## 数学代写|信息论代写information theory代考|FEO3350

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information 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。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论information theory相关的作业也就用不着说。

## 数学代写|信息论代写information theory代考|The SMI of a System of Interacting Particles in Pairs Only

In this section we consider a special case of a system of interacting particles. We start with an ideal gas-i.e. system for which we can neglect all intermolecular interactions. Strictly speaking, such a system does not exist. However, if the gas is very dilute such that the average intermolecular distance is very large the system behaves as if there are no interactions among the particle.

Next, we increase the density of the particles. At first we shall find that pairinteractions affect the thermodynamics of the system. Increasing further the density, triplets, quadruplets, and so on interactions, will also affect the behavior of the system. In the following we provide a very brief description of the first order deviation from ideal gas; systems for which one must take into account pair-interactions but neglect triplet and higher order interactions. The reader who is not interested in the details of the derivation can go directly to the result in Eq. (2.51) and the following analysis of the MI.

We start with the general configurational PF of the system, Eq. (2.31) which we rewrite in the form:
$$Z_N=\int d R^N \prod_{i<j} \exp \left[-\beta U_{i j}\right]$$
where $U_{i j}$ is the pair potential between particles $i$ and $j$. It is assumed that the total potential energy is pairwise additive.
Define the so-called Mayer $f$-function, by:
$$f_{i j}=\exp \left(-\beta U_{i j}\right)-1$$
We can rewrite $Z_N$ as:
$$Z_N=\int d R^N \prod_{i<j}\left(f_{i j}+1\right)=\int d R^N\left[1+\sum_{i<j} f_{i j}+\sum f_{i j} f_{j k}+\cdots\right]$$
Neglecting all terms beyond the first sum, we obtain:
$$Z_N=V^N+\frac{N(N-1)}{2} \int f_{12} d R^N=V^N+\frac{N(N-1)}{2} V^{N-2} \int f_{12} d R_1 d R_2$$

## 数学代写|信息论代写information theory代考|Entropy-Change in Phase Transition

In this section, we shall discuss the entropy-changes associated with phase transitions. Here, by entropy we mean thermodynamic entropy, the units of which are cal/(deg $\mathrm{mol}$ ). However, as we have seen in Chap. 5 of Ben-Naim [1]. The entropy is up to a multiplicative constant an SMI defined on the distribution of locations and velocities (or momenta) of all particles in the system at equilibrium. To convert from entropy to SMI one has to divide the entropy by the factor $k_B \log _e 2$, where $k_B$ is the Boltzmann constant, and $\log _e 2$ is the natural $\log$ arithm of 2 , which we denote by $\ln 2$. Once we do this conversion from entropy to SMI we obtain the SMI in units of bits. In this section we shall discuss mainly the transitions between gases, liquids and solids. Figure 2.9 shows a typical phase diagram of a one-component system. For more details on phase diagrams, see Ben-Naim and Casadei [8].

It is well-known that solid has a lower entropy than liquid, and liquid has a lower entropy of a gas. These facts are usually interpreted in terms of order-disorder. This interpretation of entropy is invalid; more on this in Ben-Naim [6]. Although, it is true that a solid is viewed as more ordered than liquid, it is difficult to argue that a liquid is more ordered or less ordered than a gas.

In the following we shall interpret entropy as an SMI, and different entropies in terms of different MI due to different intermolecular interactions. We shall discuss changes of phases at constant temperature. Therefore, all changes in SMI (hence, in entropy) will be due to locational distributions; no changes in the momenta distribution.

The line SG in Fig. 2.9 is the line along in which solid and gas coexist. The slope of this curve is given by:
$$\left(\frac{d P}{d T}\right)_{e q}=\frac{\Delta S_s}{\Delta V_s}$$
In the process of sublimation ( $s$, the entropy-change and the volume change for both are always positive. We denoted by $\Delta V_s$ the change in the volume of one mole of the substance, when it is transferred from the solid to the gaseous phase. This volume change is always positive. The reason is that a mole of the substance occupies a much larger volume in the gaseous phase than in the liquid phase (at the same temperature and pressure).

The entropy-change $\Delta S_s$ is also positive. This entropy-change is traditionally interpreted in terms of transition from an ordered phase (solid) to a disordered (gaseous) phase. However, the more correct interpretation is that the entropy-change is due to two factors; the huge increase in the accessible volume available to each particle and the decrease in the extent of the intermolecular interaction. Note that the slope of the SG curve is quite small (but positive) due to the large $\Delta V_s$.

# 信息论代写

## 数学代写|信息论代写information theory代考|The SMI of a System of Interacting Particles in Pairs Only

$$Z_N=\int d R^N \prod_{i<j} \exp \left[-\beta U_{i j}\right]$$

$$f_{i j}=\exp \left(-\beta U_{i j}\right)-1$$

$$Z_N=\int d R^N \prod_{i<j}\left(f_{i j}+1\right)=\int d R^N\left[1+\sum_{i<j} f_{i j}+\sum f_{i j} f_{j k}+\cdots\right]$$

$$Z_N=V^N+\frac{N(N-1)}{2} \int f_{12} d R^N=V^N+\frac{N(N-1)}{2} V^{N-2} \int f_{12} d R_1 d R_2$$

## 数学代写|信息论代写information theory代考|Entropy-Change in Phase Transition

$$\left(\frac{d P}{d T}\right)_{e q}=\frac{\Delta S_s}{\Delta V_s}$$

## 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。

## 数学代写|信息论代写information theory代考|EE430

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information 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。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论information theory相关的作业也就用不着说。

## 数学代写|信息论代写information theory代考|The Forth Step: The SMI of Locations and Momentaof N Independent Particles in a Box of Volume V.Adding a Correction Due to Indistinguishabilityof the Particles

The final step is to proceed from a single particle in a box, to $N$ independent particles in a box of volume $V$, Fig. 2.4.

We say that we know the microstate of the particle, when we know the location $(x, y, z)$, and the momentum $\left(p_x, p_y, p_z\right)$ of one particle within the box. For a system of $N$ independent particles in a box, we can write the SMI of the system as $N$ times the SMI of one particle, i.e., we write:
$$\mathrm{SMI}(N \text { independent particles })=N \times \mathrm{SMI} \text { (one particle) }$$
This is the SMI for $N$ independent particles. In reality, there could be correlation among the microstates of all the particles. We shall mention here correlations due to the indistinguishability of the particles, and correlations is due to intermolecular interactions among all the particles. We shall discuss these two sources of correlation separately. Recall that the microstate of a single particle includes the location and the momentum of that particle. Let us focus on the location of one particle in a box of volume $V$. We write the locational SMI as:
$$H_{\max }(\text { location })=\log V$$
For $N$ independent particles, we write the locational SMI as:
$$H_{\max } \text { (locations of N particles) }=\sum_{i=1}^N H_{\max }(\text { one particle })$$
Since in reality, the particles are indistinguishable, we must correct Eq. (2.22). We define the mutual information corresponding to the correlation between the particles as:

$$I(1 ; 2 ; \ldots ; N)=\ln N !$$
Hence, instead of (2.22), for the SMI of $N$ indistinguishable particles, will write:
$$H(\text { Nparticles })=\sum_{i=1}^N H(\text { oneparticle })-\ln N !$$
A detailed justification for introducing $\ln N$ ! as a correction due to indistinguishability of the particle is discussed in Sect. 5.2 of Ben-Naim [1]. Here we write the final result for the SMI of $N$ indistinguishable (but non-interacting) particles as:
$$H(N \text { indistinguishable particles })=N \log V\left(\frac{2 \pi m e k_B T}{h^2}\right)^{3 / 2}-\log N !$$

## 数学代写|信息论代写information theory代考|The Entropy of a System of Interacting Particles. Correlations Due to Intermolecular Interactions

