### 数学代写|matlab代写| Reed-Muller Codes with Maple

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## 数学代写|matlab代写|Reed-Muller Codes with Maple

In this section, we will show how Maple can be used to construct and correct errors in Reed-Muller codes. We will consider the $(16,8)$ Reed-Muller code.
We will begin by generating the normalized Hadamard matrix $H_{16}$ used to construct the code.
$>$ with(LinearAlgebra) :
$>H 1:=\operatorname{Matrix}([[1]]):$
$>H 2:=\operatorname{Matrix}([[\mathrm{H} 1, \mathrm{H} 1],[\mathrm{H} 1,-\mathrm{H1}]])$ :
$>H 4:=\operatorname{Matrix}([\mathrm{H} 2, \mathrm{H} 2],[\mathrm{H} 2,-\mathrm{H} 2]]):$
$>H 8:=$ Matrix $([\mathrm{H} 4, \mathrm{H} 4],[\mathrm{H} 4,-\mathrm{H} 4]]):$
$>H 16:=\operatorname{Matrix}([\mathrm{H8}, \mathrm{H8}],[\mathrm{H} 8,-\mathrm{H8}]]):$
We can then obtain the incidence matrix $A$ that results from $H_{16}$ by entering the following sequence of commands.
$>\mathrm{A}:=$ DeleteRow $(\mathrm{H} 16,1)$ :
$>A:=\operatorname{DeleteColumn}(A, 1):$
$>f:=x \rightarrow$ if $x=-1$ then 0 else $1 f i:$
$>A:=\operatorname{map}(f, A)$;
$>\mathrm{A}:=$ DeleteRow $(\mathrm{H} 16,1):$
$>\mathrm{A}:=$ DeleteColumn $(\mathrm{A}, 1):$
$>\mathrm{f}:=\mathrm{x} \rightarrow \mathrm{if} \mathrm{} \mathrm{x}=-1$ then 0 else $1 \mathrm{fi:}$
$>\mathrm{A}:=\operatorname{map}(\mathrm{f}, \mathrm{A}) ;$
$A:=\left[\begin{array}{c}15 x 15 \text { Matrix } \ \text { Data Type: anything } \ \text { Storage: rectangular } \ \text { Order: Fortran_order }\end{array}\right]$
$A:=\left[\begin{array}{c}15 \text { x } 15 \text { Matrix } \ \text { Data Type: anything } \ \text { Storage: rectangular } \ \text { Order: Fortran_order }\end{array}\right]$
By default, Maple only displays matrices of size $10 \times 10$ and smaller. As demonstrated in the preceding command, matrices that are larger than $10 \times 10$ are displayed with a placeholder. The Maple interface function with the rtablesize option can be used to display matrices that are larger than $10 \times 10$. Once a matrix dimension is specified with this function, all matrices of this size and smaller will be displayed until the Maple session is closed or until the function is entered again with a different dimension. By entering the following command, we cause Maple to display all matrices of size $50 \times 50$ and smaller throughout the remainder of this Maple session.
$>$ interface $($ rtablesize $=50)$ :
We can now see the incidence matrix that is stored as the variable $A$ by entering the following command.

## 数学代写|matlab代写|Reed-Muller Codes with MATLAB

In this section, we will show how MATLAB can be used to construct and correct errors in Reed-Muller codes. We will consider the $(16,8)$ ReedMuller code.

We will begin by generating the normalized Hadamard matrix $H_{16}$ used to construct the code.
$>\mathrm{H} 1=[1]$;
$>H 2=\left[\begin{array}{lll}\mathrm{H} 1 & \mathrm{H} 1 & \mathrm{H} 1\end{array}\right.$
$>\mathrm{H}{4}=\left[\begin{array}{llll}\mathrm{H} 2 & \mathrm{H} 2 ; & \mathrm{H} 2 & -\mathrm{H} 2\end{array}\right]$; $>\mathrm{HB}=[\mathrm{H} 4 \mathrm{H} 4 ; \mathrm{H} 4-\mathrm{H} 4]$; $>\mathrm{H}^{16}=[\mathrm{H8} \mathrm{H8} ; \mathrm{H8}-\mathrm{H8}]$; We can then obtain the incidence matrix $A$ that results from $H{16}$ by entering the following sequence of commands.
\begin{aligned} &\Rightarrow A=H 16 ; \ &>A(:, 1)=[] ; \ &\Rightarrow A(1,:)=[] ; \ &\Rightarrow A=f \text { ind }(A<0) \ &\Rightarrow A(f)=0 \ &A= \end{aligned}Next, to form the matrix $B$ in Theorem $3.5$, we need to interchange all zeros and ones in $A$. To do this, we will first create a matrix with the same size as $A$ but containing a zero in every position. Since the MATLAB size function returns the number of rows and columns in a matrix, the following command assigns the number of rows and columns in $A$ as the variables rowdimA and coldimA, respectively.

