\documentclass[a4j]{jarticle}
\title{$B@~7ABe?t$@$1$GM}2r$9$kBe?t4v2?Id9f(B}
\author{$B>>K\(B $BN4B@O:(B\\
$BEl5~9)6HBg3X(B $BEE5$EE;R9)3X2J(B $B:dDm8&5f<<(B\\
$B")(B152-8552 $BL\9u6hBg2,;3(B 2-12-1\\
E-mail: ryutaroh@ss.titech.ac.jp\\
URL: http://tsk-www.ss.titech.ac.jp/{\textasciitilde}ryutaroh/}
\date{1999$BG/(B12$B7n(B11$BF|(B\thanks{2000$BG/(B6$B7n(B1$BF|$K;29MJ88%$N(B ``to appear'' $B$K$J$C$F$$$kItJ,$r<B:]$N4,?t(B, $B%Z!<%8?t$K99?7$7$?(B.}}

% $B>.$5$$(BPostScript $B$r:n$k>l9g(B
%\usepackage{times,mathptmx}
%\DeclareMathAlphabet{\bm}{OT1}{ptm}{b}{it}

\DeclareMathAlphabet{\bm}{OML}{cmm}{b}{it}

\newcommand{\fq}{\mathbf{F}_q}
\newcommand{\startproof}{{\noindent \textbf{$B>ZL@(B: }}}
\newcommand{\qed}{\hspace*{\fill}\rule{1.3ex}{1.3ex}}
\newcommand{\rank}{\mathrm{rank}\,}
\newcommand{\n}{\mathbf{N}_0}

\newtheorem{definition}{$BDj5A(B}[section]
\newtheorem{proposition}[definition]{$BL?Bj(B}
\newtheorem{example}[definition]{$BNc(B}
\newtheorem{lemma}[definition]{$BJdBj(B}
\newtheorem{remark}[definition]{$BCm(B}
\newtheorem{algorithm}[definition]{$B%"%k%4%j%:%`(B}

%\usepackage{makeidx}
%\makeindex
% $B0J2<$N%^%/%m$O:w0z$K>h$;$kC18l$r;XDj$9$k%^%/%m$G$"$k(B.
% $BBh0l0z?t$KC18l$r;XDj$7(B, $BBhFs0z?t$KFI$_$r%"%k%U%!%Y%C%H$HJ?2>L>$G;XDj$9$k(B
% $BC18l$OB@;z$G=PNO$5$l$k(B
%\newcommand{\keyword}[2]{{\bfseries\boldmath #1}\index{#2@#1}}
\newcommand{\keyword}[2]{{\bfseries\boldmath #1}}

\begin{document}
\sloppy
\maketitle
\section{$B$O$8$a$K(B}
Goppa$B$K$h$j(B1981$BG/$KDs0F$5$l$?(B
$BBe?t4v2?Id9f(B \cite{goppa81} $B$OJ8;zDL$jBe?t4v2?$N35G0$r;H$C$FDj5A$5$l$k$N$G(B, 
$BBe?t4v2?$KFk@w$_$NL5$$?M$K$OBgJQ<h$CIU$-$K$/$$J,Ln$G$"$C$?(B.
$B$7$+$7(B, H\o holdt$B$i$K$h$jF3F~$5$l$?=E$_4X?t$N35G0$rMQ$$$k$3$H$K$h$j(B,
$BBe?t4v2?Id9f$rBe?t4v2?$rMQ$$$:$H$b$4$^$+$7L5$7$KM}2r$9$k$3$H$,2DG=$K$J$C$?(B.
$BK\9F$G$OBe?t4v2?Id9f$N<gMW$J35G0$r@~7ABe?t$H=iEy4DO@$N$_$rMQ$$$F2r@b$9$k(B.
$BL?Bj$K$O$9$Y$F40A4$J>ZL@$rIU$1$F$$$k(B.
$BBh(B2$B@a$G$OBe?t4v2?Id9f$N(B Feng-Rao $BI|9f%"%k%4%j%:%`$H(B Feng-Rao $B8B3&$r(B
$BG$0U$N@~7AId9f$KE,MQ$G$-$k$h$&$K(B, 
$B@~7ABe?t$@$1$rMQ$$$FDj5A$7D>$9(B. 
$BBh(B3$B@a$G$O(B H\o holdt $B$i$K$h$jDs0F$5$l$?@~7ABe?t$H=iEy4DO@$@$1$rMQ$$$?(B
$BBe?t4v2?Id9f$NDj5A$r>R2p$9$k(B. 