In this section we derive the most general relationship between the SMI (or the entropy) of a system of interacting particles, and the corresponding mutual information (MI). Later on in this chapter we shall apply this general result to some specific cases. The implication of this result is very important in interpreting the concept of entropy in terms of SMI. In other words, the “informational interpretation” of entropy is effectively extended for all systems of interacting particles at equilibrium.
We start with some basic concepts from classical statistical mechanics [7]. The classical canonical partition function (PF) of a system characterized by the variable $T, V, N$, is:
$$Q(T, V, N)=\frac{Z_N}{N ! \Lambda^{3 N}}$$
where $\Lambda^3$ is called the momentum partition function (or the de Broglie wavelength), and $Z_N$ is the configurational PF of the system”
$$Z_N=\int \cdots \int d R^N \exp \left[-\beta U_N\left(R^N\right)\right]$$
Here, $U_N\left(R^N\right)$ is the total interaction energy among the $N$ particles at a configuration $R^N=R_1, \cdots, R_N$. Statistical thermodynamics provides the probability density for finding the particles at a specific configuration $R^N=R_1, \cdots, R_N$, which is:
$$P\left(R^N\right)=\frac{\exp \left[-\beta U_N\left(R^N\right)\right]}{Z_N}$$
where $\beta=\left(k_B T\right)^{-1}$ and $T$ the absolute temperature. In the following we chose $k_B=1$. This will facilitate the connection between the entropy-change and the change in the SMI. When there are no intermolecular interactions (ideal gas), the configurational $\mathrm{PF}$ is $Z_N=V^N$, and the corresponding partition function is reduced to:
$$Q^{i g}(T, V, N)=\frac{V^N}{N ! \Lambda^{3 N}}$$
Next we define the change in the Helmholtz energy $(A)$ due to the interactions as:
$$\Delta A=A-A^{i g}=-T \ln \frac{Q(T, V, N)}{Q^{i g}(T, V, N)}=-T \ln \frac{Z_N}{V^N}$$
This change in Helmholtz energy corresponds to the process of “turning-on” the interaction among all the particles at constant $(T, V, N)$, Fig. 2.5.
The corresponding change in the entropy is:
\begin{aligned} \Delta S & =-\frac{\partial \Delta A}{\partial T}=\ln \frac{Z_N}{V^N}+T \frac{1}{Z_N} \frac{\partial Z_N}{\partial T} \ & =\ln Z_N-N \ln \mathrm{V}+\frac{1}{T} \int d R^N P\left(R^N\right) U_N\left(R^N\right) \end{aligned}
We now substitute $U_N\left(R^N\right)$ from (2.36) into (2.35) to obtain the expression for the change in entropy corresponding to “turning on” the interactions:
$$\Delta S=-N \ln V-\int P\left(R^N\right) \ln P\left(R^N\right) d R^N$$

# 信息论代写

## 数学代写|信息论代写information theory代考|The Forth Step: The SMI of Locations and Momentaof N Independent Particles in a Box of Volume V.Adding a Correction Due to Indistinguishabilityof the Particles

$$\mathrm{SMI}(N \text { independent particles })=N \times \mathrm{SMI} \text { (one particle) }$$

$$H_{\max }(\text { location })=\log V$$

$$H_{\max } \text { (locations of N particles) }=\sum_{i=1}^N H_{\max }(\text { one particle })$$

$$I(1 ; 2 ; \ldots ; N)=\ln N !$$

$$H(\text { Nparticles })=\sum_{i=1}^N H(\text { oneparticle })-\ln N !$$

$$H(N \text { indistinguishable particles })=N \log V\left(\frac{2 \pi m e k_B T}{h^2}\right)^{3 / 2}-\log N !$$

## 数学代写|信息论代写information theory代考|The Entropy of a System of Interacting Particles. Correlations Due to Intermolecular Interactions

$$Q(T, V, N)=\frac{Z_N}{N ! \Lambda^{3 N}}$$

$$Z_N=\int \cdots \int d R^N \exp \left[-\beta U_N\left(R^N\right)\right]$$

$$P\left(R^N\right)=\frac{\exp \left[-\beta U_N\left(R^N\right)\right]}{Z_N}$$

$$Q^{i g}(T, V, N)=\frac{V^N}{N ! \Lambda^{3 N}}$$

$$\Delta A=A-A^{i g}=-T \ln \frac{Q(T, V, N)}{Q^{i g}(T, V, N)}=-T \ln \frac{Z_N}{V^N}$$

\begin{aligned} \Delta S & =-\frac{\partial \Delta A}{\partial T}=\ln \frac{Z_N}{V^N}+T \frac{1}{Z_N} \frac{\partial Z_N}{\partial T} \ & =\ln Z_N-N \ln \mathrm{V}+\frac{1}{T} \int d R^N P\left(R^N\right) U_N\left(R^N\right) \end{aligned}

$$\Delta S=-N \ln V-\int P\left(R^N\right) \ln P\left(R^N\right) d R^N$$

## 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。

## 数学代写|信息论代写information theory代考|COMP2610

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information 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。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论information theory相关的作业也就用不着说。

## 数学代写|信息论代写information theory代考|Third Step: Combining the SMI for the Location and Momentum of a Particle in a $1 D$ System. Addition of Correction Due to Uncertainty

If the location and the momentum (or velocity) of the particles were independent events, then the joint SMI of location and momentum would be the sum of the two SMIs in Eqs. (2.4) and (2.12). Therefore, for this case we write:
\begin{aligned} H_{\max }(\text { location and momentum }) & =H_{\max }(\text { location })+H_{\max }(\text { momentum }) \ & =\log \left[\frac{L \sqrt{2 \pi e m k_B T}}{h_x h_p}\right] \end{aligned}
It should be noted that in the very writing of Eq. (2.14), the assumption is made that the location and the momentum of the particle are independent. However, quantum mechanics imposes restriction on the accuracy in determining both the location $x$ and the corresponding momentum $p_x$. Originally, the two quantities $h_x$ and $h_p$ that we defined above, were introduced because we did not care to determine the location and the momentum with an accuracy better than $h_x$ and $h_p$, respectively. Now, we must acknowledge that quantum mechanics imposes upon us the uncertainty condition, about the accuracy with which we can determine simultaneously both the location and the corresponding momentum of a particle. This means that in Eq. (2.14), $h_x$ and $h_p$ cannot both be arbitrarily small; their product must be of the order of Planck constant $h=6.626 \times 10^{-34} \mathrm{Js}$. Therefore, we introduce a new parameter $h$, which replaces the product:
$$h_x h_p \approx h$$
Accordingly, we modify Eq. (2.14) to:
$$H_{\max }(\text { location and momentum })=\log \left[\frac{L \sqrt{2 \pi e m k_B T}}{h}\right]$$

## 数学代写|信息论代写information theory代考|The SMI of One Particle in a Box of Volume $\mathrm{V}$

Figure 2.3 shows one simple particle in a cubic box of volume $V$.
To proceed from the 1D to the 3D system, we assume that the locations of the particle along the three axes $x, y$ and $z$ are independent. With this assumption, we can write the SMI of the location of the particle in a cube of edges $L$, as a sum of the SMI along $x, y$, and $z$, i.e.
$$H(\text { location in } 3 \mathrm{D})=3 H_{\max } \text { (location in 1D) }$$
We can do the same for the momentum of the particle if we assume that the momentum (or the velocity) along the three axes $x, y$ and $z$ are independent. Hence, we can write the SMI of the momentum as:
$$H_{\max }(\text { momentum in } 3 \mathrm{D})=3 H_{\max }(\text { momentum in 1D) }$$
We can now combine the SMI of the locations and momenta of one particle in a box of volume $V$, taking into account the uncertainty principle, to obtain the result:
$$H_{\max }(\text { location and momentum in } 3 \mathrm{D})=3 \log \left[\frac{L \sqrt{2 \pi e m k_B T}}{h}\right]$$

