### 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|The Dimensions of BCH Codes

The dimension of the BCH code $\mathcal{C}_{(q, n, \delta, b)}$ with defining set $T(b, \delta)$ in $(2.2)$ is $n-|T(b, \delta)|$. Since $|T(b, \delta)|$ may have a very complicated relation with $n, q, b$ and $\delta$, the dimension of the BCH code cannot be given exactly in terms of these parameters. The best one can do in general is to develop tight lower bounds on the dimension of $\mathrm{BCH}$ codes. The next theorem introduces such bounds [1008, Theorem 5.1.7].

Theorem 2.6.8 Let $\mathcal{C}$ be an $[n, \kappa] B C H$ code over $\mathbb{F}{q}$ of designed distance $\delta$. Then the following statements hold. (a) $\kappa \geq n-\operatorname{ord}{n}(q)(\delta-1)$.
(b) If $q=2$ and $\mathcal{C}$ is a narrow-sense $B C H$ code, then $\delta$ can be assumed odd; furthermore if $\delta=2 w+1$, then $\kappa \geq n-\operatorname{ord}_{n}(q) w$.

The bounds in Theorem 2.6.8 may not be improved for the general case, as demonstrated by the following example. However, in some special cases, they could be improved.

Example 2.6.9 Note that $m=\operatorname{ord}{15}(2)=4$, and the 2-cyclotomic cosets modulo 15 are \begin{aligned} &C{0}={0}, C_{1}={1,2,4,8}, C_{3}={3,6,9,12}, \ &C_{5}={5,10}, C_{7}={7,11,13,14} . \end{aligned}
Let $\gamma$ be a generator of $\mathbb{F}_{2^{4}}^{*}$ with $\gamma^{4}+\gamma+1=0$ and let $\alpha=\gamma^{\left(2^{4}-1\right) / 15}=\gamma$ be the primitive $15^{\text {th }}$ root of unity.

When $(b, \delta)=(0,3)$, the defining set $T(b, \delta)={0,1,2,4,8}$, and the binary cyclic code has parameters $[15,10,4]$ and generator polynomial $x^{5}+x^{4}+x^{2}+1$. In this case, the actual minimum weight is more than the designed distance, and the dimension is larger than the bound in Theorem 2.6.8(a).

When $(b, \delta)=(1,3)$, the defining set $T(b, \delta)={1,2,4,8}$, and the binary cyclic code has parameters $[15,11,3]$ and generator polynomial $x^{4}+x+1$. It is a narrow-sense BCH code. In this case, the actual minimum weight is equal to the designed distance, and the dimension reaches the bound in Theorem $2.6 .8(\mathrm{~b})$.

## 数学代写|编码理论代写Coding theory代考|Other Aspects of BCH Codes

The automorphism groups of BCH codes in most cases are open, but are known in some cases [161]. The weight distributions of the cosets of some BCH codes were considered in $[386,387,388]$. This problem is as hard as the determination of the weight distributions of $\mathrm{BCH}$ codes. The dual of a BCH code may not be a BCH code. An interesting problem is to characterise those $\mathrm{BCH}$ codes whose duals are also $\mathrm{BCH}$ codes.

Almost all references on BCH codes are about the primitive case. Only a few references on BCH codes with lengths $n=\left(q^{m}-1\right) /(q-1)$ or $n=q^{\ell}+1$ exist in the literature $[1246,1247,1277]$. Most BCH codes have never been investigated. This is due to the fact that the $q$-cyclotomic cosets modulo $n$ are very irregular and behave very badly in most cases. For example, in most cases it is extremely difficult to determine the largest coset leader, not to mention the dimension and minimum distance of a $\mathrm{BCH}$ code. This partially explains the difficulty in researching into $\mathrm{BCH}$ codes. A characteristic of $\mathrm{BCH}$ codes is that it is hard in general to determine both the dimension and minimum distance of a BCH code.

Duadic codes are a family of cyclic codes and are generalizations of the quadratic residue codes. Binary duadic codes were defined in [1220] and were generalized to arbitrary finite fields in $[1517,1519]$. Some duadic codes have very good parameters, while some have very bad parameters. The objective of this section is to give a brief introduction of duadic codes.
As before, let $n$ be a positive integer and $q$ a prime power with $\operatorname{gcd}(n, q)=1$. Let $S_{1}$ and $S_{2}$ be two subsets of $\mathbb{Z}_{n}$ such that

• $S_{1} \cap S_{2}=\emptyset$ and $S_{1} \cup S_{2}=\mathbb{Z}_{n} \backslash{0}$, and
• both $S_{1}$ and $S_{2}$ are a union of some $q$-cyclotomic cosets modulo $n$.
If there is a unit $\mu \in \mathbb{Z}{n}$ such that $S{1} \mu=S_{2}$ and $S_{2} \mu=S_{1}$, then $\left(S_{1}, S_{2}, \mu\right)$ is called a splitting of $\mathbb{Z}_{n}$.

