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）