# 生成矩阵

## 术语

G 为一矩阵，它生成线性码 C码字英语codeword的方式为，

w = s G,

${\displaystyle G={\begin{bmatrix}I_{k}|P\end{bmatrix}}}$,

${\displaystyle H={\begin{bmatrix}-P^{\top }|I_{n-k}\end{bmatrix}}}$,

1. 任意排列码的位置
2. 将固定位置上的做置换

## 注释

1. ^ MacKay, David, J.C. Information Theory, Inference, and Learning Algorithms (PDF). Cambridge University Press. 2003: 9. ISBN 9780521642989. Because the Hamming code is a linear code, it can be written compactly in terms of matrices as follows. The transmitted codeword ${\displaystyle \mathbf {t} }$ is obtained from the source sequence ${\displaystyle \mathbf {s} }$ by a linear operation,

${\displaystyle \mathbf {t} =\mathbf {G} ^{\intercal }\mathbf {s} }$

where ${\displaystyle \mathbf {G} }$ is the generator matrix of the code... I have assumed that ${\displaystyle \mathbf {s} }$ and ${\displaystyle \mathbf {t} }$ are column vectors. If instead they are row vectors, then this equation is replaced by

${\displaystyle \mathbf {t} =\mathbf {sG} }$

The rows of the generator matrix can be viewed as defining the basis vectors. 参数|quote=值左起第130位存在删除符 (帮助)
2. ^ Ling & Xing 2004，p. 52
3. ^ Roman 1992，p. 198
4. ^ Roman 1992，p. 200
5. ^ Pless 1998，p. 8