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# 网课代修|编码理论代写Coding theory代写|Basics of Coding Theory

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## 网课代修|编码理论代写Coding theory代写|Introduction

Coding theory had it genesis in the late 1940 s with the publication of works by Claude Shannon, Marcel Golay, and Richard Hamming. In 1948 Shannon published a landmark paper A mathematical theory of communication which marked the beginning of both information theory and coding theory. Given a communication channel, over which information is transmitted and possibly corrupted, Shannon identified a number called the ‘channel capacity’ and proved that arbitrarily reliable communication is possible at any rate below the channel capacity. For example, when transmitting images of planets from deep space, it is impractical to retransmit the images that have been altered by noise during transmission. Shannon’s Theorem guarantees that the data can be encoded before transmission so that the altered data can be decoded to the original, up to a specified degree of accuracy. Other examples of communication channels include wireless communication devices and storage systems such as DVDs or Blue-ray discs. In 1947 Hamming developed a code, now bearing his name, in an attempt to correct errors that arose in the Bell Telephone Laboratories’ mechanical relay computer; his work was circulated through a series of memoranda at Bell Labs and eventually published in . Both Shannon and Golay [820] published Hamming’s code, with Golay generalizing it. Additionally, Golay presented two of the four codes that now bear his name. A monograph by T. M. Thompson 1801 races the early development of coding theory.

## 网课代修|编码理论代写Coding theory代写|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 $\mathbb{F}$ 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 $\beta \cdot \beta^{}=1$.

## 网课代修|编码理论代写CODING THEORY代写|FINITE FIELDS

(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 $\beta \cdot \beta^{}=1$.

## 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。