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• 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|GENERAL PROPERTIES OF CYCLIC CODES

For both single- and double-error-correcting BCH codes, we were able to find a polynomial $g(x)$ which had the property that a binary polynomial $C(x)$, of degree $<n$, represents a codeword iff $C(x)$ is a multiple of $g(x)$. This polynomial, $g(x)$, is called the generator polynomial of the cyclic code. Let the polynomial $C(x)$ be divided by some multiple of $g(x)$, say $g(x) h(x)$. This gives a quotient $q(x)$ and a remainder $r(x)$, so $C=q g h+r$, and $C$ is a multiple of $g$ iff $r$ is a multiple of $g$. Thus, if the polynomial $C(x)$ represents a codeword, then any multiple of $C(x) \bmod g(x) h(x)$ represents another codeword. In particular, if $g$ is the product of distinct irreducible polynomials of degrees dividing $m$, and $g \neq x$, then $g$ divides $x^{2^{m}-1}-1=x^{n}-1$. Taking $h(x)=\left(x^{n}-1\right) / g(x)$, we conclude that if the polynomial $C(x)$ represents a codeword, then so does every multiple of $C(x) \bmod x^{n}-1$.

If the polynomial $C(x)=C_{0}+C_{1} x+C_{2} x^{2}+\cdots+C_{n-1} x^{n-1}$ is multiplied by $x \bmod x^{n}-1$, the result is $C_{n-1}+C_{0} x+\cdots+C_{n-2} x^{n-1}$. The codeword represented by the polynomial $x C(x) \bmod x^{n}-1$ is seen to be a cyclic shift of the codeword represented by the polynomial $C(x)$. Since every cyclic shift of a codeword therefore gives another codeword, we say that the code is a cyclic code. Evidently, if $g(x)$ is any divisor of $x^{n}-1$, then the set of multiples of $g(x) \bmod x^{n}-1$ forms a linear cyclic code. This is true even if $n+1$ is not a power of 2 .

Conversely, if the codewords of a linear cyclic code are represented by a set of polynomials, then for every code polynomial $C(x)$, the code contains all cyclic shifls of $C(x)$. The $k$ th cyclic right shift of the polynomial $C(x)$ is the polynomial $x^{k} C(x) \bmod x^{n}-1$. Since the sum of codewords of a linear code is also a codeword of the same linear code, the code must contain all multiples of $C(x) \bmod x^{n}-1$. Using Euclid’s algorithm, we may write the ged of $C(x)$ and $x^{n}-1$ as a multiple of $C(x) \bmod x^{n}-1$. Hence, a linear cyclic code contains the ged of every codeword polynomial and $x^{n}-1$. If we let $g(x)$ denote the codeword polynomial of lowest degree, we conclude that $g(x)$ must divide $x^{n}-1$ and that every codeword polynomial must be a multiple of $g(x)$.

We conclude that the codeword polynomials of any linear cyclic code consist of the multiples of some generator polynomial mod $x^{n}-1$, where the generator polynomial $g(x)$ is a divisor of $x^{n}-1$.

## 数学代写|编码理论代写Coding theory代考|THE CHIEN SEARCH

In Sec. $5.1$ we found that even after the decoder knows the error locations, it is difficult to correct the errors immediately. It turns out to be simpler to wait until the erroneous digits leave the received-word buffer, and then to correct them as they leave.

One approach to the correction problem, due to Chien (1964), avoids the explicit solution of the algebraic equation $\sigma\left(z^{-1}\right)=0$. As the digit at location $X_{i}$ leaves the buffer, the decoder may calculate the polynomial $\sigma\left(X_{i}^{-1}\right)$ to see whether or not this is zero. If $\sigma\left(X_{i}^{-1}\right) \neq 0$, then the digit at location $X_{i}$ is unchanged; if $\sigma\left(X_{i}^{-1}\right)=0$, then the error at location $X_{i}$ is corrected. This method is not restricted to single- and doubleerror-correcting codes. Once the decoder has found the coetticients of the error locator $\sigma(z)=\sum_{i=0}^{t} \sigma_{i} z^{i}$, whose roots are the reciprocals of the error locations, then Chien’s search may be used to test each of the locations $\alpha^{-1}, \alpha^{-2}, \alpha^{-3}, \ldots, 1$, to see if the digit in the position now leaving the buffer is a reciprocal root of the error locator. For double-errorcorrecting binary BCH codes, the error locator $\sigma(z)$ may be computed according to Eq. (1.47); for multiple-error-correcting BCH codes, the error locator may be computed by more involved procedures which we shall present in Chap. $7 .$

## 数学代写|编码理论代写Coding theory代考|OUTLINE OF GENERAL DECODER FOR ANY CYCLIC BINARY CODE

A sketch of an overall design for any binary cyclic code is shown in Fig. 5.14. The decoder consists of four principal parts, a buffer of $2 n$ digits, shift registers wired to divide the incoming word by each irreducible factor of the generator polynomial, a central Galois field processor, and a Chien searcher. At a typical instant of time, the buffer will hold parts of three successive blocks, as shown in Fig. 5.15. The first $i$ digits of the buffer hold the first $i$ digits of the incoming word; the next $n$ digits of the buffer hold the entire buffered word; the last $n-i$ digits of the buffer hold the last $n-i$ digits of the outgoing word. The Chien searcher is in the process of computing $\sigma\left(\alpha^{i}\right)$ in order to determine whether or not the next digit to leave the buffer should be corrected. The shift registers wired to divide the incoming word by each irreducible factor of the generator polynomial are busily doing just that. The central processor is engaged in trying to find the error-locator polynomial for the buffered word.

When all the $n$ digits of the incoming block have been received, then all the digits of the outgoing block have left. The buffer then appears as in Fig. 5.16. The buffered block then becomes the outgoing block, and the incoming block becomes the buffered block. The coefficients of the error-locator polynomial are read out of the central GF processor and into the Chien searcher, as the remainders of the received word divided by each irreducible factor of the generator polynomial are read into the central GF processor. As the next $n$ digits of the incoming word arrive from the channel, the central GF computer must compute the coefficients of the error-locator polynomial for the buffered word.

## 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。

assignmentutor™您的专属作业导师
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