### 数学代写|编码理论代写Coding theory代考|ELEC5507

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

Finite fields play an essential role in coding theory. The theory and construction of finite fields can be found, for example, in [1254] and [1408, Chapter 2]. Finite fields, as related specifically to codes, are described in [1008, 1323, 1602]. In this section we give a brief introduction.

Definition 1.2.1 A field $\mathbb{F}$ is a nonempty set with two binary operations, denoted $+$ and $\because$, satisfying the following properties.
(a) For all $\alpha, \beta, \gamma \in \mathbb{F}, \alpha+\beta \in \mathbb{F}, \alpha \cdot \beta \in \mathbb{F}, \alpha+\beta=\beta+\alpha, \alpha \cdot \beta=\beta \cdot \alpha, \alpha+(\beta+\gamma)=(\alpha+\beta)+\gamma$, $\alpha \cdot(\beta \cdot \gamma)=(\alpha \cdot \beta) \cdot \gamma$, and $\alpha \cdot(\beta+\gamma)=\alpha \cdot \beta+\alpha \cdot \gamma$.
(b) $\mathbb{F}$ possesses an additive identity or zero, denoted 0 , and a multiplicative identity or unity, denoted 1 , such that $\alpha+0=\alpha$ and $\alpha \cdot 1=\alpha$ for all $\alpha \in \mathbb{F}_{q}$.
(c) For all $\alpha \in \mathbb{F}$ and all $\beta \in \mathbb{F}$ with $\beta \neq 0$, there exists $\alpha^{\prime} \in \mathbb{F}$, called the additive inverse of $\alpha$, and $\beta^{} \in \mathbb{F}$, called the multiplicative inverse of $\beta$, such that $\alpha+\alpha^{\prime}=0$ and $\beta \cdot \beta^{}=1 .$

The additive inverse of $\alpha$ will be denoted $-\alpha$, and the multiplicative inverse of $\beta$ will be denoted $\beta^{-1}$. Usually the multiplication operation will be suppressed; that is, $\alpha \cdot \beta$ will be denoted $\alpha \beta$. If $n$ is a positive integer and $\alpha \in \mathbb{F}, n \alpha=\alpha+\alpha+\cdots+\alpha$ ( $n$ times), $\alpha^{n}=\alpha \alpha \cdots \alpha$ ( $n$ times), and $\alpha^{-n}=\alpha^{-1} \alpha^{-1} \cdots \alpha^{-1}$ ( $n$ times when $\alpha \neq 0$ ). Also $\alpha^{0}=1$ if $\alpha \neq 0$. The usual rules of exponentiation hold. If $\mathbb{F}$ is a finite set with $q$ elements, $\mathbb{F}$ is called a finite field of order $q$ and denoted $\mathbb{F}_{q}$.

Example 1.2.2 Fields include the rational numbers $\mathbb{Q}$, the real numbers $\mathbb{R}$, and the complex numbers $\mathbb{C}$. Finite fields include $\mathbb{Z}_{p}$, the set of integers modulo $p$, where $p$ is a prime.

## 数学代写|编码理论代写Coding theory代考|Codes

In this section we introduce the concept of codes over finite fields. We begin with some notation.

The set of $n$-tuples with entries in $\mathbb{F}{q}$ forms an $n$-dimensional vector space, denoted $\mathbb{F}{q}^{n}=\left{x_{1} x_{2} \cdots x_{n} \mid x_{i} \in \mathbb{F}{q}, 1 \leq i \leq n\right}$, under componentwise addition of $n$-tuples and componentwise multiplication of $n$-tuples by scalars in $\mathbb{F}{q}$. The vectors in $\mathbb{F}{q}^{n}$ will often be denoted using bold Roman characters $\mathbf{x}=x{1} x_{2} \cdots x_{n}$. The vector $\mathbf{0}=00 \cdots 0$ is the zero vector in $\mathbb{F}_{q}^{n}$.

For positive integers $m$ and $n, \mathbb{F}{q}^{m \times n}$ denotes the set of all $m \times n$ matrices with entries in $\mathbb{F}{q}$. The matrix in $\mathbb{F}{q}^{m \times n}$ with all entries 0 is the zero matrix denoted $\mathbf{0}{m \times n}$. The identity matrix of $\mathbb{F}{q}^{n \times n}$ will be denoted $I{n}$. If $A \in \mathbb{F}{q}^{m \times n}, A^{\mathrm{T}} \in \mathbb{F}{q}^{n \times m}$ will denote the transpose of $A$. If $\mathbf{x} \in \mathbb{F}{q}^{m}, \mathbf{x}^{\top}$ will denote $\mathbf{x}$ as a column vector of length $m$, that is, an $m \times 1$ matrix. The column vector $\mathbf{0}^{\top}$ and the $m \times 1$ matrix $\mathbf{0}{m \times 1}$ are the same.
If $S$ is any finite set, its order or size is denoted $|S|$.
Definition 1.3.1 A subset $\mathcal{C} \subseteq \mathbb{F}{q}^{n}$ is called a code of length $n$ over $\mathbb{F}{q} ; \mathbb{F}{q}$ is called the alphabet of $\mathcal{C}$, and $\mathbb{F}{q}^{n}$ is the ambient space of $\mathcal{C}$. Codes over $\mathbb{F}{q}$ are also called $q$-ary codes. If the alphabet is $\mathbb{F}{2}, \mathcal{C}$ is binary. If the alphabet is $\mathbb{F}{3}, \mathcal{C}$ is ternary. The vectors in $\mathcal{C}$ are the codewords of $\mathcal{C}$. If $\mathcal{C}$ has $M$ codewords (that is, $|\mathcal{C}|=M) \mathcal{C}$ is denoted an $(n, M){q}$ code, or, more simply, an $(n, M)$ code when the alphabet $\mathbb{F}{q}$ is understood. If $\mathcal{C}$ is a linear subspace of $\mathbb{F}{q}^{n}$, that is $\mathcal{C}$ is closed under vector addition and scalar multiplication, $\mathcal{C}$ is called a linear code of length $n$ over $\mathbb{F}{q}$. If the dimension of the linear code $\mathcal{C}$ is $k, \mathcal{C}$ is denoted an $[n, k]{q}$ code, or, more simply, an $[n, k]$ code. An $(n, M){q}$ code that is also linear is an $[n, k]{q}$ code where $M=q^{k}$. An $(n, M){q}$ code may be referred to as an unrestricted code; a specific unrestricted code may be either linear or nonlinear. When referring to a code, expressions such as $(n, M),(n, M){q},[n, k]$, or $[n, k]_{q}$ are called the parameters of the code.