# 信息论代写

## 数学代写|信息论代写information theory代考|Third Step: Combining the SMI for the Location and Momentum of a Particle in a $1 D$ System. Addition of Correction Due to Uncertainty

$$H[f(x)]=-\int f(x) \log f(x) d x$$

$$f_{e q}(x)=\frac{1}{L}$$

$$H(\text { locations in } 1 D)=\log L$$

$$H\left(\text { locations in 1D) }=\log L-\log h_x\right.$$

## 数学代写|信息论代写information theory代考|The SMI of One Particle in a Box of Volume $\mathrm{V}$

$$H(\text { location in } 3 \mathrm{D})=3 H_{\max } \text { (location in 1D) }$$

$$H_{\max }(\text { momentum in } 3 \mathrm{D})=3 H_{\max }(\text { momentum in 1D) }$$

$$H_{\max }(\text { location and momentum in } 3 \mathrm{D})=3 \log \left[\frac{L \sqrt{2 \pi e m k_B T}}{h}\right]$$

## 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。

## 计算机代写|编码理论代写Coding theory代考|ELEN90030

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

## 计算机代写|编码理论代写Coding theory代考|Distance and Weight

The error-correcting capability of a code is keyed directly to the concepts of Hamming distance and Hamming weight. ${ }^3$

Definition 1.6.1 The (Hamming) distance between two vectors $\mathbf{x}, \mathbf{y} \in \mathbb{F}q^n$, denoted $\mathrm{d}{\mathbf{H}}(\mathbf{x}, \mathbf{y})$, is the number of coordinates in which $\mathbf{x}$ and $\mathbf{y}$ differ. The (Hamming) weight of $\mathbf{x} \in \mathbb{F}q^n$, denoted $\mathrm{wt}{\mathrm{H}}(\mathbf{x})$, is the number of coordinates in which $\mathbf{x}$ is nonzero.
Theorem 1.6.2 ([1008, Chapter 1.4]) The following hold.
(a) (nonnegativity) $\mathrm{d}{\mathrm{H}}(\mathbf{x}, \mathbf{y}) \geq 0$ for all $\mathbf{x}, \mathbf{y} \in \mathbb{F}_q^n$. (b) $\mathrm{d}{\mathrm{H}}(\mathbf{x}, \mathbf{y})=0$ if and only if $\mathbf{x}=\mathbf{y}$.
(c) (symmetry) $\mathrm{d}{\mathrm{H}}(\mathbf{x}, \mathbf{y})=\mathrm{d}{\mathrm{H}}(\mathbf{y}, \mathbf{x})$ for all $\mathbf{x}, \mathbf{y} \in \mathbb{F}q^n$. (d) (triangle inequality) $\mathrm{d}{\mathrm{H}}(\mathbf{x}, \mathbf{z}) \leq \mathrm{d}{\mathrm{H}}(\mathbf{x}, \mathbf{y})+\mathrm{d}{\mathrm{H}}(\mathbf{y}, \mathbf{z})$ for all $\mathbf{x}, \mathbf{y}, \mathbf{z} \in \mathbb{F}q^n$. (e) $\mathrm{d}{\mathrm{H}}(\mathbf{x}, \mathbf{y})=\mathrm{wt}{\mathrm{H}}(\mathbf{x}-\mathbf{y})$ for all $\mathbf{x}, \mathbf{y} \in \mathbb{F}_q^n$. (f) If $\mathbf{x}, \mathbf{y} \in \mathbb{F}_2^n$, then $$w t_H(\mathbf{x}+\mathbf{y})=w{\mathrm{H}}(\mathbf{x})+w_{\mathrm{H}}(\mathbf{y})-2 w t_{\mathrm{H}}(\mathbf{x} \star \mathbf{y})$$
where $\mathbf{x} \star \mathbf{y}$ is the vector in $\mathbb{F}2^n$ which has 1 s precisely in those coordinates where both $\mathbf{x}$ and $\mathbf{y}$ have $1 s$. (g) If $\mathbf{x}, \mathbf{y} \in \mathbb{F}_2^n$, then $\mathrm{wt}{\mathrm{H}}(\mathbf{x} \star \mathbf{y}) \equiv \mathbf{x} \cdot \mathbf{y}(\bmod 2)$. In particular, $\mathrm{wt}_{\mathrm{H}}(\mathbf{x}) \equiv \mathbf{x} \cdot \mathbf{x}(\bmod 2)$.

## 计算机代写|编码理论代写Coding theory代考|Puncturing, Extending, and Shortening Codes

There are several methods to obtain a longer or shorter code from a given code; while this can be done for both linear and nonlinear codes, we focus on linear ones. Two codes can be combined into a single code, for example as described in Section 1.11.

Definition 1.7.1 Let $\mathcal{C}$ be an $[n, k, d]q$ linear code with generator matrix $G$ and parity check matrix $H$. (a) For some $i$ with $1 \leq i \leq n$, let $\mathcal{C}^$ be the codewords of $\mathcal{C}$ with the $i^{\text {th }}$ component deleted. The resulting code, called a punctured code, is an $\left[n-1, k^, d^\right]$ code. If $d>1, k^=k$, and $d^=d$ unless $\mathcal{C}$ has a minimum weight codeword that is nonzero on coordinate $i$, in which case $d^=d-1$. If $d=1, k^=k$ and $d^=1$ unless $\mathcal{C}$ has a weight 1 codeword that is nonzero on coordinate $i$, in which case $k^=k-1$ and $d^ \geq 1$ as long as $\mathcal{C}^$ is nonzero. A generator matrix for $\mathcal{C}^$ is obtained from $G$ by deleting column $i ; G^$ will have dependent rows if $d^=1$ and $k^*=k-1$. Puncturing is often done on multiple coordinates in an analogous manner, one coordinate at a time.
(b) Define $\widehat{\mathcal{C}}=\left{c_1 c_2 \cdots c{n+1} \in \mathbb{F}q^{n+1} \mid c_1 c_2 \cdots c_n \in \mathcal{C}\right.$ where $\left.\sum{i=1}^{n+1} c_i=0\right}$, called the extended code. This is an $[n+1, k, \widehat{d}]_q$ code where $\widehat{d}=d$ or $d+1$. A generator matrix $\widehat{G}$ for $\widehat{\mathcal{C}}$ is obtained by adding a column on the right of $G$ so that every row sum in this $k \times(n+1)$ matrix is 0 . A parity check matrix $\widehat{H}$ for $\widehat{\mathcal{C}}$ is
$$\widehat{H}=\left[\begin{array}{ccc|c} 1 & \cdots & 1 & 1 \ \hline & & 0 \ & H & & \vdots \ & & & 0 \end{array}\right] .$$
(c) Let $S$ be any set of $s$ coordinates. Let $\mathcal{C}(S)$ be all codewords in $\mathcal{C}$ that are zero on $S$. Puncturing $\mathcal{C}(S)$ on $S$ results in the $\left[n-s, k_S, d_S\right]_q$ shortened code $\mathcal{C}_S$ where $d_S \geq d$. If $\mathcal{C}^{\perp}$ has minimum weight $d^{\perp}$ and $s<d^{\perp}$, then $k_S=k-s$.

