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

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

The code rate $R=k / n$ and the minimum Hamming distance $d$ are important parameters of a linear block code $\mathrm{B}(n, k, d)$. It is therefore useful to know whether a linear block code theoretically exists for a given combination of $R$ and $d$. In particular, for a given minimum Hamming distance $d$ we will consider a linear block code to be better than another linear block code with the same minimum Hamming distance $d$ if it has a higher code rate $R$. There exist several theoretical bounds which we will briefly discuss in the following (Berlekamp, 1984; Bossert, 1999; Ling and Xing, 2004; van Lint, 1999) (see also Figure 2.22). Block codes that fulfil a theoretical bound with equality are called optimal codes.
Singleton Bound
The simplest bound is the so-called Singleton bound. For a linear block code $\mathrm{B}(n, k, d)$ it is given by
$$k \leq n-d+1 .$$
A linear block code that fulfils the Singleton bound with equality according to $k=n-d+$ 1 is called MDS (maximum distance separable). Important representatives of MDS codes are Reed-Solomon codes which are used, for example, in the channel coding scheme within the audio compact disc (see also Section 2.3.7).
Sphere Packing Bound
The so-called sphere packing bound or Hamming bound can be derived for a linear $e_{\text {cor }}=$ $\lfloor(d-1) / 2\rfloor$-error correcting $q$-nary block code $\mathrm{B}(n, k, d)$ by considering the correction balls within the code space $\mathbb{F}{q}^{n}$. Each correction ball encompasses a total of $$\sum{i=0}^{e_{\mathrm{cor}}}\left(\begin{array}{l} n \ i \end{array}\right)(q-1)^{i}$$

vectors in the code space $F_{q}^{n}$. Since there is exactly one correction ball for each code word, we have $M=q^{k}$ correction balls. The total number of vectors within any correction ball of radius $e_{\text {cor }}$ is then given by
$$q^{k} \sum_{i=0}^{e_{\text {cor }}}\left(\begin{array}{l} n \ i \end{array}\right)(q-1)^{i}$$
Because this number must be smaller than or equal to the maximum number $\left|\mathbb{F}{q}^{n}\right|=q^{n}$ of vectors in the finite vector space $\mathbb{F}{q}^{n}$, the sphere packing bound
$$q^{k} \sum_{i=0}^{e_{\operatorname{cor}}}\left(\begin{array}{l} n \ i \end{array}\right)(q-1)^{i} \leq q^{n}$$

## 数学代写|编码理论作业代写Coding Theory代考|Plotkin’s (u|v) Code Construction

In contrast to the aforementioned code constructions, the (u|v) code construction originally proposed by Plotkin takes two linear block codes $\mathrm{B}\left(n_{1}, k_{1}, d_{1}\right)$ and $\mathrm{B}\left(n_{2}, k_{2}, d_{2}\right)$ with code word lengths $n_{1}$ and $n_{2}$, information word lengths $k_{1}$ and $k_{2}$ and minimum Hamming distances $d_{1}$ and $d_{2}$. For simplification, we fill the code words of the block code with smaller code word length by an appropriate number of zeros. This step is called zero padding. Let $\mathbf{u}$ and $\mathbf{v}$ be two arbitrary code words thus obtained from the linear block codes $\mathrm{B}\left(n_{1}, k_{1}, d_{1}\right)$ and $\mathrm{B}\left(n_{2}, k_{2}, d_{2}\right)$ respectively. ${ }^{12}$ Then we have
$$\mathbf{u}= \begin{cases}\left(\mathbf{b}{1} \mid 0, \ldots, 0\right), & n{1}<n_{2} \ \mathbf{b}{1}, & n{1} \geq n_{2}\end{cases}$$
and
$$\mathbf{v}= \begin{cases}\mathbf{b}{2}, & n{1}<n_{2} \ \left(\mathbf{b}{2} \mid 0, \ldots, 0\right), & n{1} \geq n_{2}\end{cases}$$
with the arbitrarily chosen code words $\mathbf{b}{1} \in \mathbb{B}\left(n{1}, k_{1}, d_{1}\right)$ and $\mathbf{b}{2} \in \mathbb{B}\left(n{2}, k_{2}, d_{2}\right)$. We now identify the code words $\mathbf{u}$ and $\mathbf{v}$ with the original codes $\mathrm{B}\left(n_{1}, k_{1}, d_{1}\right)$ and $\mathrm{B}\left(n_{2}, k_{2}, d_{2}\right)$, i.e. we write $\mathbf{u} \in \mathbb{B}\left(n_{1}, k_{1}, d_{1}\right)$ and $\mathbf{v} \in \mathbb{B}\left(n_{2}, k_{2}, d_{2}\right)$. The code resulting from Plotkin’s (u|v) code construction is then given by
$$\mathbb{B}^{\prime}\left(n^{\prime}, k^{\prime}, d^{\prime}\right)=\left{(\mathbf{u} \mid \mathbf{u}+\mathbf{v}): \mathbf{u} \in \mathbb{B}\left(n_{1}, k_{1}, d_{1}\right) \wedge \mathbf{v} \in \mathbb{B}\left(n_{2}, k_{2}, d_{2}\right)\right}$$
This code construction creates all vectors of the form $(\mathbf{u} \mid \mathbf{u}+\mathbf{v})$ by concatenating all possible vectors $\mathbf{u}$ and $\mathbf{u}+\mathbf{v}$ with $\mathbf{u} \in \mathbb{B}\left(n_{1}, k_{1}, d_{1}\right)$ and $\mathbf{v} \in \mathbb{B}\left(n_{2}, k_{2}, d_{2}\right)$. The resulting code is characterised by the following code parameters
\begin{aligned} &n^{\prime}=2 \max \left{n_{1}, n_{2}\right} \ &k^{\prime}=k_{1}+k_{2}, \ &d^{\prime}=\min \left{2 d_{1}, d_{2}\right} \end{aligned}

