# 计算机代写|编码理论代写Coding theory代考|MATH597

## 计算机代写|编码理论代写Coding theory代考|Shannon’s Theorem

Shannon’s Channel Coding Theorem [1661] guarantees that good codes exist making reliable communication possible. We will discuss this theorem in the context of binary linear codes for which maximum likelihood decoding over a BSC is used. Note however that the theorem can be stated in a more general setting.

Assume thát thẽ communication channeel is a BSC with crossover probabbility @ and that syndrome decoding is used as the implementation of ML decoding to decode an $[n, k, d]2$ code $\mathcal{C}$. The word error rate $P{\text {err }}$ for this channel and decoding scheme is the probability that the decoder makes an error, averaged over all codewords of $\mathcal{C}$; for simplicity assume that each codeword of $\mathcal{C}$ is equally likely to be sent. A decoder error occurs when $\widetilde{\mathbf{c}}=$ syndrome decoder makes a correct decision if $\mathbf{y}-\mathbf{c}$ is a coset leader. The probability that the decoder makes a correct decision is
$$\varrho^{w t_H(\mathbf{y}-\mathbf{c})}(1-\varrho)^{n-w t_H(\mathbf{y}-\mathbf{c})}$$
by (1.5). Therefore the probability that the syndrome decoder makes a correct decision averaged over all equally likely transmitted codewords is $\sum_{i=0}^n \alpha_i \varrho^i(1-\varrho)^{n-i}$ where $\alpha_i$ is the number of coset leaders of weight $i$. Thus
$$\mathrm{P}{\mathrm{err}}=1-\sum{i=0}^n \alpha_i \varrho^i(1-\varrho)^{n-i} .$$
Example 1.18.1 Suppose binary messages of length $k$ are sent unencoded over a BSC with crossover probability $\varrho$. This in effect is the same as transmitting codewords from the $[k, k, 1]2$ code $\mathcal{C}=\mathbb{F}_2^k$. This code has a unique coset, the code itself, and its leader is the zero codeword of weight 0 . Hence $\alpha_0=1$ and $\alpha_i=0$ for $i>0$. Therefore (1.6) shows that the probability of decoder error is $$\mathrm{P}{\mathrm{err}}=1-\varrho^0(1-\varrho)^k=1-(1-\varrho)^k .$$
This is precisely what we expect as the probability of no decoding error is the probability $(1-\varrho)^k$ that the $k$ bits are received without error. For instance if $\varrho=0.01$ and $k=4$, Perr without coding the length 4 messages is $0.03940399$.

## 计算机代写|编码理论代写Coding theory代考|Notation and Introduction

A brief introduction to cyclic codes over finite fields was given in Section $1.12$. The objective of this chapter is to introduce several important families of cyclic codes over finite fields. We will follow the notation of Chapter 1 as closely as possible.

By an $[n, \kappa, d]_q$ code, we mean a linear code over $\mathbb{F}_q$ with length $n$, dimension $\kappa$ and minimum distance $d$. Notice that the minimum distance of a linear code is equal to the minimum nonzern weight of the code. By the parameters of a linear code, we mean its length, dimension and minimum distance. An $[n, \kappa, d]_q$ code is said to be distance-optimal (respectively dimension-optimal) if there is no $[n, \kappa, d+1]_q$ (respectively $[n, \kappa+1, d]_q$ ) code. By the best known parameters of $[n, \kappa]$ linear codes over $\mathbb{F}_q$ we mean an $[n, \kappa, d]_q$ code with the largest known $d$ reported in the tables of linear codes maintained at [845].

In this chapter, we deal with cyclic codes of length $n$ over $\mathbb{F}q$ and always assume that $\operatorname{gcd}(n, q)=1$. Under this assumption, $x^n-1$ has no repeated factors over $\mathbb{F}_q$. Denote by $C_i$ the $q$-cyclotomic coset modulo $n$ that contains $i$ for $0 \leq i \leq n-1$. Put $m=\operatorname{ord}_n(q)$, and let $\gamma$ be a generator of $\mathbb{F}{q^m}^*:=\mathbb{F}{q^m} \backslash{0}$. Define $\alpha=\gamma^{\left(q^m-1\right) / n}$. Then $\alpha$ is a primitive $n^{\text {th }}$ root of unity. The canonical factorization of $x^n-1$ over $\mathbb{F}_q$ is given by $$x^n-1=M{\alpha^{i_0}}(x) M_{\alpha^{i_1}}(x) \cdots M_{\alpha^{i_t}}(x),$$

where $i_0, i_1, \ldots, i_t$ are representatives of the $q$-cyclotomic cosets modulo $n$, and
$$M_{\alpha^{i_j}}(x)=\prod_{h \in C_{i_j}}\left(x-\alpha^h\right),$$
which is the minimal polynomial of $\alpha^{i_j}$ over $\mathbb{F}q$ and is irreducible over $\mathbb{F}_q$. Throughout this chapter, we define $\mathcal{R}{(n, q)}=\mathbb{F}q[x] /\left\langle x^n-1\right\rangle$ and use $\operatorname{Tr}{q^m / q}$ to denote the trace function from $\mathbb{F}{q^m}$ to $\mathbb{F}_q$ defined by $\operatorname{Tr}{q^m / q}(x)=\sum_{j=0}^{m-1} x^{q^j}$. The ring of integers modulo $n$ is denoted by $\mathbb{Z}_n={0,1, \ldots, n-1}$.

Cyclic codes form an important subclass of linear codes over finite fields. Their algebraic structure is richer. Because of their cyclic structure, they are closely related to number theory. In addition, they have efficient encoding and decoding algorithms and are the most studied linear codes. In fact, most of the important families of linear codes are either cyclic codes or extended cyclic codes.

# 编码理论代考

## 计算机代写|编码理论代写Coding theory代考|Shannon’s Theorem

Shannon 的信道编码定理 [1661] 保证存在良好的代码，从而使可靠的通信成为可能。我们将在使用 BSC 上的最大似然解码的二进制线性码的上下文中讨论这个定理。但是请注意，该定理可以在更一般的设置中 陈述。

$$\varrho^{w t_H(\mathbf{y}-\mathbf{c})}(1-\varrho)^{n-w t_H(\mathbf{y}-\mathbf{c})}$$

$$\text { Perr }=1-\sum i=0^n \alpha_i \varrho^i(1-\varrho)^{n-i} .$$

$$\text { Perr }=1-\varrho^0(1-\varrho)^k=1-(1-\varrho)^k .$$

## 计算机代写|编码理论代写Coding theory代考|Notation and Introduction

$$M_{\alpha^{i_j}}(x)=\prod_{h \in C_{i_j}}\left(x-\alpha^h\right),$$

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