$B>.J8$rDL$7$F(B, $\fq$ $B$rMWAG?t(B $q$ $B$NM-8BBN$H$7(B, 
$\fq^n$ $B$rM-8BBN$NMWAG$r(B $n$ $B8DJB$Y$?%Y%/%H%k$+$i$J$k@~7A6u4V$H$9$k(B.
$B$^$?(B $\{\bm{u}_1, \ldots, \bm{u}_n\},
\{\bm{v}_1, \ldots, \bm{v}_n\},
\{\bm{w}_1,\ldots,\bm{w}_n\}$
$B$r(B $\fq^n$ $B$N4pDl$H$9$k(B. 
$W$ $B$r(B $\{\bm{w}_1,\ldots,\bm{w}_n\}$
$B$N6u$G$J$$??ItJ,=89g$H$7(B, $W$ $B$NMWAG$r8!::9TNs$N9T$K;}$D(B
$B@~7AId9f$r(B $C(W)$ $B$GI=$9(B.

\section{Feng-Rao $B8B3&$H(B Feng-Rao $BI|9f%"%k%4%j%:%`(B}
$B$3$N@a$G$O(B1993$BG/$K(BFeng$B$H(BRao$B$K$h$jDs0F$5$l$?I|9f%"%k%4%j%:%`(B \cite{fengrao93},
$B$*$h$SId9f$N8m$jD{@5G=NO$rB;$J$o$:$K8!::%7%s%\%k?t$r8:$i$9J}K!(B \cite{feng95}
$B$r>R2p$9$k(B. $B$3$3$G>R2p$9$kFbMF$O;01:$NO@J8(B \cite[3$B>O(B]{miurathesis},
\cite{miura98a} $B$rB?>/0lHL2=$7$?$b$N$G$"$k(B.

\subsection{Feng-Rao $B8B3&(B}
$BK\>.@a$G$O(B $C(W)$ $B$N:G>.5wN%$N2<3&$rM?$($k(B.
$B%Y%/%H%k(B $\bm{a},\bm{b}\in \fq^n$ $B$KBP$7(B
$\bm{a} * \bm{b}$ $B$O(B $\bm{a}$ $B$H(B $\bm{b}$ $B$r:BI8$4$H$K3]$1$FF@$i$l$k(B
$B%Y%/%H%k$H$9$k(B. 

\begin{definition}\normalfont 
$\mathcal{W}_i$ $B$r(B $\bm{w}_1,\ldots,\bm{w}_i$ $B$GD%$i$l$k@~7A(B
$B6u4V$H$7(B $\mathcal{W}_0 = \{\mathbf{0}\}$ $B$H$9$k(B. 
$(\bm{u}_i,\bm{v}_j)$ $B$,(B, $B$"$k(B $1\leq s \leq n$ $B$KBP$7(B
$\bm{u}_i * \bm{v}_j \in \mathcal{W}_s \setminus \mathcal{W}_{s-1}$
$B$+$D$9$Y$F$N(B $1\leq u \leq i$, $1 \leq v\leq j$, $(u,v) \neq (i,j)$
$B$K$D$$$F(B $\bm{u}_u * \bm{v}_v\in \mathcal{W}_{s-1}$
$B$H$J$k$H$-(B, \keyword{$B?6Iq$$$,NI$$(B}{$B$U$k$^$$$,$h$$(B}$B$H$$$&(B. 
$1 \leq s \leq n$ $B$K$D$$$F(B $M(s) := \{
(i,j) \mid \bm{u}_i * \bm{v}_j \in \mathcal{W}_s \setminus
\mathcal{W}_{s-1}$ $B$+$D(B $(\bm{u}_i,\bm{v}_j)$ $B$O?6Iq$$$,NI$$(B $\}$
$B$HDj5A$9$k(B. 
\end{definition}

