) 找出码字的个数C.
7. (c) 构造码字在G在C由向量产生的在=(10110101110110111110111000)
8. (d) 将向量 (1101011100110110110101011110111) 修正为C.
9. (e) 将向量 (1101010110111010110011000011101) 修正为C.
10. 让C成为[63,57]汉明码。
11. (a) 构造奇偶校验矩阵和生成矩阵C.
12. (b) 找出码字的个数C.
13. (c) 构造其中的两个码字C.
14. 考虑第 83 页的 Maple 命令，其中我们使用了 convert 函数来转换r从整数的二进制表达式转换为整数的正常基数 10 表达式，从而揭示在r包含一个错误。回想一下，我们也可以确定在r通过查找中的列号包含错误H符合的综合症r. 在 Maple 中编写一个例程或命令序列来替换第 83 页上的转换命令，您可以在其中找到位置r通过查找中的列号包含错误H匹配的综合症r.
15. 考虑第 88 页上的 MATLAB 命令，其中我们使用 bin2dec 函数来转换r从整数的二进制表达式转换为整数的正常基数 10 表达式，从而揭示在r包含一个错误。回想一下，我们也可以确定在r通过查找中的列号包含错误H符合的综合症r. 在 MATLAB 中编写一个例程或命令序列来替换第 88 页上的 bin2dec 命令，您可以在其中找到位置r通过查找中的列号包含错误H匹配的综合症r.

## 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。