## 计算机代写|编码理论代写Coding theory代考|Distance and Weight

(a) (非消极性) $\mathrm{dH}(\mathbf{x}, \mathbf{y}) \geq 0$ 对所有人 $\mathbf{x}, \mathbf{y} \in \mathbb{F}q^n$. (二) $\mathrm{dH}(\mathbf{x}, \mathbf{y})=0$ 当且仅当 $\mathbf{x}=\mathbf{y}$. (c) (对称) $\mathrm{dH}(\mathbf{x}, \mathbf{y})=\mathrm{dH}(\mathbf{y}, \mathbf{x})$ 对所有人 $\mathbf{x}, \mathbf{y} \in \mathbb{F} q^n$. (d) (三角不等式) $\mathrm{dH}(\mathbf{x}, \mathbf{z}) \leq \mathrm{dH}(\mathbf{x}, \mathbf{y})+\mathrm{dH}(\mathbf{y}, \mathbf{z})$ 对 所有人 $\mathbf{x}, \mathbf{y}, \mathbf{z} \in \mathbb{F} q^n$. (和) $\mathrm{dH}(\mathbf{x}, \mathbf{y})=w \mathrm{wt}(\mathbf{x}-\mathbf{y})$ 对所有人 $\mathbf{x}, \mathbf{y} \in \mathbb{F}_q^n$. (f) 如果 $\mathbf{x}, \mathbf{y} \in \mathbb{F}_2^n$ ，然后 $$w t_H(\mathbf{x}+\mathbf{y})=w \mathrm{H}(\mathbf{x})+w{\mathrm{H}}(\mathbf{y})-2 w t_{\mathrm{H}}(\mathbf{x} \star \mathbf{y})$$

## 计算机代写|编码理论代写Coding theory代考|Puncturing, Extending, and Shortening Codes

(b) 定义 得 $G$ 这样每一行总和 $k \times(n+1)$ 矩阵是 0 。奇偶校验矩阵 $\widehat{H}$ 为了 $\widehat{\mathcal{C}}$ 是
(c) 让 $S$ 是任何一组 $s$ 坐标。让 $\mathcal{C}(S)$ 是所有的代码字 $\mathcal{C}$ 是零 $S$. 穿刺 $\mathcal{C}(S)$ 上 $S$ 结果是 $\left[n-s, k_S, d_S\right]_q$ 缩短的代码 $\mathcal{C}_S$ 在哪里 $d_S \geq d$. 如果 $\mathcal{C}^{\perp}$ 有最小重量 $d^{\perp}$ 和 $s<d^{\perp}$ ，然后 $k_S=k-s$.

## 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。

## 计算机代写|编码理论代写Coding theory代考|MATH597

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

## 计算机代写|编码理论代写Coding theory代考|Generator and Parity Check Matrices

When choosing between linear and nonlinear codes, the added algebraic structure of linear codes often makes them easier to describe and use. Generally, a linear code is defined by giving either a generator or a parity check matrix.

Definition 1.4.1 Let $\mathcal{C}$ be an $[n, k]_q$ linear code. A generator matrix $G$ for $\mathcal{C}$ is any $G \in \mathbb{F}_q^{k \times n}$ whose row span is $\mathcal{C}$. Because any $k$-dimensional subspace of $\mathbb{F}_q^n$ is the kernel of some linear transformation from $\mathbb{F}_q^n$ onto $\mathbb{F}_q^{n-k}$, there exists $H \in \mathbb{F}_q^{(n-k) \times n}$, with independent many, is called a parity check matrix of $\mathcal{C}$.

Example 1.4.2 Continuing with Example 1.3.2, there are several generator matrices for $\mathcal{C}_1$ including
$$G_1=\left[\begin{array}{llll} 1 & 0 & 0 & 1 \ 0 & 1 & 0 & 1 \ 0 & 0 & 1 & 1 \end{array}\right], G_1^{\prime}=\left[\begin{array}{cccc} 1 & 1 & 1 & 1 \ 1 & 1 & 0 & 0 \ 0 & 1 & 1 & 0 \end{array}\right] \text {, and } G_1^{\prime \prime}=\left[\begin{array}{cccc} 1 & 1 & 0 & 0 \ 0 & 1 & 1 & 0 \ 0 & 0 & 1 & 1 \end{array}\right] .$$
Remark 1.4.3 Any matrix obtained by elementary row operations from a generator matrix for a code remains a generator matrix of that code.

Remark 1.4.4 By Definition 1.4.1, the rows of $G$ form a basis of $\mathcal{C}$, and the rows of $H$ are independent. At times, the requirement may be relaxed so that the rows of $G$ are only required to span $\mathcal{C}$. Similarly, the requirement that the rows of $H$ be independent may be dropped as long as $\mathcal{C}=\left{\mathbf{c} \in \mathbb{F}_q^n \mid H \mathbf{c}^{\top}=0^{\top}\right}$ remains true.

## 计算机代写|编码理论代写Coding theory代考|Orthogonality

In this section we introduce the concept of codes over finite fields. We begin with some notation.

The set of $n$-tuples with entries in $\mathbb{F}_q$ forms an $n$-dimensional vector space, denoted $\mathbb{F}_q^n=\left{x_1 x_2 \cdots x_n \mid x_i \in \mathbb{F}_q, 1 \leq i \leq n\right}$, under componentwise addition of $n$-tuples and componentwise multiplication of $n$-tuples by scalars in $\mathbb{F}_q$. The vectors in $\mathbb{F}_q^n$ will often be denoted using bold Roman characters $\mathbf{x}=x_1 x_2 \cdots x_n$. The vector $\mathbf{0}=00 \cdots 0$ is the zero vector in $\mathbb{F}_q^n$.

There is a natural inner product on $\mathbb{F}q^n$ that often proves useful in the study of codes. ${ }^2$ Definition 1.5.1 The ordinary inner product, also called the Euclidean inner product, on $\mathbb{F}_q^n$ is defined by $\mathbf{x} \cdot \mathbf{y}=\sum{i=1}^n x_i y_i$ where $\mathbf{x}=x_1 x_2 \cdots x_n$ and $\mathbf{y}=y_1 y_2 \cdots y_n$. Two vectors $\mathbf{x}, \mathbf{y} \in \mathbb{F}_q^n$ are orthogonal if $\mathbf{x} \cdot \mathbf{y}=0$. If $\mathcal{C}$ is an $[n, k]_q$ code,
$$\mathcal{C}^{\perp}=\left{\mathbf{x} \in \mathbb{F}_q^n \mid \mathbf{x} \cdot \mathbf{c}=0 \text { for all } \mathbf{c} \in \mathcal{C}\right}$$ is the orthogonal code or dual code of $\mathcal{C}$. $\mathcal{C}$ is self-orthogonal if $\mathcal{C} \subseteq \mathcal{C}^{\perp}$ and self-dual if $\mathcal{C}=\mathcal{C}^{\perp}$.

Theorem 1.5.2 ([1323, Chapter 1.8 $])$ Let $\mathcal{C}$ be an $[n, k]_q$ code with generator and parity check matrices $G$ and $H$, respectively. Then $\mathcal{C}^{\perp}$ is an $[n, n-k]_q$ code with generator and parity check matrices $H$ and $G$, respectively. Additionally $\left(\mathcal{C}^{\perp}\right)^{\perp}=\mathcal{C}$. Furthermore $\mathcal{C}$ is self-dual if and only if $\mathcal{C}$ is self-orthogonal and $k=\frac{n}{2}$.

Example 1.5.3 $\mathcal{C}2$ from Example $1.4 .8$ is a $[4,2]_2$ self-dual code with generator and parity check matrices both equal to $$\left[\begin{array}{llll} 1 & 1 & 0 & 0 \ 0 & 0 & 1 & 1 \end{array}\right] \text {. }$$ The dual of the Hamming $[7,4]_2$ code in Example 1.4.9 is a $[7,3]_2$ code $\mathcal{H}{3,2}^{\perp} . H_{3,2}$ is a generator matrix of $\mathcal{H}{3,2}^{\perp}$. As every row of $H{3,2}$ is orthogonal to itself and every other row of $H_{3,2}, \mathcal{H}{3,2}^{\perp}$ is self-orthogonal. As $\mathcal{H}{3,2}^{\perp}$ has dimension 3 and $\left(\mathcal{H}{3,2}^{\perp}\right)^{\perp}=\mathcal{H}{3,2}$ has dimension 4, $\mathcal{H}_{3,2}^{\perp}$ is not self-dual.

## 计算机代写|编码理论代写Coding theory代考|Generator and Parity Check Matrices

$\mathcal{C}$. 同样，要求的行 $H$ 只要是独立的就可以被丟弃

## 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。

## 计算机代写|编码理论代写Coding theory代考|CS294-226

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

## 计算机代写|编码理论代写Coding theory代考|Finite Fields

Finite fields play an essential role in coding theory. The theory and construction of finite fields can be found, for example, in [1254] and [1408, Chapter 2]. Finite fields, as related specifically to codes, are described in [1008, 1323, 1602]. In this section we give a brief introduction.