\begin{definition}\normalfont 
$C(W)$ $B$KBP$9$k(B\keyword{Feng-Rao$B8B3&(B}{Feng-Rao$B$2$s$+$$(B}$B$O(B
\[
\delta_\mathrm{FR}(W) := \min\{\sharp M(s) \mid \bm{w}_s \notin W \}
\]
$B$GDj5A$5$l$k(B. $BC"$7(B, $\sharp M$ $B$O=89g(B $M$ $B$NMWAG?t$rI=$9(B.
\end{definition}

\begin{definition}\normalfont 
$B%Y%/%H%k(B $\bm{y} = (y_1,\ldots,y_n) \in \fq^n$ $B$KBP$7(B
\keyword{$B%7%s%I%m!<%`9TNs(B}{$B$7$s$I$m!<$`$.$g$&$l$D(B}
$S(\bm{y})$ $B$r(B
\[
S(\bm{y}) := \left(
\begin{array}{c}
\bm{u}_1\\
\vdots\\
\bm{u}_n
\end{array}\right)
\left(
\begin{array}{ccc}
y_1&&\\
&\ddots&\\
&&y_n
\end{array}\right)
\left(
\begin{array}{c}
\bm{v}_1\\
\vdots\\
\bm{v}_n
\end{array}\right)^T
\]
$B$GDj5A$9$k(B. $B$^$?(B $w(\bm{y})$ $B$G(B $\bm{y}$ $B$N%O%_%s%0=E$_$rI=$9(B.
\end{definition}

\begin{proposition}\normalfont 
$B%Y%/%H%k(B $\bm{y}$ $B$KBP$7(B $\rank S(\bm{y}) = w(\bm{y})$.
$B$^$?(B $S(\bm{y})$ $B$N(B $(i,j)$ $B@.J,$O(B $\langle \bm{y},
\bm{u}_i*\bm{v}_j\rangle$ $B$GM?$($i$l$k(B. 
$BC"$7(B $\langle,\rangle$ $B$O%Y%/%H%k$NFb@Q$rI=$9(B. 
\end{proposition}
\startproof
$B:G=i$N<gD%$O(B $S(\bm{y})$ $B$NDj5A$h$jL@$i$+(B. $B$^$?(B,
\[
S(\bm{y}) = 
\left(
\begin{array}{c}
\bm{y}*\bm{u}_1\\
\vdots\\
\bm{y}*\bm{u}_n
\end{array}\right)
\left(
\begin{array}{c}
\bm{v}_1\\
\vdots\\
\bm{v}_n
\end{array}\right)^T
\]
$B$G$"$k$N$G(B, 
$S(\bm{y})$ $B$N(B $(i,j)$ $B@.J,$O(B $\langle \bm{y}*\bm{u}_i, \bm{v}_j\rangle$
$B$K$J$k$,(B, $B$3$l$O(B $\langle \bm{y}, \bm{u}_i * \bm{v}_j \rangle$
$B$KEy$7$$(B. 
\qed

\begin{proposition}\normalfont 
$C(W)$ $B$N:G>.5wN%$O(B $\delta_\mathrm{FR}(W)$ $B0J>e$G$"$k(B.
\end{proposition}
\startproof
$\bm{y}$ $B$r(B $C(W)$ $B$N(B $\mathbf{0}$ $B$G$J$$Id9f8l$H$7(B,
$B$"$k@0?t(B $s$ $B$KBP$7(B $\langle \bm{y},\bm{w}_1\rangle = \cdots
= \langle \bm{y},\bm{w}_{s-1}\rangle = 0$,
$\langle \bm{y},\bm{w}_s\rangle \neq 0$ $B$,@.N)$7$F$$$k$H$9$k(B. 
$B$3$N$H$-(B $\bm{w}_s \notin W$ $B$G$"$k(B. $B0lJ}(B, $BG$0U$N(B $(i,j) \in M(s)$ $B$KBP$7(B
$S(\bm{y})$ $B$N(B $(i,j)$ $B@.J,$OHsNm$G$"$k(B. 
$B$J$<$J$i$P(B $\bm{u}_i*\bm{v}_j$ $B$O(B $\bm{w}_1,\ldots,\bm{w}_s$
$B$N@~7A7k9g$G=q$1(B, $B@~7A7k9g$N(B $\bm{w}_s$ $B$N78?t$OHsNm$J$N$G(B, 
\begin{eqnarray*}
\langle \bm{y}, \bm{u}_i*\bm{v}_j\rangle &=&
\left\langle \bm{y}, \sum_{\ell=1}^s c_\ell\bm{w}_\ell \right\rangle\\
&=& \sum_{\ell=1}^{s} c_\ell \langle\bm{y},\bm{w}_\ell\rangle\\
&=& c_s \langle \bm{y},\bm{w}_s \rangle\\
&\neq&0.
\end{eqnarray*}
$B$^$?F1MM$K$9$Y$F$N(B $1\leq u \leq i, 1\leq v\leq j, (u,v)\neq (i,j)$
$B$K$D$$$F(B $S(\bm{y})$ $B$N(B $(u,v)$ $BMWAG$O(B $0$ $B$G$"$k(B. 
$M(s)$ $B$NMWAG$O9TNs$N1&>e$+$i:82<$K$+$1$FJB$s$G$$$k$N$G(B, 
$\rank S(\bm{y}) \geq \sharp M(s) \geq \delta_\mathrm{FR}(W)$ $B$rF@$k(B. 
\qed

