### 数学代写|编码理论代写Coding theory代考|ECE4042

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|Punctured Generalized Reed-Muller Codes

Binary Reed-Muller codes were introduced in Section 1.11. It is known that these codes are equivalent to the extended codes of some cyclic codes. In other words, after puncturing the binary Reed-Muller codes at a proper coordinate, the obtained codes are permutation equivalent to some cyclic codes. The purpose of this section is to introduce a family of cyclic codes of length $n=q^{m}-1$ over $\mathbb{F}{q}$ whose extended codes are the generalized Reed-Muller code over $\mathbb{F}{q}$.

Let $q$ be a prime power as before. For any integer $j=\sum_{i=0}^{m-1} j_{i} q^{i}$, where $0 \leq j_{i} \leq q-1$ for all $0 \leq i \leq m-1$ and $m$ is a positive integer, we define
$$\omega_{q}(j)=\sum_{i=0}^{m-1} j_{i}$$
where the sum is taken over the ring of integers, and is called the $q$-weight of $j$.
Let $\ell$ be a positive integer with $1 \leq \ell<(q-1) m$. The $\ell^{\text {th }}$ order punctured generalized Reed-Muller code $\mathcal{R} \mathcal{M}{q}(\ell, m)^{*}$ over $\mathbb{F}{q}$ is the cyclic code of length $n=q^{m}-1$ with generator polynomial
$$g(x)=\sum_{\substack{1 \leq j \leq n-1 \ \omega_{q}(j)<(q-1) m-\ell}}\left(x-\alpha^{j}\right),$$
where $\alpha$ is a generator of $\mathbb{F}{q^{m}}$. Since $\omega{q}(j)$ is a constant function on each $q$-cyclotomic coset modulo $n=q^{m}-1, g(x)$ is a polynomial over $\mathbb{F}_{q}$.

The parameters of the punctured generalized Reed-Muller code $\mathcal{R} \mathcal{M}_{q}(\ell, m)^{*}$ are known and summarized in the next theorem [71, Section 5.5].

Theorem 2.8.1 For any $\ell$ with $0 \leq \ell<(q-1) m, \mathcal{R} \mathcal{M}{q}(\ell, m)^{*}$ is a cyclic code over $\mathbb{F}{q}$ with length $n=q^{m}-1$, dimension
$$\kappa=\sum_{i=0}^{\ell} \sum_{j=0}^{m}(-1)^{j}\left(\begin{array}{c} m \ j \end{array}\right)\left(\begin{array}{c} i-j q+m-1 \ i-j q \end{array}\right)$$
and minimum weight $d=\left(q-\ell_{0}\right) q^{m-\ell_{1}-1}-1$, where $\ell=\ell_{1}(q-1)+\ell_{0}$ and $0 \leq \ell_{0}<q-1$.

## 数学代写|编码理论代写Coding theory代考|Another Generalization of the Punctured Binary Reed-Muller Codes

The punctured generalized Reed-Muller codes are a generalization of the classical punctured binary Reed-Muller codes, and were introduced in the previous section. A new generalization of the classical punctured binary Reed-Muller codes was given recently in [561]. The task of this section is to introduce the newly generalized cyclic codes.

Let $n=q^{m}-1$. For any integer $a$ with $0 \leq a \leq n-1$, we have the following $q$-adic expansion
$$a=\sum_{j=0}^{m-1} a_{j} q^{j}$$
where $0 \leq a_{j} \leq q-1$. The Hamming weight of $a$, denoted by wt $\mathrm{H}{\mathrm{H}}(a)$, is the number of nonzero coordinates in the vector $\left(a{0}, a_{1}, \ldots, a_{m-1}\right)$.
Let $\alpha$ be a generator of $\mathbb{F}{q^{m}}^{*}$. For any $1 \leq h \leq m$, we define a polynomial $$g(q, m, h)(x)=\prod{\substack{1 \leq n \leq n-1 \ 1 \leq w^{t} H(a) \leq h}}\left(x-\alpha^{\alpha}\right) .$$ polynomial over $\mathbb{F}{q}$. By definition, $g{(q, m, h)}(x)$ is a divisor of $x^{n}-1$.
Let $\sigma(q, m, h)$ denote the cyclic code over $\mathbb{F}{q}$ with length $n$ and generator polynomial $g{(m, q, h)}(x)$. By definition, $g_{(q, m, m)}(x)=\left(x^{n}-1\right) /(x-1)$. Therefore, the code $\gamma(q, m, m)$ is trivial, as it has parameters $[n, 1, n]$ and is spanned by the all-1 vector. Below we consider the code $\delta(q, m, h)$ for $1 \leq h \leq m-1$ only.

Theorem 2.9.1 Let $m \geq 2$ and $1 \leq h \leq m-1$. Then $\mathcal{V}(q, m, h)$ has parameters $\left[q^{m}-\right.$ $1, \kappa, d]$, where
$$\kappa=q^{m}-\sum_{i=0}^{h}\left(\begin{array}{c} m \ i \end{array}\right)(q-1)^{i}$$
and
$$\frac{q^{h+1}-1}{q-1} \leq d \leq 2 q^{h}-1 .$$
When $q=2$, the code $\delta(q, m, h)$ clearly becomes the classical punctured binary ReedMuller code $\mathcal{R} \mathcal{M}(m-1-h, m)^{*}$. Hence, $\delta(q, m, h)$ is indeed a generalization of the original punctured binary Reed-Muller code. In addition, when $q=2$, the lower bound and the upper bound in (2.3) become identical. It is conjectured that the lower bound on $d$ is the actual minimum distance.

