### 数学代写|编码理论作业代写Coding Theory代考|Error Detection and Error Correction

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## 数学代写|编码理论作业代写Coding Theory代考|Error Detection and Error Correction

Based on the minimum distance decoding rule and the code space concept, we can assess the error detection and error correction capabilities of a given channel code. To this end, let $\mathbf{b}$ and $\mathbf{b}^{\prime}$ be two code words of an $(n, k)$ block code $\mathrm{B}(n, k, d)$. The distance of these code words shall be equal to the minimum Hamming distance, i.e. $\operatorname{dist}\left(\mathbf{b}, \mathbf{b}^{\prime}\right)=d$. We are able to detect errors as long as the erroneously received word $\mathbf{r}$ is not equal to a code word different from the transmitted code word. This error detection capability is guaranteed as long as the number of errors is smaller than the minimum Hamming distance $d$, because another code word (e.g. $\mathbf{b}^{\prime}$ ) can be reached from a given code word (e.g. b) merely by changing at least $d$ components. For an ( $n, k)$ block code $\mathrm{B}(n, k, d)$ with minimum Hamming distance $d$, the number of detectable errors is therefore given by (Bossert, 1999; Lin and Costello, 2004; Ling and Xing, 2004; van Lint, 1999)
$$e_{\text {det }}=d-1 .$$
For the analysis of the error correction capabilities of the $(n, k)$ block code $\mathbb{B}(n, k, d)$ we define for each code word $\mathbf{b}$ the corresponding correction ball of radius $\varrho$ as the subset of all words that are closer to the code word $\mathbf{b}$ than to any other code word $\mathbf{b}^{\prime}$ of the block code $\mathrm{B}(n, k, d)$ (see Figure 2.10). As we have seen in the last section, for minimum distance decoding, all received words within a particular correction ball are decoded into the respective code word $\mathbf{b}$. According to the radius $\varrho$ of the correction balls, besides the code word $\mathbf{b}$, all words that differ in $1,2, \ldots, \varrho$ components from $\mathbf{b}$ are elements of the corresponding correction ball. We can uniquely decode all elements of a correction ball into the corresponding code word $\mathbf{b}$ as long as the correction balls do not intersect. This condition is true if $\varrho<\frac{d}{2}$ holds. Therefore, the number of correctable errors of a block code $\mathbb{B}(n, k, d)$ with minimum Hamming distance $d$ is given by (Bossert, 1999; Lin and Costello, 2004; Ling and Xing, 2004; van Lint, 1999) ${ }^{5}$
$$e_{\mathrm{cor}}=\left\lfloor\frac{d-1}{2}\right\rfloor .$$

For the binary symmetric channel the number of errors $w$ within the $n$-dimensional transmitted code word is binomially distributed according to $\operatorname{Pr}{w$ errors $}=\left(\begin{array}{c}n \ w\end{array}\right) \varepsilon^{w}(1-$ remaining detection error probability is bounded by
$$p_{\mathrm{det}} \leq \sum_{w=e_{\mathrm{det}}+1}^{n}\left(\begin{array}{l} n \ w \end{array}\right) \varepsilon^{w}(1-\varepsilon)^{n-w}=1-\sum_{w=0}^{e_{\text {det }}}\left(\begin{array}{l} n \ w \end{array}\right) \varepsilon^{w}(1-\varepsilon)^{n-w}$$ for a binary symmetric channel can be similarly bounded by
$$p_{\mathrm{err}} \leq \sum_{w=e_{\mathrm{cor}}+1}^{n}\left(\begin{array}{l} n \ w \end{array}\right) \varepsilon^{w}(1-\varepsilon)^{n-w}=1-\sum_{w=0}^{e_{\mathrm{cor}}}\left(\begin{array}{l} n \ w \end{array}\right) \varepsilon^{w}(1-\varepsilon)^{n-w}$$

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

The $(n, k)$ block code $\mathbb{B}(n, k, d)$ with minimum Hamming distance $d$ over the finite field $\mathrm{F}{q}$ is called linear, if $\mathrm{B}(n, k, d)$ is a subspace of the vector space $\mathrm{F}{q}^{n}$ of dimension $k$ (Lin and Costello, 2004; Ling and Xing, 2004). The number of code words is then given by
$$M=q^{k}$$
according to the code rate
$$R=\frac{k}{n} .$$
Because of the linearity property, an arbitrary superposition of code words again leads to a valid code word of the linear block code $\mathrm{B}(n, k, d)$, i.e.
$$\alpha_{2} \mathbf{b}{1}+\alpha{2} \mathbf{b}{2}+\cdots+\alpha{l} \mathbf{b}{l} \in \mathbb{B}(n, k, d)$$ with $\alpha{1}, \alpha_{2}, \ldots, \alpha_{l} \in \mathbb{F}{q}$ and $\mathbf{b}{1}, \mathbf{b}{2}, \ldots, \mathbf{b}{l} \in \mathrm{B}(n, k, d)$. Owing to the linearity, the $n$ dimensional zero row vector $\mathbf{0}=(0,0, \ldots, 0)$ consisting of $n$ zeros is always a valid code word. It can be shown that the minimum Hamming distance of a linear block code $\mathrm{B}(n, k, d)$ is equal to the minimum weight of all non-zero code words, i.e.
$$d=\min {\mathbf{b} \neq \mathbf{b}^{\prime}} \operatorname{dist}\left(\mathbf{b}, \mathbf{b}^{\prime}\right)=\min {\mathbf{b} \neq \mathbf{0}} \mathbf{w t}(\mathbf{b}) .$$
These properties are summarised in Figure 2.11. As a simple example of a linear block code, the binary parity-check code is described in Figure $2.12$ (Bossert, 1999).