\begin{example}[Reed-Solomon$BId9f(B]\normalfont 
Reed-Solomon $BId9f$N:G>.5wN%$,(B Feng-Rao $B8B3&$NFC<l$J>l9g$H$7$F(B
$BF@$i$l$k$3$H$rNc$r5s$2$F<($9(B.
$\alpha$ $B$r(B $\fq$ $B$N86;O85(B,
$i=1,\ldots,n (= q-1)$ $B$KBP$7(B
$\bm{u}_i = \bm{v}_i = \bm{w}_i = (1^i, \alpha^i, \ldots, \alpha^{i(q-2)})$
$B$H$9$k(B.
$B$3$N$H$-(B $\bm{u}_i * \bm{v}_j = \bm{w}_{i+j-1}$ $B$J$N$G(B, 
$M(s) = \{ (1,s),(2,s-1),\ldots,(s,1) \}$ $B$H$J$k(B. 
$W=\{\bm{w}_1,\ldots,\bm{w}_r\}$ $B$H<h$k(B.
$B$3$N$H$-(B $C(W)$ $B$O(B $[q-1, q-1-r]$ Reed-Solomon$BId9f$K$J$j(B,
$\delta_\mathrm{FR}(W) = r+1$ $B$O$=$N:G>.5wN%$KEy$7$$(B.
\end{example}

$W$ $B$H$7$FE:$(;z$,>.$5$$=g$K(B $\{\bm{w}_1,\ldots,\bm{w}_r\}$
$B$H<h$k$N$,=>Mh$NBe?t4v2?Id9f$KAjEv$9$k$,(B,
Feng$B$H(BRao$B$O@_7W5wN%(B $d$ $B$r<B8=$9$k$?$a$K=89g(B $W$ $B$r(B
\[
W(d) := \{ \bm{w}_s \mid \sharp M(s) \leq d-1 \}
\]
$B$N$h$&$K<h$k$H8!::%7%s%\%k?t$r$h$j>/$J$/$G$-$k$3$H$rL@$i$+$K$7$?(B \cite{feng95}.
$B$3$N9=@.K!$O(B\keyword{$B2~NIBe?t4v2?Id9f(B}{$B$+$$$j$g$&$@$$$9$&$-$+$U$4$&(B}$B$H8F$P$l$k(B. 

\subsection{Feng-Rao $BI|9f%"%k%4%j%:%`(B}
$BId9f$r<BMQ$K6!$9$k$?$a$K$O8zN(E*$JI|9f%"%k%4%j%:%`$,IT2D7g$G$"$k(B. 
$BK\>.@a$G$O(B $C(W)$ $B$KDL?.$N2aDx$G@8$8$?(B $\lfloor (\delta_{FR}(W)-1)/2
\rfloor$ $B8D0J2<$N8m$j$r(B $O(n^3)$ $B2s$NM-8BBN$N1i;;$GD{@5$9$k(B
\keyword{Feng-Rao$BI|9f%"%k%4%j%:%`(B}{Feng-Rao$B$U$/$4$&$"$k$4$j$:$`(B}$B$r<($9(B. 