Example 1.3.2 Let $\mathcal{C}={1100,1010,1001,0110,0101,0011} \subseteq \mathbb{F}{2}^{4}$. Then $\mathcal{C}$ is a $(4,6){2}$ binary nonlinear code. Let $\mathcal{C}{1}=\mathcal{C} \cup{0000,1111}$. Then $\mathcal{C}{1}$ is a $(4,8){2}$ binary linear code. As $\mathcal{C}{1}$ is a subspace of $\mathbb{F}{2}^{4}$ of dimension $3, \mathcal{C}{1}$ is also a $[4,3]_{2}$ code.

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

There is a natural inner product on $\mathbb{F}{q}^{n}$ that often proves useful in the study of codes. ${ }^{2}$ Definition 1.5.1 The ordinary inner product, also called the Euclidean inner product, on $\mathbb{F}{q}^{n}$ is defined by $\mathbf{x} \cdot \mathbf{y}=\sum_{i=1}^{n} x_{i} y_{i}$ where $\mathbf{x}=x_{1} x_{2} \cdots x_{n}$ and $\mathbf{y}=y_{1} y_{2} \cdots y_{n}$. Two vectors $\mathbf{x}, \mathbf{y} \in \mathbb{F}{q}^{n}$ are orthogonal if $\mathbf{x} \cdot \mathbf{y}=0$. If $\mathcal{C}$ is an $[n, k]{q}$ code,
$$\mathcal{C}^{\perp}=\left{\mathbf{x} \in \mathbb{F}_{q}^{n} \mid \mathbf{x} \cdot \mathbf{c}=0 \text { for all } \mathbf{c} \in \mathcal{C}\right}$$

is the orthogonal code or dual code of $\mathcal{C}$. $\mathcal{C}$ is self-orthogonal if $\mathcal{C} \subseteq \mathcal{C}^{\perp}$ and self-dual if $\mathcal{C}=\mathcal{C}^{\perp}$.

Theorem 1.5.2 ([1323, Chapter 1.8]) Let $\mathcal{C}$ be an $[n, k]{q}$ code with generator and parity check matrices $G$ and $H$, respectively. Then $\mathcal{C}^{\perp}$ is an $[n, n-k]{q}$ code with generator and parity check matrices $H$ and $G$, respectively. Additionally $\left(\mathcal{C}^{\perp}\right)^{\perp}=\mathcal{C}$. Furthermore $\mathcal{C}$ is self-dual if and only if $\mathcal{C}$ is self-orthogonal and $k=\frac{n}{2}$.

Example $1.5 .3 \mathcal{C}{2}$ from Example $1.4 .8$ is a $[4,2]{2}$ self-dual code with generator and parity check matrices both equal to
$$\left[\begin{array}{llll} 1 & 1 & 0 & 0 \ 0 & 0 & 1 & 1 \end{array}\right] \text {. }$$
The dual of the Hamming $[7,4]{2}$ code in Example $1.4 .9$ is a $[7,3]{2}$ code $\mathcal{H}{3,2}^{\perp} . H{3,2}$ is a generator matrix of $\mathcal{H}{3,2}^{\perp}$. As every row of $H{3,2}$ is orthogonal to itself and every other row of $H_{3,2}, \mathcal{H}{3,2}^{\perp}$ is self-orthogonal. As $\mathcal{H}{3,2}^{\perp}$ has dimension 3 and $\left(\mathcal{H}{3,2}^{\perp}\right)^{\perp}=\mathcal{H}{3,2}$ has dimension 4, $\mathcal{H}_{3,2}^{\perp}$ is not self-dual.

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

(a) 对所有人一个,b,C∈F,一个+b∈F,一个⋅b∈F,一个+b=b+一个,一个⋅b=b⋅一个,一个+(b+C)=(一个+b)+C, 一个⋅(b⋅C)=(一个⋅b)⋅C， 和一个⋅(b+C)=一个⋅b+一个⋅C.
(二)F拥有一个加法单位或零，表示为 0 ，和一个乘法单位或单位，表示为 1 ，使得一个+0=一个和一个⋅1=一个对所有人一个∈Fq.
(c) 对所有人一个∈F和所有b∈F和b≠0， 那里存在一个′∈F，称为加法逆一个， 和b∈F，称为乘法逆b, 这样一个+一个′=0和b⋅b=1.

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

\mathcal{C}^{\perp}=\left{\mathbf{x} \in \mathbb{F}_{q}^{n} \mid \mathbf{x} \cdot \mathbf{c}=0 \text { 对于所有 } \mathbf{c} \in \mathcal{C}\right}\mathcal{C}^{\perp}=\left{\mathbf{x} \in \mathbb{F}_{q}^{n} \mid \mathbf{x} \cdot \mathbf{c}=0 \text { 对于所有 } \mathbf{c} \in \mathcal{C}\right}

[1100 0011].

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## MATLAB代写

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