For each linear block code an equivalent code can be found by rearranging the code word symbols. ${ }^{7}$ This equivalent code is characterised by the same code parameters as the original code, i.e. the equivalent code has the same dimension $k$ and the same minimum Hamming distance $d$.

## 数学代写|编码理论作业代写Coding Theory代考|Parity-Check Matrix

With the help of the generator matrix $\mathbf{G}=\left(\mathbf{I}{k} \mid \mathbf{A}{k, n-k}\right)$, the following $(n-k) \times n$ matrix $-$ the so-called parity-check matrix – can be defined (Bossert, 1999; Lin and Costello, 2004; Ling and Xing, 2004)
$$\mathbf{H}=\left(\mathbf{B}{n-k, k} \mid \mathbf{I}{n-k}\right)$$
with the $(n-k) \times(n-k)$ identity matrix $\mathbf{I}{n-k}$. The $(n-k) \times k$ matrix $\mathbf{B}{n-k, k}$ is given by
$$\mathbf{B}{n-k, k}=-\mathbf{A}{k, n-k^{*}}^{\mathrm{T}}$$
For the matrices $\mathbf{G}$ and $\mathbf{H}$ the following property can be derived
$$\mathbf{H} \mathbf{G}^{\mathrm{T}}=\mathbf{B}{n-k, k}+\mathbf{A}{k, n-k}^{\mathrm{T}}=\mathbf{0}{n-k, k}$$ with the $(n-k) \times k$ zero matrix $\mathbf{0}{n-k, k}$. The generator matrix $\mathbf{G}$ and the parity-check matrix $\mathbf{H}$ are orthogonal, i.e. all row vectors of $\mathbf{G}$ are orthogonal to all row vectors of $\mathbf{H}$.
Using the $n$-dimensional basis vectors $\mathbf{g}{0}, \mathbf{g}{1}, \ldots, \mathbf{g}{k-1}$ and the transpose of the generator matrix $\mathbf{G}^{\mathrm{T}}=\left(\mathrm{g}{0}^{\mathrm{T}}, \mathbf{g}{1}^{\mathrm{T}}, \ldots, \mathrm{g}{k-1}^{\mathrm{T}}\right)$, we obtain
$$\mathbf{H} \mathbf{G}^{\mathrm{T}}=\mathbf{H}\left(\mathrm{g}{0}^{\mathrm{T}}, \mathbf{g}{1}^{\mathrm{T}}, \ldots, \mathbf{g}{k-1}^{\mathrm{T}}\right)=\left(\mathbf{H g}{0}^{\mathrm{T}}, \mathbf{H} \mathbf{g}{1}^{\mathrm{T}}, \ldots, \mathbf{H} \mathbf{g}{k-1}^{\mathrm{T}}\right)=(\mathbf{0}, \mathbf{0}, \ldots, \mathbf{0})$$
with the $(n-k)$-dimensional all-zero column vector $0=(0,0, \ldots, 0)^{\mathrm{T}}$. This is equivalent to $\mathbf{H} \mathbf{g}{i}^{\mathrm{T}}=\mathbf{0}$ for $0 \leq i \leq k-1$. Since each code vector $\mathbf{b} \in \mathbb{B}(n, k, d)$ can be written as $$\mathbf{b}=\mathbf{u} \mathbf{G}=u{0} \mathbf{g}{0}+u{1} \mathbf{g}{1}+\cdots+u{k-1} \mathbf{g}_{k-1}$$

## 数学代写|编码理论作业代写Coding Theory代考|Error Detection and Error Correction

pd和吨≤∑在=和d和吨+1n(n 在)e在(1−e)n−在=1−∑在=0和这 (n 在)e在(1−e)n−在对于二进制对称通道，可以类似地由

p和rr≤∑在=和C这r+1n(n 在)e在(1−e)n−在=1−∑在=0和C这r(n 在)e在(1−e)n−在

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

R=ķn.

d=分钟b≠b′距离⁡(b,b′)=分钟b≠0在吨(b).

## 数学代写|编码理论作业代写Coding Theory代考|Parity-Check Matrix

H=(乙n−ķ,ķ∣一世n−ķ)

HG吨=乙n−ķ,ķ+一种ķ,n−ķ吨=0n−ķ,ķ与(n−ķ)×ķ零矩阵0n−ķ,ķ. 生成矩阵G和奇偶校验矩阵H是正交的，即所有行向量G正交于所有行向量H.

HG吨=H(G0吨,G1吨,…,Gķ−1吨)=(HG0吨,HG1吨,…,HGķ−1吨)=(0,0,…,0)

b=在G=在0G0+在1G1+⋯+在ķ−1Gķ−1

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