### 数学代写|编码理论代写Coding theory代考|MTH 3007

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## 数学代写|编码理论代写Coding theory代考|The Minimum Distances of Cyclic Codes

The length of a cyclic code is clear from its definition. However, determining the dimensions and minimum distances of cyclic codes is nontrivial. If a cyclic code $\mathcal{C}$ of length $n$ is defined by its generator polynomial $g(x)$, then the dimension of $\mathcal{C}$ equals $n-\operatorname{deg}(g)$. But it may be hard to find the degree of $g(x)$ when $g(x)$ is given as the least common multiple of a number of polynomials. If a cyclic code is defined in the trace form, it may also be difficult to determine the dimension. Determining the exact minimum distance of a cyclic code is more difficult. In the case that the minimum distance of a cyclic code cannot be settled, the best one could expect is to develop a good lower bound on the minimum distance. Unlike many other subclasses of linear codes, cyclic codes have some lower bounds on their

minimum distances. Some of the bounds are easy to use, while others are hard to employ. Below we introduce a few effective lower bounds on the minimum distances of cyclic codes.
Let $\mathcal{C}$ be a cyclic code of length $n$ over $\mathbb{F}{q}$ with generator polynomial $$g(x)=\prod{i \in T}\left(x-\alpha^{i}\right)$$
where $T$ is the union of some $q$-cyclotomic cosets modulo $n$, and is called the defining set of $\mathcal{C}$ relative to $\alpha$. The following is a simple but very useful lower bound ([248] and [968]).
Theorem 2.4.1 (BCH Bound) Let $\mathcal{C}$ be a cyclic code of length $n$ over $\mathbb{F}_{q}$ with defining set $T$ and minimum distance $d$. Assume $T$ contains $\delta-1$ consecutive integers for some integer $\delta$. Then $d \geq \delta$.

The BCH Bound depends on the choice of the primitive $n^{\text {th }}$ root of unity $\alpha$. Different choices of the primitive root may yield different lower bounds. When applying the BCH Bound, it is crucial to choose the right primitive root. However, it is open how to choose such a primitive root. In many cases the BCH Bound may be far away from the actual minimum distance. In such cases, the lower bound given in the following theorem may be much better. It was discovered by Hartmann and Tzeng [914]. To introduce this bound, we define
$$A+B={a+b \mid a \in A, b \in B},$$
where $A$ and $B$ are two subsets of the ring $\mathbb{Z}_{n}, n$ is a positive integer, and + denotes the integer addition modulo $n$.

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

Let $\mathcal{C}(q, n, i)$ denote the cyclic code of length $n$ over $\mathbb{F}{q}$ with parity check polynomial $M{\alpha^{i}}(x)$, which is the minimal polynomial of $\alpha^{i}$ over $\mathbb{F}{q}$, and where $\alpha$ is a primitive $n^{\text {th }}$ root of unity over an extension field of $\mathbb{F}{q}$. These $\mathcal{C}(q, n, i)$ are called irreducible cyclic codes. Since the ideals $\left\langle\left(x^{n}-1\right) / M_{\alpha^{i}}(x)\right\rangle$ of $\mathcal{R}{(n, q)}$ are minimal, these $\mathcal{C}(q, n, i)$ are also called minimal cyclic codes. By Theorem 2.3.5, $\mathcal{C}(q, n, i)$ has the following trace representation: $$\mathcal{C}(q, n, i)=\left{\left(\operatorname{Tr}{q^{m_{i}} / q}\left(a \beta^{0}\right), \operatorname{Tr}{q^{m{i}} / q}(a \beta), \ldots, \operatorname{Tr}{q^{m{i}} / q}\left(a \beta^{n-1}\right)\right) \mid a \in \mathbb{F}{q^{m{i}}}\right},$$
where $\beta=\alpha^{-i} \in \mathbb{F}{q^{m{i}}}$ and $m_{i}=\left|C_{i}\right|$.
Example 2.5.1 Let $n=\left(q^{m}-1\right) /(q-1)$ and $\alpha=\gamma^{q-1}$, where $\gamma$ is a generator of $\mathbb{F}_{q^{m}}^{*}$. If $\operatorname{gcd}(q-1, m)=1$, then $\mathcal{C}(q, n, 1)$ has parameters $\left[n, m, q^{m-1}\right]$ and is equivalent to the simplex code whose dual is the Hamming code. Hence, when $\operatorname{gcd}(q-1, m)=1$, the Hamming code is equivalent to a cyclic code.