$BId9f8l(B $\bm{c} \in C(W)$ $B$,Aw?.$5$l(B $\bm{c} + \bm{e}$ $B$,<u?.$5$l$?$H$9$k(B. 
$\langle \bm{e}, \bm{w}_1\rangle, \ldots, \langle\bm{e},
\bm{w}_n\rangle$ $B$,J,$+$l$P9TNs$N3]$1;;$K$h$j(B
$O(n^2)$ $B2s$N1i;;$G8m$j%Y%/%H%k(B $\bm{e}$ $B$r5a$a$k$3$H$,$G$-$k(B.

$\bm{w}_s\in W$ $B$J$i(B $\langle \bm{c}+\bm{e},\bm{w}_s\rangle
= \langle\bm{e},\bm{w}_s\rangle$ $B$G$"$k(B. $B=>$C$F(B, 
$\bm{u}_i*\bm{v}_j$ $B$,(B $W$ $B$NMWAG$N@~7A7k9g$G=q$1$k>l9g(B,
$B<u?.8l$+$i(B $S(\bm{e})$ $B$N(B $(i,j)$ $B@.J,$rCN$k$3$H$,$G$-$k(B. 
$S(\bm{e})$ $B$N4{CN$NItJ,$+$iL$CN$NItJ,$r@5$7$/?dB,$9$k%"%k%4%j%:%`$r0J2<$K(B
$B<($9(B. 

$A$ $B$rBP3Q@.J,$,$9$Y$F(B $1$ $B$N(B $n \times n$ $B2<;03Q9TNs$H$9$k(B.
$B := A S(\bm{e})$ $B$H$*$/(B. 
$B$ $B$N3F!9$N9T(B $i$ $B$K$D$-(B $A$ $B$N:82<ItJ,$rE,Ev$K@_Dj$7$F(B
$\min\{ j \mid B$ $B$N(B $(i,j)$ $B@.J,$OHsNm(B $\}$ $B$r:GBg2=$9$k$3$H$r9M$($k(B. 
$B:#$"$k9T(B $i$ $B$K$D$$$F(B $\min\{ j' \mid B$ $B$N(B $(i,j')$ $B@.J,$OHsNm(B $\}$
$B$r:GBg2=$7(B $j$ $B$K$J$C$?$H$9$k(B. $B$3$N$H$-(B $(i,j)$ $B$r(B $S(\bm{e})$ $B$N(B
\keyword{discrepancy}{discrepancy}$B$H8F$V(B.
$B0J2<(B $S(\bm{e})$ $B$N$9$Y$F$NMWAG$,J,$+$C$F$$$k$H$$$&2>Dj$N$b$H$G(B
discrepancy$B$r5a$a$k%"%k%4%j%:%`$r<($9(B.
$B$ $B$N(B $(i,j)$ $B@.J,$,Nm$K$J$k$h$&$K(B,
$B$ $B$N(B $i$ $B9T$K(B $i' (< i)$ $B9T$NDj?tG\$r2C$($kA`:n$r(B,
$B$9$Y$F$N(B $(i,j)$ $B$K$D$$$F9T$&$@$1$N%"%k%4%j%:%`$G$"$k(B.
\begin{algorithm}\label{alga}\normalfont