Definition 1.2.1 A field $F$ is a nonempty set with two binary operations, denoted $+$ and , satisfying the following properties.
(a) For all $\alpha, \beta, \gamma \in \mathbb{F}, \alpha+\beta \in \mathbb{F}, \alpha \cdot \beta \in \mathbb{F}, \alpha+\beta=\beta+\alpha, \alpha \cdot \beta=\beta \cdot \alpha, \alpha+(\beta+\gamma)=(\alpha+\beta)+\gamma$, $\alpha \cdot(\beta \cdot \gamma)=(\alpha \cdot \beta) \cdot \gamma$, and $\alpha \cdot(\beta+\gamma)=\alpha \cdot \beta+\alpha \cdot \gamma$.
(b) $\mathbb{F}$ possesses an additive identity or zero, denoted 0 , and a multiplicative identity or unity, denoted 1 , such that $\alpha+0=\alpha$ and $\alpha \cdot 1=\alpha$ for all $\alpha \in \mathbb{F}_q$.
(c) For all $\alpha \in \mathbb{F}$ and all $\beta \in \mathbb{F}$ with $\beta \neq 0$, there exists $\alpha^{\prime} \in \mathbb{F}$, called the additive inverse of $\alpha$, and $\beta^* \in \mathbb{F}$, called the multiplicative inverse of $\beta$, such that $\alpha+\alpha^{\prime}=0$ and $\beta \cdot \beta^*=1$

The additive inverse of $\alpha$ will be denoted $-\alpha$, and the multiplicative inverse of $\beta$ will be denoted $\beta^{-1}$. Usually the multiplication operation will be suppressed; that is, $\alpha \cdot \beta$ will be denoted $\alpha \beta$. If $n$ is a positive integer and $\alpha \in \mathbb{F}, n \alpha=\alpha+\alpha+\cdots+\alpha\left(n\right.$ times), $\alpha^n=\alpha \alpha \cdots \alpha$ ( $n$ times), and $\alpha^{-n}=\alpha^{-1} \alpha^{-1} \cdots \alpha^{-1}$ ( $n$ times when $\alpha \neq 0$ ). Also $\alpha^0=1$ if $\alpha \neq 0$. The usual rules of exponentiation hold. If $\mathbb{F}$ is a finite set with $q$ elements, $\mathbb{F}$ is called a finite field of order $q$ and denoted $\mathbb{F}_q$.

Example 1.2.2 Fields include the rational numbers $\mathbb{Q}$, the real numbers $\mathbb{R}$, and the complex numbers $\mathbb{C}$. Finite fields include $\mathbb{Z}_p$, the set of integers modulo $p$, where $p$ is a prime.

## 计算机代写|编码理论代写Coding theory代考|Codes

In this section we introduce the concept of codes over finite fields. We begin with some notation.

The set of $n$-tuples with entries in $\mathbb{F}_q$ forms an $n$-dimensional vector space, denoted $\mathbb{F}_q^n=\left{x_1 x_2 \cdots x_n \mid x_i \in \mathbb{F}_q, 1 \leq i \leq n\right}$, under componentwise addition of $n$-tuples and componentwise multiplication of $n$-tuples by scalars in $\mathbb{F}_q$. The vectors in $\mathbb{F}_q^n$ will often be denoted using bold Roman characters $\mathbf{x}=x_1 x_2 \cdots x_n$. The vector $\mathbf{0}=00 \cdots 0$ is the zero vector in $\mathbb{F}_q^n$.

For positive integers $m$ and $n, \mathbb{F}q^{m \times n}$ denotes the set of all $m \times n$ matrices with entries in $\mathbb{F}_q$. The matrix in $\mathbb{F}_q^{m \times n}$ with all entries 0 is the zero matrix denoted $\mathbf{0}{m \times n}$. The identity matrix of $\mathbb{F}q^{n \times n}$ will be denoted $I_n$. If $A \in \mathbb{F}_q^{m \times n}, A^{\top} \in \mathbb{F}_q^{n \times m}$ will denote the transpose of $A$. If $\mathbf{x} \in \mathbb{F}_q^m$, $\mathbf{x}^{\top}$ will denote $\mathbf{x}$ as a column vector of length $m$, that is, an $m \times 1$ matrix. The column vector $\mathbf{0}^{\top}$ and the $m \times 1$ matrix $\mathbf{0}{m \times 1}$ are the same.
If $S$ is any finite set, its order or size is denoted $|S|$.
Definition 1.3.1 A subset $\mathcal{C} \subseteq \mathbb{F}_q^n$ is called a code of length $n$ over $\mathbb{F}_q ; \mathbb{F}_q$ is called the alphabet of $\mathcal{C}$, and $\mathbb{F}_q^n$ is the ambient space of $\mathcal{C}$. Codes over $\mathbb{F}_q$ are also called $q$-ary codes. If the alphabet is $\mathbb{F}_2, \mathcal{C}$ is binary. If the alphabet is $\mathbb{F}_3, \mathcal{C}$ is ternary. The vectors in $\mathcal{C}$ are the codewords of $\mathcal{C}$. If $\mathcal{C}$ has $M$ codewords (that is, $|\mathcal{C}|=M$ ) $\mathcal{C}$ is denoted an $(n, M)_q$ code, or, more simply, an $(n, M)$ code when the alphabet $\mathbb{F}_q$ is understood. If $\mathcal{C}$ is a linear subspace of $\mathbb{F}_q^n$, that is $\mathcal{C}$ is closed under vector addition and scalar multiplication, $\mathcal{C}$ is called a linear code of length $n$ over $\mathbb{F}_q$. If the dimension of the linear code $\mathcal{C}$ is $k, \mathcal{C}$ is denoted an $[n, k]_q$ code, or, more simply, an $[n, k]$ code. An $(n, M)_q$ code that is also linear is an $[n, k]_q$ code where $M=q^k$. An $(n, M)_q$ code may be referred to as an unrestricted code; a specific unrestricted code may be either linear or nonlinear. When referring to a code, expressions such as $(n, M),(n, M)_q,[n, k]$, or $[n, k]_q$ are called the parameters of the coodé.

Example 1.3.2 Let $\mathcal{C}={1100,1010,1001,0110,0101,0011} \subseteq \mathbb{F}_2^4$. Then $\mathcal{C}$ is a $(4,6)_2$ binary nonlinear code. Let $\mathcal{C}_1=\mathcal{C} \cup{0000,1111}$. Then $\mathcal{C}_1$ is a $(4,8)_2$ binary linear code. As $\mathcal{C}_1$ is a subspace of $\mathbb{F}_2^4$ of dimension $3, \mathcal{C}_1$ is also a $[4,3]_2$ code.

## 计算机代写|编码理论代写Coding theory代考|Finite Fields

(a) 对所有人
$\alpha, \beta, \gamma \in \mathbb{F}, \alpha+\beta \in \mathbb{F}, \alpha \cdot \beta \in \mathbb{F}, \alpha+\beta=\beta+\alpha, \alpha \cdot \beta=\beta \cdot \alpha, \alpha+(\beta+\gamma)=(\alpha+\beta)+\gamma$ ，
$\alpha \cdot(\beta \cdot \gamma)=(\alpha \cdot \beta) \cdot \gamma ，$ 和 $\alpha \cdot(\beta+\gamma)=\alpha \cdot \beta+\alpha \cdot \gamma$.
(二) $\mathbb{F}$ 拥有一个加法单位或零，表示为 0 ，和一个乘法单位或单位，表示为 1 ，使得 $\alpha+0=\alpha$ 和 $\alpha \cdot 1=\alpha$ 对所 有人 $\alpha \in \mathbb{F}_q$.
(c) 对所有人 $\alpha \in \mathbb{F}$ 和所有 $\beta \in \mathbb{F}$ 和 $\beta \neq 0$ ，那里存在 $\alpha^{\prime} \in \mathbb{F}$ ，称为加法逆 $\alpha$ ，和 $\beta^* \in \mathbb{F}$ ，称为乘法逆 $\beta$ ，这样 $\alpha+\alpha^{\prime}=0$ 和 $\beta \cdot \beta^*=1$

