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# Cyclic code

`(7, 4)`循环码的生多项式 $$g(x)=x^3+x^2+1$$ , 则`D(1010)`的系循环码是多少？

1\.

$$
\begin{align\*}
1010 \Rightarrow d(x) = x^3 + x^1
\end{align\*}
$$

2\.

$$
\begin{align\*}
\&x^{7 - 4} \cdot d(x)
\ \\
\=\&x^3 \cdot (x^3 + x)
\ \\
\=\&x^6 + x^4
\end{align\*}
$$

3\.

$$
\begin{align\*}
\&rem\[(x^6 + x^4), g(x)] = 1
\ \\
&\text{rem( a , b ) returns the remainder after division of a by b}
\end{align\*}
$$

4\.

$$
\begin{align\*}
c(x) &= x^6 + x^4 + 1
\\
&\Downarrow
\\
10&10001
\end{align\*}
$$

## hamming code

$$n = 2^r - 1$$

`n` represents the length of codes

`r` represents the 监督码（不传递信息，只为纠错而加的东西）