\noindent
\begin{tabbing}
$B$"(B\= $B$"(B\= $B$"(B\= $B$"(B\=\kill
$A$ $B$rC10L9TNs$K=i4|2=$9$k(B.\\
\textbf{for} $i = 1,\ldots, n$ \textbf{do} \\
\>\textbf{for} $j = 1,\ldots, n$ \textbf{do} \\
\>\>\textbf{if} $B_{ij} \neq 0$ $B$+$D(B\\
\>\>$B$9$Y$F$N(B $j' < j$ $B$K$D$$$F(B $B_{ij'} = 0$ \textbf{then}\\
\>\>\>\textbf{if} $B$"$k(B $i'<i$ $B$,B8:_$7$F(B $B_{i'j} \neq 0$ $B$+$D(B\\
\>\>\>$B$9$Y$F$N(B $j'<j$ $B$K$D$$$F(B $B_{i'j'}=0$ \textbf{then}\\
\>\>\>\>\parbox{0.83\linewidth}{$A$ $B$N(B $i$ $B9T$+$i(B $A$ $B$N(B $i'$ $B9T$r(B $B_{ij}/B_{i'j}$ $BG\$7$F0z$/(B.
$B$3$N$H$-(B $B_{ij} = 0$ $B$H$J$j(B $(i,j)$ $B$O(Bdiscrepancy$B$G$O$J$$(B.}\\
\>\>\>\textbf{else}\\
\>\>\>\>\parbox{0.83\linewidth}{$B:#(B $B_{ij} \neq 0$ $B$+$D$9$Y$F$N(B $i'<i$,
$j'<j$ $B$K$D$$$F(B $B_{i'j} = B_{ij'} = 0$ $B$G$"$k(B.
$B_{ij} = \langle A$ $B$N(B $i$ $B9T(B, $S(\bm{e})$ $B$N(B $j$ $BNs(B $\rangle$
$B$G$"$k(B.
$B$^$?(B, $i$ $BHVL\$,(B $1$ $B$G(B $i+1, \ldots, n$ $BHVL\$,(B $0$ $B$G$"$k$9$Y$F$N2#(B
$B%Y%/%H%k$O(B $A$ $B$N(B $1, \ldots, i$ $B9T$N@~7A7k9g$G$+$1$k$,(B,
$A$ $B$N(B $i$ $B9T$r$I$N$h$&$J(B$A$ $B$N(B $1, \ldots, i$ $B9T$N@~7A7k9g(B ($BC"$7(B $A$ $B$N(B $i$ $B9T$N78?t$O(B $1$)
$B$GCV$-49$($F$b(B $B_{ij}$ $B$r(B $0$ $B$K$9$k$3$H$O$G$-$J$$(B.
$B$J$<$J$i$P$9$Y$F$N(B $j'<j$ $B$K$D$$$F(B $B_{ij'} = 0$ $B$@$+$i$G$"$k(B.
$B$7$?$,$C$F(B $(i,j)$ $B$O(Bdiscrepancy$B$G$"$k(B.}\\
\>\>\>\textbf{endif}\\
\>\>\textbf{else}\\
\>\>\>$B_{ij}=0$ $B$^$?$O(B $(i,j)$ $B$N:8$K4{$K(B discrepancy $B$,(B\\
\>\>\>$B$"$k$N$G(B, $(i,j)$ $B$O(Bdiscrepancy$B$G$O$J$$(B.\\
\>\>\textbf{endif}\\
\>\textbf{endfor}\\
\textbf{endfor}
\end{tabbing}
\end{algorithm}
$B>e5-$N%"%k%4%j%:%`$+$i(B
$B$9$Y$F$N9T(B $i$ $B$K$D$$$F(B $\min\{ j \mid B$ $B$N(B $(i,j)$ $B@.J,$OHsNm(B $\}$
$B$r:GBg$K$9$k9TNs(B $A$ $B$,B8:_$9$k$3$H$H(B,
$B3F9T3FNs$K$D$-(Bdiscrepancy$B$O9b!9(B1$B8D$7$+$J$$$3$H$,$o$+$k(B.
$B%"%k%4%j%:%`(B\ref{alga}$B<B9T8e(B $B$ $B$N(Bdiscrepancy$B$KAjEv$9$k@.J,$N(B
$B:8$K$"$k@.J,$*$h$S>e$K$"$k@.J,$O$9$Y$F(B $0$ $B$G$"$k(B.
$B$7$?$,$C$F(B $B$ $B$N3,?t$O(B discrepancy $B$NAm?t$KEy$7$/(B, 
$S(\bm{e})$ $B$N3,?t$O(B discrepancy $B$NAm?t$KEy$7$$(B. 