## 数学代写|编码理论代写Coding theory代考|BCH Codes and Their Properties

BCH codes are a subclass of cyclic codes with special properties and are important in both theory and practice. Experimental data shows that binary and ternary BCH codes of certain lengths are the best cyclic codes in almost all cases; see [549, Appendix A]. BCH codes were briefly introduced in Section 1.14. This section treats $\mathrm{BCH}$ codes further and summarizes their basic properties.

Let $\delta$ be an integer with $2 \leq \delta \leq n$ and let $b$ be an integer. A BCH code over $\mathbb{F}{q}$ of length $n$ and designed distance $\delta$, denoted by $\mathcal{C}{(q, n, \delta, b)}$, is a cyclic code with defining set
$$T(b, \delta)=C_{b} \cup C_{b+1} \cup \cdots \cup C_{b+\delta-2}$$
relative to the primitive $n^{\text {th }}$ root of unity $\alpha$, where $C_{i}$ is the $q$-cyclotomic coset modulo $n$ containing $i$.

When $b=1$, the code $\mathcal{C}{(q, n, \delta, b)}$ with defining set in (2.2) is called a narrow-sense $\mathrm{BCH}$ code. If $n=q^{m}-1$, then $\mathcal{C}{(q, n, \delta, b)}$ is referred to as a primitive BCH code. The Reed-Solomon code introduced in Section $1.14$ is a primitive BCH code.

Sometimes $T\left(b_{1}, \delta_{1}\right)=T\left(b_{2}, \delta_{2}\right)$ for two distinct pairs $\left(b_{1}, \delta_{1}\right)$ and $\left(b_{2}, \delta_{2}\right)$. The maximum designed distance of a $\mathrm{BCH}$ code is defined to be the largest $\delta$ such that the set $T(b, \delta)$ in (2.2) defines the code for some $b \geq 0$. The maximum designed distance of a BCH code is also called the Bose distance.

Given the canonical factorization of $x^{n}-1$ over $\mathbb{F}{q}$ in (2.1), we know that the total number of nonzero cyclic codes of length $n$ over $\mathbb{F}{q}$ is $2^{t+1}-1$. Then the following two natural questions arise:

1. How many of the $2^{t+1}-1$ cyclic codes are BCH codes?
2. Which of the $2^{t+1}-1$ cyclic codes are BCH codes?The first question is open. Regarding the second question, we have the next result whose proof is straightforward.

## 数学代写|编码理论代写Coding theory代考|The Minimum Distances of Cyclic Codes

G(X)=∏一世∈吨(X−一个一世)

BCH Bound 取决于原语的选择nth 团结之根一个. 原根的不同选择可能产生不同的下界。应用 BCH Bound 时，选择正确的原始根至关重要。但是，如何选择这样的原始根是开放的。在许多情况下，BCH Bound 可能远离实际的最小距离。在这种情况下，以下定理中给出的下限可能要好得多。它是由 Hartmann 和 Tzeng [914] 发现的。为了引入这个界限，我们定义

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

\mathcal{C}(q, n, i)=\left{\left(\operatorname{Tr}{q^{m_{i}} / q}\left(a \beta^{0}\right), \operatorname{Tr}{q^{m{i}} / q}(a \beta), \ldots, \operatorname{Tr}{q^{m{i}} / q}\left(a \beta^ {n-1}\right)\right) \mid a \in \mathbb{F}{q^{m{i}}}\right},\mathcal{C}(q, n, i)=\left{\left(\operatorname{Tr}{q^{m_{i}} / q}\left(a \beta^{0}\right), \operatorname{Tr}{q^{m{i}} / q}(a \beta), \ldots, \operatorname{Tr}{q^{m{i}} / q}\left(a \beta^ {n-1}\right)\right) \mid a \in \mathbb{F}{q^{m{i}}}\right},

## 数学代写|编码理论代写Coding theory代考|BCH Codes and Their Properties

BCH码是具有特殊性质的循环码的子类，在理论和实践中都很重要。实验数据表明，在几乎所有情况下，一定长度的二进制和三进制 BCH 码都是最好的循环码；见 [549，附录 A]。1.14 节简要介绍了 BCH 码。本节处理乙CH进一步编码并总结它们的基本属性。

1. 其中有多少2吨+1−1循环码是BCH码吗？
2. 哪一个2吨+1−1循环码是 BCH 码吗？第一个问题是开放的。关于第二个问题，我们有下一个证明很简单的结果。

## 有限元方法代写

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## MATLAB代写

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