### 数学代写|编码理论作业代写Coding Theory代考|Cyclic Codes

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## 数学代写|编码理论作业代写Coding Theory代考|Definition of Cyclic Codes

A cyclic code is characterised as a linear block code $\mathrm{B}(n, k, d)$ with the additional property that for each code word
$$\mathbf{b}=\left(b_{0}, b_{1}, \ldots, b_{n-2}, b_{n-1}\right)$$

all cyclically shifted words
\begin{aligned} &\left(b_{n-1}, b_{0}, \ldots, b_{n-3}, b_{n-2}\right) \ &\left(b_{n-2}, b_{n-1}, \ldots, b_{n-4}, b_{n-3}\right) \ &\vdots \ &\left(b_{2}, b_{3}, \ldots, b_{0}, b_{1}\right) \ &\left(b_{1}, b_{2}, \ldots, b_{n-1}, b_{0}\right) \end{aligned}
are also valid code words of $\mathrm{B}(n, k, d)$ (Lin and Costello, 2004; Ling and Xing, 2004). This property can be formulated concisely if a code word $\mathbf{b} \in \mathbb{F}{q}^{n}$ is represented as a polynomial $$b(z)=b{0}+b_{1} z+\cdots+b_{n-2} z^{n-2}+b_{n-1} z^{n-1}$$
over the finite field $\mathbb{F}{q} \cdot{ }^{16}$ A cyclic shift $$\left(b{0}, b_{1}, \ldots, b_{n-2}, b_{n-1}\right) \mapsto\left(b_{n-1}, b_{0}, b_{1}, \ldots, b_{n-2}\right)$$
of the code polynomial $b(z) \in \mathrm{F}{q}[z]$ can then be expressed as $$b{0}+b_{1} z+\cdots+b_{n-2} z^{n-2}+b_{n-1} z^{n-1} \mapsto b_{n-1}+b_{0} z+b_{1} z^{2}+\cdots+b_{n-2} z^{n-1} .$$
Because of
$$b_{n-1}+b_{0} z+b_{1} z^{2}+\cdots+b_{n-2} z^{n-1}=z b(z)-b_{n-1}\left(z^{n}-1\right)$$
and by observing that a code polynomial $b(z)$ is of maximal degree $n-1$, we represent the cyclically shifted code polynomial modulo $z^{n}-1$, i.e.
$$b_{n-1}+b_{0} z+b_{1} z^{2}+\cdots+b_{n-2} z^{n-1} \equiv z b(z) \bmod z^{n}-1 .$$
Cyclic codes $\mathrm{B}(n, k, d)$ therefore fulfil the following algebraic property
$$b(z) \in \mathrm{B}(n, k, d) \quad \Leftrightarrow \quad z b(z) \bmod z^{n}-1 \in \mathbb{B}(n, k, d) .$$
For that reason – if not otherwise stated – we consider polynomials in the factorial ring $F_{q}[z] /\left(z^{n}-1\right)$. Figure $2.38$ summarises the definition of cyclic codes.

Similarly to general linear block codes, which can be defined by the generator matrix G or the corresponding parity-check matrix $\mathbf{H}$, cyclic codes can be characterised by the generator polynomial $g(z)$ and the parity-check polynomial $h(z)$, as we will show in the following (Berlekamp, 1984; Bossert, 1999; Lin and Costello, 2004; Ling and Xing, 2004).

## 数学代写|编码理论作业代写Coding Theory代考|Generator Polynomial

A linear block code $\mathbb{B}(n, k, d)$ is defined by the $k \times n$ generator matrix
$$\mathbf{G}=\left(\begin{array}{c} \mathbf{g}{0} \ \mathbf{g}{1} \ \vdots \ \mathbf{g}{k-1} \end{array}\right)=\left(\begin{array}{cccc} g{0,0} & g_{0,1} & \cdots & g_{0, n-1} \ g_{1,0} & g_{1,1} & \cdots & g_{1, n-1} \ \vdots & \vdots & \ddots & \vdots \ g_{k-1,0} & g_{k-1,1} & \cdots & g_{k-1, n-1} \end{array}\right)$$