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

In this section we will present some important linear block codes. In particular, we will consider the already introduced repetition codes, parity-check codes and Hamming codes. Furthermore, simplex codes and Reed-Muller codes and their relationships are discussed (Berlekamp, 1984; Bossert, 1999; Lin and Costello, 2004; Ling and Xing, 2004).

We have already introduced the binary triple repetition code in Section $1.3$ as a simple introductory example of linear block codes. In this particular code the binary information symbol 0 or 1 is transmitted with the binary code word 000 or 111 respectively. In general, a repetition code over the alphabet $\mathrm{F}{q}$ assigns to the $q$-nary information symbol $u{0}$ the $n$-dimensional code word $\mathbf{b}=\left(u_{0}, u_{0}, \ldots, u_{0}\right)$. Trivially, this block code is linear. The minimum Hamming distance is
$$d=n .$$
Therefore, the repetition code in Figure $2.27$ is a linear block code $\mathrm{B}(n, 1, n)$ that is able to detect $e_{\text {det }}=d-1=n-1$ errors or to correct $e_{\text {cor }}=\lfloor(d-1) / 2\rfloor=\lfloor(n-1) / 2\rfloor$ errors. The code rate is
$$R=\frac{1}{n} .$$
For the purpose of error correction, the minimum distance decoding can be implemented by a majority decoding scheme. The decoder emits the information symbol $\hat{u}_{0}$ which appears most often in the received word.
The weight distribution of a repetition code is simply given by
$$W(x)=1+(q-1) x^{n}$$
Although this repetition code is characterised by a low code rate $R$, it is used in some communication standards owing to its simplicity. For example, in the short-range wireless communication system Bluetooth ${ }^{\mathrm{TM}}$ a triple repetition code is used as part of the coding scheme of the packet header of a transmitted baseband packet (Bluetooth, 2004).

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

ķ≤n−d+1.

Sphere Packing Bound

∑一世=0和C这r(n 一世)(q−1)一世

qķ∑一世=0和心电图 (n 一世)(q−1)一世

qķ∑一世=0和心电图(n 一世)(q−1)一世≤qn

## 数学代写|编码理论作业代写Coding Theory代考|Plotkin’s (u|v) Code Construction

\mathbb{B}^{\prime}\left(n^{\prime}, k^{\prime}, d^{\prime}\right)=\left{(\mathbf{u} \mid \mathbf {u}+\mathbf{v}): \mathbf{u} \in \mathbb{B}\left(n_{1}, k_{1}, d_{1}\right) \wedge \mathbf{v} \in \mathbb{B}\left(n_{2}, k_{2}, d_{2}\right)\right}\mathbb{B}^{\prime}\left(n^{\prime}, k^{\prime}, d^{\prime}\right)=\left{(\mathbf{u} \mid \mathbf {u}+\mathbf{v}): \mathbf{u} \in \mathbb{B}\left(n_{1}, k_{1}, d_{1}\right) \wedge \mathbf{v} \in \mathbb{B}\left(n_{2}, k_{2}, d_{2}\right)\right}

\begin{对齐} &n^{\prime}=2 \max \left{n_{1}, n_{2}\right} \ &k^{\prime}=k_{1}+k_{2}, \ &d ^{\prime}=\min \left{2 d_{1}, d_{2}\right} \end{aligned}\begin{对齐} &n^{\prime}=2 \max \left{n_{1}, n_{2}\right} \ &k^{\prime}=k_{1}+k_{2}, \ &d ^{\prime}=\min \left{2 d_{1}, d_{2}\right} \end{aligned}

d=n.

R=1n.

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