### 数学代写|编码理论代写Coding theory代考|COMP2610

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## 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.

Example 2.5.2 The celebrated Golay codes introduced in Section $1.13$ are also irreducible cyclic codes and the binary $[24,12,8]$ extended Golay code was used on the Voyager 1 and Voyager 2 missions to Jupiter, Saturn, and their moons.

By definition, the dimension of $\mathcal{C}(q, n, i)$ equals $\operatorname{deg}\left(M_{\alpha^{i}}(x)\right)$, which is a divisor of $m:=\operatorname{ord}_{n}(q)$. The determination of the weight enumerators of irreducible cyclic codes is equivalent to the evaluation of Gaussian periods, which is extremely difficult in general. However, in a small number of cases, the weight enumerator of some irreducible cyclic codes is known. One-weight, two-weight and three-weight irreducible cyclic codes exist. It is in general hard to determine the minimum distance of an irreducible cyclic code. A lower bound on the minimum distances of irreducible cyclic codes has been developed. The reader is referred to $[568]$ for detailed information.

Irreducible cyclic codes are very important for many reasons. First of all, every cyclic code is the direct sum of a number of irreducible cyclic codes. Secondly, the automorphism group of some irreducible codes (Golay codes) has high transitivity. Thirdly, some irreducible codes can be employed to construct maximal arcs, elliptic quadrics (ovoids), inversive planes, and $t$-designs. Hence, irreducible cyclic codes are closely related to group theory, finite geometry and combinatorics. In addition, irreducible cyclic codes also have a number of applications in engineering.

## 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 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 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 $\mathrm{BCH}$ 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 $B C H$ codes?
2. Which of the $2^{t+1}-1$ cyclic codes are BCH codes?

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

It follows from Theorem 2.4.1 that a cyclic code with designed distance $\delta$ has minimum weight at least $\delta$. It is possible that the actual minimum distance is equal to the designed distance. Sometimes the actual minimum distance is much larger than the designed distance.
A codeword $\left(c_{0}, \ldots, c_{n-1}\right)$ of a linear code $\mathcal{C}$ is even-like if $\sum_{j=0}^{n-1} c_{j}=0$, and oddlike otherwise. The weight of an even-like (respectively odd-like) codeword is called an even-like weight (respectively odd-like weight). Let $\mathcal{C}$ be a primitive narrow-sense BCH code of length $n=q^{m}-1$ over $F_{q}$ with designed distance $\delta$. The defining set is then $T(1, \delta)=C_{1} \cup C_{2} \cup \cdots \cup C_{\delta-1}$. The following theorem provides useful information on the minimum weight of narrow-sense primitive $\mathrm{BCH}$ codes.

Theorem 2.6.4 Let $\mathcal{C}$ be the narrow-sense primitive $B C H$ code of length $n=q^{m}-1$ over $\mathbb{F}{q}$ with designed distance $\delta$. Then the minimum weight of $\mathcal{C}$ is its minimum odd-like weight. The coordinates of the narrow-sense primitive BCH code $\mathcal{C}$ of length $n=q^{m}-1$ over $\mathbb{F}{q}$ with designed distance $\delta$ can be indexed by the elements of $\mathbb{F}{q^{m}}$, and the extended coordinate in the extended code $\hat{\mathcal{C}}$ can be indexed by the zero element of $\mathbb{F}{q^{m} \text {. The general }}^{\text {. }}$ affine group $\mathrm{GA}{1}\left(\mathbb{F}{q^{m}}\right)$ then acts on $\mathbb{F}{q^{m}}$ and also on $\hat{\mathcal{C}}$ doubly transitively, where $$\mathrm{GA}{1}\left(\mathbb{F}{q^{m}}\right)=\left{a x+b \mid a \in \mathbb{F}{q^{m}}^{*}, b \in \mathbb{F}{q^{m}}\right}$$ Since $\mathrm{GA}{1}\left(\mathbb{F}{q^{m}}\right)$ is transitive on $\mathbb{F}{q^{m}}$, it is a subgroup of the permutation automorphism group of $\widehat{\mathcal{C}}$. Theorem $2.6 .4$ then follows.

In the following cases, the minimum distance of the $\mathrm{BCH}$ code $\mathcal{C}_{(q, n, \delta, b)}$ is known. We first have the following $[1323, \mathrm{p} .260]$.

## 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。本节进一步处理 BCH 码并总结其基本属性。

1. 其中有多少2吨+1−1循环码是乙CH代码？
2. 哪一个2吨+1−1循环码是BCH码吗？

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

\mathrm{GA}{1}\left(\mathbb{F}{q^{m}}\right)=\left{a x+b \mid a \in \mathbb{F}{q^{m} }^{*}, b \in \mathbb{F}{q^{m}}\right}\mathrm{GA}{1}\left(\mathbb{F}{q^{m}}\right)=\left{a x+b \mid a \in \mathbb{F}{q^{m} }^{*}, b \in \mathbb{F}{q^{m}}\right}自从G一个1(Fq米)是可传递的Fq米，它是置换自同构群的一个子群C^. 定理2.6.4然后跟随。

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