Reed-Solomon code ¤ÎÂå¿ô´ö²¿³ØŪ²ò¼á

$${\bf P}^1({\bf F}_q) := \{(x,y)\in {\bf F}_q^2-\{(0,0)\}\}/\s... ...rrow \exists \lambda\in{\bf F}_q-\{0\}, (x',y')=\lambda (x,y)$$

(x,y)¤ÎƱÃÍÎà¤ò(x:y)¤È½ñ¤¯¤³¤È¤Ë¤¹¤ë¤È,

$${\bf F}_q\ni x \longmapsto (x:1)\in{\bf P}^1({\bf F}_q)$$

¤È¤¹¤ë¤³¤È¤Ë¤è¤Ã¤Æ, ${\bf F}_q$¤Ï ${\bf P}^1({\bf F}_q)$¤ÎÉôʬ½¸¹ç¤È¤Ê¤ê,

$${\bf P}^1({\bf F}_q)={\bf F}_q\cup \{(1:0)\}$$

¤È¤Ê¤ë. ¤·¤¿¤¬¤Ã¤Æ, ${\bf P}^1({\bf F}_q)$¤Ï, ${\bf F}_q$¤Ë̵¸Â±óÅÀ¤òÉÕ¤±²Ã¤¨¤Æ¥³¥ó¥Ñ¥¯¥È²½¤·¤¿¤è¤¦¤Ê¤â¤Î¤È»×¤¦¤³¤È¤¬¤Ç¤­¤ë(Ê£ÁÇÊ¿Ì̤Υ³¥ó¥Ñ¥¯¥È²½¤Î¥¢¥Ê¥í¥¸¡¼). ¤³¤Î(1:0)¤ò$\infty$¤È¤«¤¯¤³¤È¤Ë¤¹¤ë.

$$\begin{align*}P_1, P_2, \cdots , &P_n , Q \in{\bf P}^1({\bf F}_q), \quad P_i\n... ...̰ʲ¼¤Î¶Ë¤ò»ý¤Ä${\bf P}^1$ ¾å¤ÎÍ­Íý´Ø¿ô}\} = {\bf F}_q[x]_{<k} \\ \end{align*}$$

$$\begin{CD} \phi : L((k-1)\infty)\ni f @>>> (f(P_1), f(P_2), \cdots , f(P_n)) \in{\bf F}_q^n \\ \end{CD}$$

¤³¤¦¤·¤Æ´ö²¿³ØŪ¤Ê¤â¤Î¤È¤·¤Æ¤È¤é¤¨¤Æ¤ß¤ë¤È, Éä¹æ¤òºî¤ë¤È¤¤¤¦´ÑÅÀ¤Î¤ß ¤«¤é¤Ï, P_j ¤¬¸¶»Ïq-1¾èº¬¤ò$\beta$¤È¤·¤Æ $(\beta^j:1)$¤ÇÍ¿¤¨¤é¤ì¤ëɬÍפϤʤ¯, P_j=(x:1), $x\in{\bf F}_q$¤Ç¤¢¤ì¤Ð¤è¤¤ ¤È¤¤¤¨¤ë. (ºÇ¾®µ÷Î¥¤äÉü¹æ¤Î·×»»Î̤ÏÊÌÌäÂê)