## 数学代写|matlab代写|Linear Codes

As we have shown, Hadamard and Reed-Muller codes are easy to construct and can have significant error correction capabilities. However, because Hadamard and Reed-Muller codes do not form vector spaces, they are not ideal for situations in which a very large number of codewords are needed. Because Hadamard and Reed-Muller codes do not form vector spaces, error correction in these types of codes must be done by comparing received vectors with each of the codewords one by one. While this error correction scheme does not pose any problems in relatively small codes, it would not be an efficient way to correct errors in a code with a very large number of codewords. In this section, we will present a method for constructing codes that do form vector spaces. We will then present some more efficient schemes for correcting errors in these codes.

Recall that a code that forms a vector space is called a linear code. We will describe a linear code using the parameters $[n, k]$ if the codewords in the code have length $n$ positions and the code forms a vector space of dimension $k$. In this section, we will present linear codes constructed using generator matrices. Specifically, let $W=\mathbb{Z}{2}^{k}$ and $V=\mathbb{Z}{2}^{n}$ with $k<n$, and let $G$ be a $k \times n$ matrix over $\mathbb{Z}_{2}$ of full row rank. Then $C={\mathbf{v} \in V \mid \mathbf{v}=\mathbf{w} \cdot G$ for some $\mathbf{w} \in W}$ is a subspace of $V$ of dimension $k$. Thus, the vectors in $C$ are the codewords in an $[n, k]$ linear code in $V$ with $2^{k}$ codewords. The matrix $G$ is called a generator matrix for $C$.

Note that the code $C$ in Example $3.4$ has a minimum distance of 7 . Thus, $C$ will be three-error correcting, whereas bit errors could not be corrected in $W=\mathbb{Z}_{2}^{2}$. Of course, the vectors in $C$ are longer than the vectors in $W$, and consequently it would take more effort to transmit the vectors in $C$. However, the ability to correct up to three bit errors in $C$ should be much more valuable than the extra effort required to transmit the vectors. Furthermore, $W$ could still be used in the encoding and decoding of the actual information being transmitted. Specifically, information could be encoded as vectors in $W$, and then converted to vectors in $C$ before being transmitted. Received vectors could then be corrected to codewords in $C$ if necessary, and converted back into vectors in $W$ to be decoded. In order for this process to be valid, we must be able to convert between $W$ and $C$ uniquely. This is precisely why we required $G$ to have full row rank. With this requirement, $G$ must have a right inverse, say $B$, and then $\mathbf{w} \in W$ can be retrieved uniquely from $\mathbf{w} G \in C$ by $\mathbf{w}=\mathbf{w} G B$.

## 数学代写|matlab代写|Reed-Muller Codes with Maple

>与（线性代数）：
>H1:=矩阵⁡([[1]]):
>H2:=矩阵⁡([[H1,H1],[H1,−H1]]) :
>H4:=矩阵⁡([H2,H2],[H2,−H2]]):
>H8:=矩阵([H4,H4],[H4,−H4]]):
>H16:=矩阵⁡([H8,H8],[H8,−H8]]):

>一个:=删除行(H16,1):
>一个:=删除列⁡(一个,1):
>F:=X→如果X=−1然后 0 其他1F一世:
>一个:=地图⁡(F,一个);
>一个:=删除行(H16,1):
>一个:=删除列(一个,1):
>F:=X→一世FX=−1然后 0 其他1F一世:
>一个:=地图⁡(F,一个);

>界面(表大小=50)：

## 数学代写|matlab代写|Reed-Muller Codes with MATLAB

>H1=[1];
>H2=[H1H1H1
>H4=[H2H2;H2−H2]; >H乙=[H4H4;H4−H4];>H16=[H8H8;H8−H8]; 然后我们可以得到关联矩阵一个结果是H16通过输入以下命令序列。

⇒一个=H16; >一个(:,1)=[]; ⇒一个(1,:)=[]; ⇒一个=F 工业 (一个<0) ⇒一个(F)=0 一个=接下来，形成矩阵乙定理3.5，我们需要交换所有的零和一一个. 为此，我们将首先创建一个与一个但在每个位置都包含一个零。由于 MATLAB 的 size 函数返回矩阵中的行数和列数，因此以下命令分配矩阵中的行数和列数一个分别作为变量rowdimA和coldimA。

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## MATLAB代写

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