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$$H=\begin{pmatrix} E_m \\ P \end{pmatrix}\Longrightarrow G... ...-m-1} \end{pmatrix}\qquad P\in M((2^m-m-1)\times m,{\bf F}_2)$$

$$G\in M((2^m-m-1)\times (2^m-1),{\bf F}_2), \operatorname{rank}G=2^m-m-1, \operatorname{Im}T_G=\operatorname{Ker}T_H$$

Îã 3.1   m=3¤Î¾ì¹ç.

$$P= \begin{pmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}$$

$$G=(-P \ E_4)\in M(4\times 7,{\bf F}_2)$$

$${\mathbf y}={\mathbf x}G=(x_1\ x_2\ x_3\ x_4)(-P\ E_4)=(-{\mathbf x}P\ x_1\ x_2\ x_3\ x_4)$$

¤è¤ê,

$$\begin{cases} (y_1\ y_2\ y_3)=-{\mathbf x}P & \text{ ¥Ñ¥ê¥Æ¥... ..._6\ y_7)=(x_1\ x_2\ x_3\ x_4) & \text{ ¾ðÊó¥Ó¥Ã¥È } \end{cases}$$

¾ðÊó¥Ó¥Ã¥È	¥Ñ¥ê¥Æ¥£¸¡ºº¥Ó¥Ã¥È	¾ðÊó¥Ó¥Ã¥È	¥Ñ¥ê¥Æ¥£¸¡ºº¥Ó¥Ã¥È
(0,0,0,0)	(0,0,0)	(1,0,0,0)	(1,1,0)
(0,0,0,1)	(1,1,1)	(1,0,0,1)	(0,0,1)
(0,0,1,0)	(0,1,1)	(1,0,1,0)	(1,0,1)
(0,0,1,1)	(1,0,0)	(1,0,1,1)	(0,1,0)
(0,1,0,0)	(1,0,1)	(1,1,0,0)	(0,1,1)
(0,1,0,1)	(0,1,0)	(1,1,0,1)	(1,0,0)
(0,1,1,0)	(1,1,0)	(1,1,1,0)	(0,0,0)
(0,1,1,1)	(0,0,1)	(1,1,1,1)	(1,1,1)