## 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。

## 数学代写|编码理论代写Coding theory代考|MTH 4107

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|MULTIPLICATIVE INVERSION

Let us now consider the problem of finding the multiplicative inverse of an element in the field of residue classes mod an irreducible binary polynomial $M(x)$ of degree $m$. Given the residue class containing $r(x)$, a polynomial of degree $<m$, we wish to find the polynomial $p(x)$ of degree $<m$ such that the product satisfies
$$r(x) p(x) \equiv 1 \bmod M(x)$$
or equivalently, $r(x) p(x)+M(x) q(x)=1$ for some polynomial $q(x)$. Since. $M(x)$ is irreducible, the ged of $M$ and $r$ is 1 . We may therefore apply the continued-fractions version of Euclid’s algorithm as described in Sec. 2.1. Starting with $r^{(-2)} \equiv M, r^{(-1)} \equiv r, p^{(-2)} \equiv 0, p^{(-1)} \equiv 1$, $q^{(-2)}=1, q^{(-1)}=0$, we use the division algorithm to find $a^{(k)}$ and $r^{(k)}$ such that
$$r^{(k-2)}=a^{(k)} r^{(k-1)}+r^{(k)} \quad \operatorname{deg} r^{(k)}<\operatorname{deg} r^{(k-1)}$$
We then set
\begin{aligned} &q^{(k)}=a^{(k)} q^{(k-1)}+q^{(k-2)} \ &p^{(k)}=a^{(k)} p^{(k-1)}+p^{(k-2)} \end{aligned}

The iteration is to be continued until $r^{(n)}=0$. The solution is then given by $q=q^{(n-1)}, p=p^{(n-1)}$ with $\operatorname{deg} q<\operatorname{deg} r, \operatorname{deg} p<\operatorname{deg} M=$ $m$. Since we wish to find only $p$ (and do not particularly care about $q$ ), we may dispense with the $q$ ‘s entirely.

Before designing the logical circuits, let us work an example. Suppose $r(x)=x^{4}+x+1$ and $M(x)=x^{5}+x^{2}+1$. One method of computing successive $a$ ‘s and $r$ ‘s and $p^{\prime}$ ‘s follows.

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

When considering the multiplication of residue classes mod $M(x)$, where $M(x)$ is an irreducible binary polynomial of degree $m$, it is helpful to introduce the symbol $\alpha$ to denote the residue class containing $x$. Then $\alpha^{2}$ represents the residue class containing $x^{2}$, and, in general, if $r(x)$ is any polynomial, then $r(\alpha)$ represents the residue class containing $r(x)$. Since $M(x) \equiv 0 \bmod M(x)$, we must have $M(\alpha)=0$. The element represented by the symbol $\alpha$ is therefore a root of the polynomial $M(x)$. Hence, we have an obvious isomorphism between the field containing the $2^{m}$ residue classes $\bmod M(x)$ and the field containing the binary field and all polynomials in $\alpha$, where $\alpha$ is a root of the irreducible binary polynomial $M(x)$.

Any element $Y$ in this field may be expressed uniquely as a polynomial of degree $<m$ in $\alpha, Y=\sum_{i=0}^{m-1} Y_{i} \alpha^{i}$, where the $Y_{i}$ are binary numbers. The element $Y$ may be conveniently stored in an $m$-bit register, whose components contain the binary numbers $Y_{m-1}, Y_{m-2}, \ldots, Y_{0}$.

## 数学代写|编码理论代写Coding theory代考|MULTIPLICATION OF A REGISTER BY A WIRED CONSTANT

Let us first consider the multiplication of the field element in the $Y$ register by a constant field element $A$. We may assume that $A$ is represented by some binary polynomial in $\alpha$. Since $Y=\sum_{i=0}^{m-1} Y_{i} \alpha^{i}$, we have $Y A=\sum_{i=0}^{m-1} Y_{i}\left(A \alpha^{i}\right)$. Expressing $A \alpha^{i}$ as a polynomial of degree $<m$ in $\alpha$ gives $A \alpha^{i}=\sum_{j=0}^{m-1} A_{i, j} \alpha^{j}$, so that
\begin{aligned} Y A &=\sum_{i=0}^{m-1} Y_{i} \sum_{j=0}^{m-1} A_{i, j} \alpha^{j} \ &=\sum_{j=0}^{m-1}\left(\sum_{i=0}^{m-1} Y_{i} A_{i, j}\right) \alpha^{j} \end{aligned}
Thus, multiplication of the field element $Y$ by the field element $A$ is equivalent to multiplication of the $m$-dimensional binary row vector $\mathbf{Y}=\left[Y_{m-1}, Y_{m-2}, \ldots, Y_{0}\right]$ by the $m \times m$ matrix whose components are $A_{i, j}$. The rows of this matrix represent the products $A \alpha^{m-1}, A \alpha^{m-2}$, $\cdots, A$.

For example, let $M(x)=x^{5}+x^{2}+1$. Suppose we wish to multiply the contents of the $Y$ register by the field element $A=\alpha^{3}+\alpha$. We first compute
\begin{aligned} A \alpha &=\alpha^{4}+\alpha^{2} \ A \alpha^{2} &=\alpha^{5}+\alpha^{3}=\alpha^{3}+\alpha^{2}+1 \ A \alpha^{3} &=\alpha^{4}+\alpha^{3}+\alpha \ A \alpha^{4} &=\alpha^{5}+\alpha^{4}+\alpha^{2}=\alpha^{4}+1 \end{aligned}
The multiplication $Z=Y A$ is equivalent to
$$\left[Z_{4}, Z_{3}, Z_{2}, Z_{1}, Z_{0}\right]=\left[Y_{4}, Y_{3}, Y_{2}, Y_{1}, Y_{0}\right]\left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & 1 \ 1 & 1 & 0 & 1 & 0 \ 0 & 1 & 1 & 0 & 1 \ 1 & 0 & 1 & 0 & 0 \ 0 & 1 & 0 & 1 & 0 \end{array}\right]$$
This multiplication may readily be accomplished by the circuit of Fig. 2.11.

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

$$r(x) p(x) \equiv 1 \bmod M(x)$$

$r^{(-2)} \equiv M, r^{(-1)} \equiv r, p^{(-2)} \equiv 0, p^{(-1)} \equiv 1, q^{(-2)}=1, q^{(-1)}=0$ ，我们使用除法算法找到 $a^{(k)}$ 和 $r^{(k)}$ 这样
$$r^{(k-2)}=a^{(k)} r^{(k-1)}+r^{(k)} \quad \operatorname{deg} r^{(k)}<\operatorname{deg} r^{(k-1)}$$

$$q^{(k)}=a^{(k)} q^{(k-1)}+q^{(k-2)} \quad p^{(k)}=a^{(k)} p^{(k-1)}+p^{(k-2)}$$

## 数学代写|编码理论代写Coding theory代考|MULTIPLICATION OF A REGISTER BY A WIRED CONSTANT

$$Y A=\sum_{i=0}^{m-1} Y_{i} \sum_{j=0}^{m-1} A_{i, j} \alpha^{j} \quad=\sum_{j=0}^{m-1}\left(\sum_{i=0}^{m-1} Y_{i} A_{i, j}\right) \alpha^{j}$$

## 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。

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

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|MANIPULATIVE INTRODUCTION TO DOUBLE-ERROR-CORRECTING BCH CODES

We have seen that a linear code is characterized by its parity-check matrix $3 C$. We have also seen that the syndrome of the received sequence is the sum of the columns of $\mathcal{F C}$ corresponding to the error positions. Hence, a linear code is capable of correcting all single-error patterns iff all columns of $3 C$ are different and nonzero. If $\exists C$ has $m$ rows and can correct single errors, then $n \leq 2^{m}-1$. The Hamming codes achieve this bound.

Each digit of a Hamming code may be labeled by a nonzero binary $m$-luple, which is equal to the corresponding column of the $\mathfrak{B C}$ matrix. The $m$ syndrome digits then reveal directly the label of the error (if there is only one) or the binary vector sum of the labels (if there are several).