## 数学代写|编码理论代写Coding theory代考|Reversible Cyclic Codes

Definition 2.10.2 A polynomial $f(x)$ over $\mathbb{F}_{q}$ is called self-reciprocal if it equals its reciprocal $f^{\perp}(x)$.

The conclusions of the following theorem are known in the literature [1323, page 206] and are easy to prove.

Theorem 2.10.3 Let $\mathcal{C}$ be a cyclic code of length $n$ over $\mathbb{F}{q}$ with generator polynomial $g(x)$. Then the following statements are equivalent. (a) $\mathcal{C}$ is reversible. (b) $g(x)$ is self-reciprocal. (c) $\beta^{-1}$ is a root of $g(x)$ for every root $\beta$ of $g(x)$ over the splitting field of $g(x)$. Furthermore, if $-1$ is a power of $q$ mod $n$, then every cyclic code over $\mathbb{F}{q}$ of length $n$ is reversible.

Now we give an exact count of reversible cyclic codes of length $n=q^{m}-1$ for odd primes $m$. Recall the $q$-cyclotomic cosets $C_{a}$ modulo $n$ given in Definition 1.12.7. It is straightforward that $-a=n-a \in C_{a}$ if and only if $a\left(1+q^{j}\right) \equiv 0(\bmod n)$ for some integer $j$. The following two lemmas are straightforward and hold whenever $\operatorname{gcd}(n, q)=1$.

Lemma 2.10.4 The irreducible polynomial $M_{\alpha^{a}}(x)$ is self-reciprocal if and only if $n-a \in$ $C_{a}$

Lemma 2.10.5 The least common multiple $\operatorname{lcm}\left(M_{\alpha^{a}}(x), M_{\alpha^{n-a}}(x)\right)$ is self-reciprocal for every $a \in \mathbb{Z}_{n}$.

Definition 2.10.6 The least nonnegative integer in a $q$-cyclotomic coset modulo $n$ is called the coset leader of this coset.
By Lemma 2.10.4, we have that
$$\operatorname{lcm}\left(M_{\alpha^{a}}(x), M_{\alpha^{n-a}}(x)\right)= \begin{cases}M_{\alpha^{a}}(x) & \text { if } n-a \in C_{a}, \ M_{\alpha^{a}}(x) M_{\alpha^{n-a}}(x) & \text { otherwise. }\end{cases}$$
Let $\Gamma_{(n, q)}$ denote the set of coset leaders of all $q$-cyclotomic cosets modulo $n$. Define
$$\Pi_{(n, q)}=\Gamma_{(n, q)} \backslash\left{\max {a, \operatorname{leader}(n-a)} \mid a \in \Gamma_{(n, q)}, n-a \notin C_{a}\right},$$
where leader $(i)$ denotes the coset leader of $C_{i}$. Then $\left{C_{a} \cup C_{n-a} \mid a \in \Pi_{(q, n)}\right}$ is a partition of $\mathbb{Z}_{n}$.

The following conclusion then follows directly from Lemmas $2.10 .4,2.10 .5$, and Theorem 2.10.3.

Theorem 2.10.7 The total number of reversible cyclic codes over $\mathbb{F}{q}$ of length $n$ is equal to $2^{\left|\Pi{(\alpha, n)}\right|}$, including the zero code and the code $\mathbb{F}{q}^{n}$. Every reversible cyclic code over $\mathbb{F}{q}$ of length $n$ is generated by a polynomial
$$g(x)=\prod_{a \in S} \operatorname{lcm}\left(M_{\alpha^{a}}(x), M_{\alpha^{n-a}}(x)\right),$$
where $S$ is a (possibly empty) subset of $\Pi_{(q, n)}$.

## 数学代写|编码理论代写Coding theory代考|Punctured Generalized Reed-Muller Codes

1.11 节介绍了二进制 Reed-Muller 码。众所周知，这些码相当于一些循环码的扩展码。换句话说，在适当的坐标处对二进制 Reed-Muller 码进行穿孔后，得到的码是等价于一些循环码的置换。本节的目的是介绍一系列长度为的循环码n=q米−1超过Fq其扩展码是广义 Reed-Muller 码Fq.

ωq(j)=∑一世=0米−1j一世

G(X)=∑1≤j≤n−1 ωq(j)<(q−1)米−ℓ(X−一个j),

ķ=∑一世=0ℓ∑j=0米(−1)j(米 j)(一世−jq+米−1 一世−jq)

## 数学代写|编码理论代写Coding theory代考|Another Generalization of the Punctured Binary Reed-Muller Codes

G(q,米,H)(X)=∏1≤n≤n−1 1≤在吨H(一个)≤H(X−一个一个).多项式Fq. 根据定义，G(q,米,H)(X)是一个除数Xn−1.

ķ=q米−∑一世=0H(米 一世)(q−1)一世

qH+1−1q−1≤d≤2qH−1.

## 数学代写|编码理论代写Coding theory代考|Reversible Cyclic Codes

\Pi_{(n, q)}=\Gamma_{(n, q)} \backslash\left{\max {a, \operatorname{leader}(na)} \mid a \in \Gamma_{(n, q )}, na \notin C_{a}\right},\Pi_{(n, q)}=\Gamma_{(n, q)} \backslash\left{\max {a, \operatorname{leader}(na)} \mid a \in \Gamma_{(n, q )}, na \notin C_{a}\right},

G(X)=∏一个∈小号厘米⁡(米一个一个(X),米一个n−一个(X)),

## 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。