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# 数学代写|抽象代数代写Abstract Algebra代考|Math417 Coset Decoding

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## 数学代写|抽象代数代写Abstract Algebra代考|Coset Decoding

There is another convenient decoding method that utilizes the fact that an $(n, k)$ linear code $C$ over a finite field $F$ is a subgroup of the additive group of $V=F^n$ . This method was devised by David Slepian in 1956 and is called coset decoding (or standard decoding). To use this method, we proceed by constructing a table, called a standard array. The first row of the table is the set $C$ of code words, beginning in column 1 with the identity $0 \cdots 0$. To form additional rows of the table, choose an element $v$ of $V$ not listed in the table thus far. Among all the elements of the coset $v+C$, choose one of minimum weight, say, v’. Complete the next row of the table by placing under the column headed by the code word $c$ the vector $v^{\prime}+c$. Continue this process until all the vectors in $V$ have been listed in the table. [Note that an $(n, k)$ linear code over a field with $q$ elements will have $|V: C|=q^{n-k}$ rows.] The words in the first column are called the coset leaders. The decoding procedure is simply to decode any received word $w$ as the code word at the head of the column containing $w$.

I EXAMPLE 11 Consider the $(6,3)$ binary linear code
$$C={000000,100110,010101,001011,110011,101101,011110,111000} .$$
The first row of a standard array is just the elements of $C$. Obviously, 100000 is not in $C$ and has minimum weight among the elements of $100000+C$, so it can be used to lead the second row. Table 29.4 is the completed table.

## 数学代写|抽象代数代写Abstract Algebra代考|Historical Note

In this “Age of Information,” no one need be reminded of the importance not only of speed but also of accuracy in the storage, retrieval, and transmission of data. Machines do make errors, and their non-man-made mistakes can turn otherwise flawless programming into worthless, even dangerous, trash. Just as architects design buildings that will remain standing even through an earthquake, their computer counterparts have come up with sophisticated techniques capable of counteracting digital disasters.

The idea for the current error-correcting techniques for everything from computer hard disk drives to CD players was first introduced in 1960 by Irving Reed and Gustave Solomon, then staff members at MIT’s Lincoln Laboratory….
“When you talk about CD players and digital audio tape and now digital television, and various other digital imaging systems that are coming-all of those need Reed-Solomon [codes] as an integral part of the system,” says Robert McEliece, a coding theorist in the electrical engineering department at Caltech.

Why? Because digital information, virtually by definition, consists of strings of “bits”-0’s and 1’s – and a physical device, no matter how capably manufactured, may occasionally confuse the two. Voyager II, for example, was transmitting data at incredibly low power-barely a whisper — over tens of millions of miles. Disk drives pack data so densely that a read/write head can (almost) be excused if it can’t tell where one bit stops and the next 1 (or 0 ) begins. Careful engineering can reduce the error rate to what may sound like a negligible level – the industry standard for hard disk drives is 1 in 10 billionbut given the volume of information processing done these days, that “negligible” level is an invitation to daily disaster. Error-correcting codes are a kind of safety net-mathematical insurance against the vagaries of an imperfect material world.
In 1960, the theory of error-correcting codes was only about a decade old. The basic theory of reliable digital communication had been set forth by Claude Shannon in the late 1940s. At the same time, Richard Hamming introduced an elegant approach to single-error correction and double-error detection. Through the 1950s, a number of researchers began experimenting with a variety of errorcorrecting codes. But with their SIAM journal paper, McEliece says, Reed and Solomon “hit the jackpot.”

## 数学代写|抽象代数代写Abstract Algebra代考|Coset Decoding

$$C=000000,100110,010101,001011,110011,101101,011110,111000 .$$

## 数学代写|抽象代数代写Abstract Algebra代考|Historical Note

1960 年，欧文·里德 (Irving Reed) 和古斯塔夫·所罗门 (Gustave Solomon) 首次提出了从计算机硬盘驱动器到 CD 播放器的当前纠错技术的想法，然后是麻省理工学院林肯实验室的工作人员……。
“当你谈论 CD 播放器和数字音频磁带以及现在的数字电视和即将到来的各种其他数字成像系统时，所有这些都需要 Reed-Solomon [代码] 作为系统的组成部分，”Robert McEliece 说，他是一名加州理工学院电气工程系的编码理论家。

1960 年，纠错码理论只有大约十年的历史。克劳德·香农 (Claude Shannon) 在 20 世纪 40 年代后期提出了可靠数字通信的基本理论。与此同时，Richard Hamming 引入了一种优雅的单错误纠正和双错误检测方法。整个 1950 年代，许多研究人员开始试验各种纠错码。但 McEliece 说，凭借他们的 SIAM 期刊论文，里德和所罗门“中了大奖”。

## 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。