with $k$ linearly independent basis vectors $\mathbf{g}{0}, \mathbf{g}{1}, \ldots, \mathbf{g}{k-1}$ which themselves are valid code vectors of the linear block code $\mathrm{B}(n, k, d)$. Owing to the algebraic properties of a cyclic code there exists a unique polynomial $$g(z)=g{0}+g_{1} z+\cdots+g_{n-k-1} z^{n-k-1}+g_{n-k} z^{n-k}$$
of minimal degree $\operatorname{deg}(g(z))=n-k$ with $g_{n-k}=1$ such that the corresponding generator matrix can be written as
$$\mathbf{G}=\left(\begin{array}{ccccccccccc} g_{0} & g_{1} & \cdots & g_{n-k} & 0 & \cdots & 0 & 0 & \cdots & 0 & 0 \ 0 & g_{0} & \cdots & g_{n-k-1} & g_{n-k} & \cdots & 0 & 0 & \cdots & 0 & 0 \ \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots \ 0 & 0 & \cdots & 0 & 0 & \cdots & g_{0} & g_{1} & \cdots & g_{n-k} & 0 \ 0 & 0 & \cdots & 0 & 0 & \cdots & 0 & g_{0} & \cdots & g_{n-k-1} & g_{n-k} \end{array}\right)$$
This polynomial $g(z)$ is called the generator polynomial of the cyclic code $\mathbb{B}(n, k, d)$ (Berlekamp, 1984; Bossert, 1999; Lin and Costello, 2004; Ling and Xing, 2004). The rows of the generator matrix $\mathbf{G}$ are obtained from the generator polynomial $g(z)$ and all cyclic shifts $z g(z), z^{2} g(z), \ldots, z^{k-1} g(z)$ which correspond to valid code words of the cyclic code. Formally, we can write the generator matrix as
$$\mathbf{G}=\left(\begin{array}{c} g(z) \ z g(z) \ \vdots \ z^{k-2} g(z) \ z^{k-1} g(z) \end{array}\right)$$

## 数学代写|编码理论作业代写Coding Theory代考|Cyclic Redundancy Check

With the help of the generator polynomial $g(z)$ of a cyclic code $\mathrm{B}(n, k, d)$, the so-called cyclic redundancy check (CRC) can be defined for the detection of errors (Lin and Costello, 2004). Besides the detection of $e_{\text {det }}=d-1$ errors by a cyclic code $\mathbb{B}(n, k, d)$ with minimum Hamming distance $d$, cyclic error bursts can also be detected. With a generator polynomial $g(z)$ of degree $\operatorname{deg}(g(z))=n-k$, all cyclic error bursts of length
$$\ell_{\text {burst }} \leq n-k$$
can be detected (Jungnickel, 1995). This can be seen by considering the error model $r(z)=$ $b(z)+e(z)$ with the received polynomial $r(z)$, the code polynomial $b(z)$ and the error polynomial $e(z)$ (see also Figure 2.46). Errors can be detected as long as the parity-check equation
$$g(z) \mid r(z) \Leftrightarrow r(z) \in \mathrm{B}(n, k, d)$$
of the cyclic code $\mathbb{B}(n, k, d)$ is fulfilled. Since $g(z) \mid b(z)$, all errors for which the error polynomial $e(z)$ is not divisible by the generator polynomial $g(z)$ can be detected. As long as the degree $\operatorname{deg}(e(z))$ is smaller than $\operatorname{deg}(g(z))=n-k$, the error polynomial $e(z)$ cannot be divided by the generator polynomial. This is also true if cyclically shifted variants $z^{i} e(z)$ of such an error polynomial are considered. Since for an error burst of length $\ell_{\text {burst }}$ the degree of the error polynomial is equal to $\ell_{\text {burst }}-1$, the error detection is possible if
$$\operatorname{deg}(e(z))=\ell_{\text {burst }}-1<n-k=\operatorname{deg}(g(z)) \text {. }$$

## 数学代写|编码理论作业代写Coding Theory代考|Definition of Cyclic Codes

b=(b0,b1,…,bn−2,bn−1)

(bn−1,b0,…,bn−3,bn−2) (bn−2,bn−1,…,bn−4,bn−3) ⋮ (b2,b3,…,b0,b1) (b1,b2,…,bn−1,b0)

b(和)=b0+b1和+⋯+bn−2和n−2+bn−1和n−1

(b0,b1,…,bn−2,bn−1)↦(bn−1,b0,b1,…,bn−2)

b0+b1和+⋯+bn−2和n−2+bn−1和n−1↦bn−1+b0和+b1和2+⋯+bn−2和n−1.

bn−1+b0和+b1和2+⋯+bn−2和n−1=和b(和)−bn−1(和n−1)

bn−1+b0和+b1和2+⋯+bn−2和n−1≡和b(和)反对和n−1.

b(和)∈乙(n,ķ,d)⇔和b(和)反对和n−1∈乙(n,ķ,d).

## 数学代写|编码理论作业代写Coding Theory代考|Generator Polynomial

G=(G0 G1 ⋮ Gķ−1)=(G0,0G0,1⋯G0,n−1 G1,0G1,1⋯G1,n−1 ⋮⋮⋱⋮ Gķ−1,0Gķ−1,1⋯Gķ−1,n−1)

G(和)=G0+G1和+⋯+Gn−ķ−1和n−ķ−1+Gn−ķ和n−ķ

G=(G0G1⋯Gn−ķ0⋯00⋯00 0G0⋯Gn−ķ−1Gn−ķ⋯00⋯00 ⋮⋮⋱⋮⋮⋱⋮⋮⋱⋮⋮ 00⋯00⋯G0G1⋯Gn−ķ0 00⋯00⋯0G0⋯Gn−ķ−1Gn−ķ)

G=(G(和) 和G(和) ⋮ 和ķ−2G(和) 和ķ−1G(和))

## 数学代写|编码理论作业代写Coding Theory代考|Cyclic Redundancy Check

ℓ爆裂 ≤n−ķ

G(和)∣r(和)⇔r(和)∈乙(n,ķ,d)

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