Recall that $m:=\operatorname{ord}{n}(q)$ and $\alpha$ is a primitive $n^{\text {th }}$ root of unity in $\mathbb{F}{q^{m}}$. Let $\left(S_{1}, S_{2}, \mu\right)$ be a splitting of $\mathbb{Z}{n}$. Define $$g{i}(x)=\prod_{i \in S_{i}}\left(x-\alpha^{i}\right) \text { and } \tilde{g}{i}(x)=(x-1) g{i}(x)$$
for $i \in{1,2}$. Since both $S_{1}$ and $S_{2}$ are unions of $q$-cyclotomic cosets modulo $n$, both $g_{1}(x)$ and $g_{2}(x)$ are polynomials over $\mathbb{F}{q}$. The pair of cyclic codes $\mathcal{C}{1}$ and $\mathcal{C}{2}$ of length $n$ over $\mathbb{F}{q}$ with generator polynomials $g_{\widetilde{r}}(x)$ and $g_{2}(x)$ are called odd-like duadic codes, and the pair of cyclic codes $\widetilde{\mathcal{C}}{1}$ and $\widetilde{\mathcal{C}}{2}$ of length $n$ over $\mathbb{F}{q}$ with generator polynomials $\tilde{g}{1}(x)$ and $\widetilde{g}_{2}(x)$ are called even-like duadic codes.

By definition, $\mathcal{C}{1}$ and $\mathcal{C}{2}$ have parameters $[n,(n+1) / 2]$ and $\widetilde{\mathcal{C}}{1}$ and $\widetilde{\mathcal{C}}{2}$ have parameters $[n,(n-1) / 2]$. For odd-like duadic codes, we have the following result [1008, Theorem 6.5.2].
Theorem 2.7.1 (Square Root Bound) Let $\mathcal{C}{1}$ and $\mathcal{C}{2}$ be a pair of odd-like duadic codes of length $n$ over $\mathbb{F}{q}$. Let $d{o}$ be their (common) minimum odd-like weight. Then the following hold.
(a) $d_{o}^{2} \geq n$.
(b) If the splitting defining the duadic codes is given by $\mu=-1$, then $d_{o}^{2}-d_{o}+1 \geq n$.
(c) Suppose $d_{o}^{2}-d_{o}+1=n$, where $d_{o}>2$, and assume that the splitting defining the duadic codes is given by $\mu=-1$. Then $d_{o}$ is the minimum weight of both $\mathcal{C}{1}$ and $\mathcal{C}{2}$.

## 数学代写|编码理论代写Coding theory代考|The Dimensions of BCH Codes

BCH码的维度C(q,n,d,b)带有定义集吨(b,d)在(2.2)是n−|吨(b,d)|. 自从|吨(b,d)|可能有很复杂的关系n,q,b和d，BCH码的维数不能根据这些参数准确给出。一般来说，最好的方法是在维度上制定严格的下界乙CH代码。下一个定理引入了这样的界限[1008，定理 5.1.7]。

(b) 如果q=2和C是狭义的乙CH代码，然后d可以假设为奇数；此外，如果d=2在+1， 然后ķ≥n−单词n⁡(q)在.

C0=0,C1=1,2,4,8,C3=3,6,9,12, C5=5,10,C7=7,11,13,14.

## 数学代写|编码理论代写Coding theory代考|Other Aspects of BCH Codes

BCH 码的自同构群在大多数情况下是开放的，但在某些情况下是已知的 [161]。一些 BCH 码的陪集的权重分布在[386,387,388]. 这个问题和确定权重分布一样困难乙CH代码。BCH 码的对偶可能不是 BCH 码。一个有趣的问题是描述那些乙CH对偶也是乙CH代码。

• 小号1∩小号2=∅和小号1∪小号2=从n∖0， 和
• 两个都小号1和小号2是一些人的联合q-分圆陪集模n.
如果有单位μ∈从n这样小号1μ=小号2和小号2μ=小号1， 然后(小号1,小号2,μ)被称为分裂从n.

G一世(X)=∏一世∈小号一世(X−一个一世) 和 G~一世(X)=(X−1)G一世(X)

（一个）d○2≥n.
(b) 如果定义二元码的分裂由下式给出μ=−1， 然后d○2−d○+1≥n.
(c) 假设d○2−d○+1=n， 在哪里d○>2，并假设定义二元码的分裂由下式给出μ=−1. 然后d○是两者的最小权重C1和C2.

## 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。