This labeling idea is so useful that we shall continue to assume that $n=2^{m}-1$
and that the columns of $\Im C$ have been labeled accordingly. Now suppose that we wish to correct all patterns of two or fewer errors. Obviously we need a greater redundancy; that is, $\mathcal{B C}$ must have more rows. Proceeding naĩvely, we suspect that we may need about twice as many parity checks to correct two errors as we need to correct one, so we shall try to find a parity-check matrix $\xi c$ with $2^{m}-1$ columns and $2 m$ rows.

## 数学代写|编码理论代写Coding theory代考|A CLOSER LOOK AT EUCLID’S ALGORITHM

In the previous section we indicated that the decoding of binary $\mathrm{BCH}$ codes requires arithmetic operations in the field of binary polynomials mod some irreducible binary polynomial $M(x)$. From both the theoretical and practical standpoints, Euclid’s algorithm plays a key role in this development.

From the theoretical standpoint, Euclid’s algorithm is used to prove that the factorization of polynomials into irreducible polynomials is unique (except for scalar multiples) over any field and that a polynomial of degree $d$ cannot have more than $d$ roots in any field. This fact is needed to prove that the error locator polynomial $\sigma(z)$ cannot have more roots than its degree. If it did, then the entire decoding procedure sketched in Sec. $1.4$ would be invalid, for several different pairs of error locations might conceivably be reciprocal roots of the same quadratic equation.

From the practical standpoint, Euclid’s algorithm is important because one of its modifications, the method of convergents of continued fractions, provides the basis for one of the most efficient methods for implementing division in finite fields. This method, apparently new, will be detailed in this section and the next.

Euclid’s algorithm is based on the observation that any divisor of $R$ and $r$ must also divide their sum and their difference. Furthermore, since any divisor of $r$ also divides any nonzero multiple of $r$, such as $a r$, then any divisor of $R$ and $r$ must also divide $R \pm a r$. Conversely, any divisor of $r$ and $R \pm a r$ must also divide $(R \pm a r) \mp a r=R$. Hence, if we let $(R, r)$ denote the greatest common divisor (hereafter called ged) of $R$ and $r$, then we have $(R, r)=(r, R \pm a r)$. Consequently, starting from an original pair of elements $R$ and $r$, we can find a new pair of elements which have the same ged. If the multiplier $a$ is judiciously chosen, the problem of finding the ged of the new pair of elements will be easier than the original problem.

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

The three basic elements used in logical design are the AND gate, the OR gate, and the inverter, which are represented as shown in Fig. 2.01. The AND and OR gates may have several inputs, each of which carries a binary signal having either the value 0 or the value 1 . The output of the AND gate is zero unless all its inputs are ones, in which case the output of the AND gate is also one. The output of the OR gate is one unless all of its inputs are zero, in which case the output of the OR gate is also zero. The inverter, in contrast to the AND and OR gates, has only one input, and its output is the opposite of its input. If its input signal has value 0 , the output has value 1 ; if the input signal has value 1 , the output has value 0 .

In practice, circuits having the logical properties of these three elements may be constructed out of transistors, resistors, diodes, vacuum tubes, and/or other components. Depending on the detailed properties t Starred sections of this book may be skimmed or omitted on first reading.of these components, the overall design will be subject to certain restrictions, called design constraints. For example, there will be maximum numbers of inputs to AND and OR gates and a maximum number of elements through which signals can propagate successively without additional amplification. Typically, every inverter is equipped with an amplifier, but AND and OR gates are not. Design constraints then specify how many AND and/or OR gates may be successively encountered between inverters and in what orders. Since the design constraints depend heavily on the properties of the components, we shall not consider design constraints much further here. If some of our circuits do not satisfy particular design constraints, it may be necessary to insert additional amplifiers (or pairs of successive inverters) into the circuits at certain crucial points.

## 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。

## 数学代写|编码理论代写Coding theory代考|COMP2610

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|REPETITION CODES AND SINGLE-PARITY-CHECK CODES

Suppose that we wish to transmit a sequence of binary digits across a noisy channel. If we send a one, a one will probably be rcecivcd; if we send a zero, a zero will probably be received. Occasionally, however, the channel noise will cause a transmitted one to be mistakenly interpreted as a zero or a transmitted zero to be mistakenly interpreted as a one. Although we are unable to prevent the channel from causing such errors, we can reduce their undesirable effects with the use of coding. The basic idea is simple. We take a set of $k$ message digits which we wish to transmit, annex to them $r$ check digits, and transmit the entire block of $n=k+r$ channel digits. Assuming that the channcl noise changes sufficiently few of these $n$ transmitted channel digits, the $r$ check digits may provide the receiver with sufficient information to enable him to detect and correct the channel errors.

Given any particular sequence of $k$ message digits, the transmitter must have some rule for selecting the $r$ check digits. This is called the encoding problem. Any particular scquence of $n$ digits which the encoder might transmit is called a codeword. Although there are $2^{n}$ different binary sequences of length $n$, only $2^{k}$ of these sequences are codewords, because the $r$ check digits within any codeword are completely determined by the $k$ message digits. The set consisting of these $2^{k}$ codewords of length $n$ is called the code.

No matter which codeword is transmitted, any of the $2^{\text {n }}$ possible binary sequences of length $n$ may be received if the channel is sufficiently noisy. Given the $n$ received digits, the decoder must attempt to decide which of the $2^{k}$ possible codewords was transmitted.

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

In a code containing several message digits and several check digits, each check digit must be some function of the message digits. In the simple case of single-parity-check codes, the single parity check was chosen to be the binary sum of all the message digits. If there are several parity checks, it is wise to set each check digit equal to the binary sum of some subset of the message digits. For example, we construct a binary code of block length $n=6$, having $k=3$ message digits and $r=3$ check digits. We shall label the three message digits $C_{1}, C_{2}$, and $C_{3}$ and the three check digits $C_{4}, C_{5}$, and $C_{6}$. We choose these check digits from the message digits according to the following rules:
$C_{4}=C_{1}+C_{2}$
$C_{5}=C_{1}+C_{3}$
$C_{6}=C_{2}+C_{3}$
or, in matrix notation,
$$\left[\begin{array}{l} C_{4} \ C_{5} \ C_{6} \end{array}\right]=\left[\begin{array}{lll} 1 & 1 & 0 \ 1 & 0 & 1 \ 0 & 1 & 1 \end{array}\right]\left[\begin{array}{l} C_{1} \ C_{2} \ C_{3} \end{array}\right]$$
The full codcword coneists of the digits $C_{1}, C_{2}, C_{3}, C_{4}, C_{8}, C_{6}$. Every codeword must satigfy the parity=eheck equations or, in matrix notation,
$$\left[\begin{array}{llllll} 1 & 1 & 0 & 1 & 0 & 0 \ 1 & 0 & 1 & 0 & 1 & 0 \ 0 & 1 & 1 & 0 & 0 & 1 \end{array}\right] \quad \mathbf{C}^{t}=\left[\begin{array}{l} 0 \ 0 \ 0 \end{array}\right]$$

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

At extremely low rates or extremely high rates, it is relatively easy to find good linear codes. In order to interpolate between these two extremes, we might adopt either of two approaches: (1) start with the low-rate codes and gradually increase $k$ by adding more and more codewords, attempting to maintain a large error-correction capability, or (2) start with good high=rate codes and gradually increase the error= correction capability, attempting to add only a few additional paritycheck constraints.

Historically, the second approach has proved more successful.
† All of the perfect singlc-error-correcting binary group codes were first discovered by Hamming. The Hamming code of length 7 was first published as an example in the paper by Shannon (1948). The generalization of this example was mentioned by Golay (1949) prior to the appearance of the paper by Hamming (1950). The Hamming codes had been anticipated by Fisher (1942) in a different context.

This is the approach we shall follow. We begin by constructing certain codes to correct single errors, the Hamming codes.

The syndrome of a linear code is related to the error pattern by the equation $\mathbf{s}^{t}=\tilde{F} E^{t}$. In general, the right side of this equation may be written as $E_{1}$ times the first column of the $F C$ matrix, plus $E_{2}$ times the second column of the $F C$ matrix, plus $E_{3}$ times the third column of the FC matrix, plus …. For example, if
$$\mathbf{s}^{t}=\left[\begin{array}{cccccc} 1 & 1 & 0 & 1 & 0 & 0 \ 1 & 0 & 1 & 0 & 1 & 0 \ 0 & 1 & 1 & 0 & 0 & 1 \end{array}\right]\left[E_{1}, E_{2}, E_{3}, E_{4}, E_{5}, E_{6}\right]^{t}$$
then
$$\left[\begin{array}{l} s_{1} \ s_{2} \ s_{3} \end{array}\right]=E_{1}\left[\begin{array}{l} 1 \ 1 \ 0 \end{array}\right]+E_{2}\left[\begin{array}{l} 1 \ 0 \ 1 \end{array}\right]+E_{3}\left[\begin{array}{l} 0 \ 1 \ 1 \end{array}\right]+E_{4}\left[\begin{array}{l} 1 \ 0 \ 0 \end{array}\right]+E_{5}\left[\begin{array}{l} 0 \ 1 \ 0 \end{array}\right]+E_{6}\left[\begin{array}{l} 0 \ 0 \ 1 \end{array}\right]$$

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

C4=C1+C2
C5=C1+C3
C6=C2+C3

[C4 C5 C6]=[110 101 011][C1 C2 C3]

[110100 101010 011001]C吨=[0 0 0]

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

† 所有完美的单次纠错二进制群码都是由 Hamming 首次发现的。长度为 7 的汉明码首先在 Shannon (1948) 的论文中作为示例发表。在 Hamming (1950) 的论文出现之前，Golay (1949) 已经提到了这个例子的推广。Fisher (1942) 在不同的背景下已经预料到了汉明码。

s吨=[110100 101010 011001][和1,和2,和3,和4,和5,和6]吨

[s1 s2 s3]=和1[1 1 0]+和2[1 0 1]+和3[0 1 1]+和4[1 0 0]+和5[0 1 0]+和6[0 0 1]

## 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。

## 数学代写|信息论代写information theory代考|ECE4042

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information 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。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论information 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|信息论代写information theory代考|Definition of entropy of a continuous random variable

Up to now we have assumed that a random variable $\xi$, with entropy $H_{\xi}$, can take values from some discrete space consisting of either a finite or a countable number of elements, for instance, messages, symbols, etc. However, continuous variables are also widespread in engineering, i.e. variables (scalar or vector), which can take values from a continuous space $X$, most often from the space of real numbers. Such a random variable $\xi$ is described by the probability density function $p(\xi)$ that assigns the probability
$$\Delta P=\int_{\xi \varepsilon \Delta X} p(\xi) d \xi \approx p(A) \Delta V \quad(A \in \Delta X)$$
of $\xi$ appearing in region $\Delta X$ of the specified space $X$ with volume $\Delta V(d \xi=d V$ is a differential of the volume).

How can we define entropy $H_{\xi}$ for such a random variable? One of many possible formal ways is the following: In the formula
$$H_{\xi}=-\sum_{\xi} P \xi \ln P(\xi)=-\mathbb{E}[\ln P(\xi)]$$
appropriate for a discrete variable we formally replace probabilities $P(\xi)$ in the argument of the logarithm by the probability density and, thereby, consider the expression
$$H_{\xi}=-\mathbb{E}[\ln p(\xi)]=-\int_{x} p(\xi) \ln p(\xi) d \xi .$$
This way of defining entropy is not well justified. It remains unclear how to define entropy in the combined case, when a continuous distribution in a continuous space coexists with concentrations of probability at single points, i.e. the probability density contains delta-shaped singularities. Entropy (1.6.2) also suffers from the drawback that it is not invariant, i.e. it changes under a non-degenerate transformation of variables $\eta=f(\xi)$ in contrast to entropy (1.6.1), which remains invariant under such transformations.

## 数学代写|信息论代写information theory代考|Properties of entropy in the generalized version

Entropy (1.6.13), (1.6.16) defined in the previous section possesses a set of properties, which are analogous to the properties of an entropy of a discrete random variable considered earlier. Such an analogy is quite natural if we take into account the interpretation of entropy (1.6.13) (provided in Section 1.6) as an asymptotic case (for large $N$ ) of entropy (1.6.1) of a discrete random variable.

The non-negativity property of entropy, which was discussed in Theorem $1.1$, is not always satisfied for entropy (1.6.13), (1.6.16) but holds true for sufficiently large $N$. The constraint
$$H_{\xi}^{P / Q} \leqslant \ln N$$
results in non-negativity of entropy $H_{\xi}$.
Now we move on to Theorem $1.2$, which considered the maximum value of entropy. In the case of entropy (1.6.13), when comparing different distributions $P$ we need to keep measure $v$ fixed. As it was mentioned, quantity (1.6.17) is non-negative and, thus, (1.6.16) entails the inequality
$$H_{\xi} \leqslant \ln N .$$
At the same time, if we suppose $P=Q$, then, evidently, we will have
$$H_{\xi}=\ln N .$$
This proves the following statement that is an analog of Theorem $1.2$.

## 数学代写|信息论代写information theory代考|Encoding of discrete information

The definition of the amount of information, given in Chapter 1, is justified when we deal with a transformation of information from one kind into another, i.e. when considering encoding of information. It is essential that the law of conservation of information amount holds under such a transformation. It is very useful to draw an analogy with the law of conservation of energy. The latter is the main argument for introducing the notion of energy. Of course, the law of conservation of information is more complex than the law of conservation of energy in two respects. The law of conservation of energy establishes an exact equality of energies, when one type of energy is transformed into another. However, in transforming information we have a more complex relation, namely ‘not greater’ $(\leqslant)$, i.e. the amount of information cannot increase. The equality sign corresponds to optimal encoding. Thus, when formulating the law of conservation of information, we have to point out that there possibly exists such an encoding, for which the equality of the amounts of information occurs.

The second complication is that the equality is not exact. It is approximate, asymptotic, valid for complex (large) messages and for composite random variables. The larger a system of messages is, the more exact such a relation becomes. The exact equality sign takes place only in the limiting case. In this respect, there is an analogy with the laws of statistical thermodynamics, which are valid for large thermodynamic systems consisting of a large number (of the order of the Avogadro number) of molecules.

When conducting encoding, we assume that a long sequence of messages $\xi_{1}, \xi_{2}$, … is given together with their probabilities, i.e. a sequence of random variables. Therefore, the amount of information (entropy $H$ ) corresponding to this sequence can be calculated. This information can be recorded and transmitted by different realizations of the sequence. If $M$ is the number of such realizations, then the law of conservation of information can be expressed by the equality $H=\ln M$, which is complicated by the two above-mentioned factors (i.e. actually. $H \leqslant \ln M$ ).

Two different approaches may be used for solving the encoding problem. One can perform encoding of an infinite sequence of messages, i.e. online (or ‘sliding’) encoding. The inverse procedure, i.e. decoding, will be performed analogously.

## 数学代写|信息论代写information theory代考|Definition of entropy of a continuous random variable

$$\Delta P=\int_{\xi \varepsilon \Delta X} p(\xi) d \xi \approx p(A) \Delta V \quad(A \in \Delta X)$$

$$H_{\xi}=-\sum_{\xi} P \xi \ln P(\xi)=-\mathbb{E}[\ln P(\xi)]$$

$$H_{\xi}=-\mathbb{E}[\ln p(\xi)]=-\int_{x} p(\xi) \ln p(\xi) d \xi$$

## 数学代写|信息论代写information theory代考|Properties of entropy in the generalized version

$$H_{\xi}^{P / Q} \leqslant \ln N$$

$$H_{\xi} \leqslant \ln N \text {. }$$

$$H_{\xi}=\ln N .$$

## 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。