$L(s) := \{ (u,v) \mid$ $B$"$k(B $t\leq s$ $B$KBP$7(B
$(i,j) \in M(t)$ $B$,B8:_$7$F(B $1\leq u\leq i$, $1\leq v\leq j$ $B$+$D(B $(u,v)\neq (i,j)$
$B$G$"$k(B $\}$ $B$H$9$k(B. 
$L(s)$ $B$KBP1~$9$k(B $S(\bm{y})$ $B$N$9$Y$F$N@.J,$,$o$+$C$F$$$k$H$9$k(B.
$B$3$N$H$-%"%k%4%j%:%`(B\ref{alga}$B$r(B $(i,j)$ $B@.J,$KBP$7$F<B9T$9$k$H$-(B
$S(\bm{e})$ $B$N$9$Y$F$N(B $(u,v)$ $B@.J,(B ($BC"$7(B $u\leq i$, $v\leq j$)
$B$,4{CN$G$"$l$P%"%k%4%j%:%`$,<B9T$G$-$k$N$G(B,
$L(s)$ $B$K4^$^$l$kMWAG$K8BDj$7$F%"%k%4%j%:%`(B\ref{alga}$B$r<B9T$9$k$3$H$,(B
$B$G$-$k(B.
$B$3$N$H$-0J2<$N$h$&$K(B $\langle \bm{e}, \bm{w}_s \rangle$ $B$NCM$r@5$7$/?dB,$9$k$3$H$,$G$-$k(B. 

$(i,j) \in M(s)$ $B$N:8$^$?$O>e$K(B discrepancy $B$,L5$+$C$?>l9g(B,
$S(\bm{e})$ $B$N(B $(i,j)$ $B@.J,$r(B $(i,j)$ $B$,(B discrepancy $B$K$J$i$J$$$h$&$K(B
$B?dB,$9$k(B. $B6qBNE*$K$O(B
\[
-\sum_{i=1}^{i-1} A_{ik} S(\bm{e})_{kj}
\]
$B$H?dB,$9$k(B. $(i,j)$ $B@.J,$KBP$9$k?dB,CM$+$i(B $\langle \bm{e}, \bm{w}_s
\rangle$ $B$N?dB,CM$,7hDj$5$l$k(B.
$B$3$N?dB,$,??$NCM$KEy$7$$(B $(i,j)$ $B$N?t$r(B $T$,
$B4V0c$C$F$$$k?t$r(B $F$, $L(s)$ $B$NCf$G4{$K8+$D$+$C$F$$$k(B discrepancy
$B$N?t$r(B $K$ $B$H$9$k(B. 
$B$3$N$H$-?dB,$,8m$C$F$$$k$N$O(B $(i,j)$ $B$,(Bdiscrepancy$B$K$J$C$F$$$k$H$-(B
$B$@$1$J$N$G(B $K+F \leq$ discrepancy$B$NAm?t(B $= \rank S(\bm{e}) = w(\bm{e})$
$B$,@.$jN)$D(B.
$B$^$??dB,$N;EJ}$h$j(B $\sharp M(s) \leq T+F+2K$ $B$,@.$jN)$D(B.
$B$b$7(B $2 w(\bm{e}) < \delta_\mathrm{FR}(W)$ ($\leq \sharp M(s)$)
$B$J$i$P$3$l$i$NITEy<0$h$j(B $T>F$ $B$,@.$jN)$D(B.
$B$3$N$H$-(B $\langle \bm{e}, \bm{w}_s\rangle$
$B$N?dB,CM$NCf$GB??t$r@j$a$k$b$N$,@5$7$$CM$K$J$k(B. 
$\langle \bm{e}, \bm{w}_s \rangle$ $B$NCM$+$i(B
$L(s+1) \setminus L(s)$ $B$K$"$?$k(B $S(\bm{e})$ $B$N@.J,$,7W;;$G$-$k(B.
$B$3$N$"$H(B $L(s+1) \setminus L(s)$ $B$N@.J,$K$D$$$F%"%k%4%j%:%`(B\ref{alga}
$B$r<B9T$7(B, $S(\bm{e})$ $B$N(B $M(s+1)$ $B$KBP1~$9$k@.J,$r?dB,$9$k(B.
$M(s)$ $B$KBP1~$9$k(B $S(\bm{e})$ $B$N@.J,$r?dB,$7$?8e(B,
$L(s)$ $B$N@.J,$K$D$$$F%"%k%4%j%:%`(B\ref{alga}$B$r<B9T$7$F$b7k2L$OJQ$o$i$J$$$N$G(B,
$B<B9T$9$kI,MW$O$J$$(B.

$B$h$j9bB.$JI|9f%"%k%4%j%:%`$K$D$$$F$O(B \cite{sakatakaisetsuA,sakatakaisetsuB}
$B$r;2>H$;$h(B.

\begin{remark}\normalfont 
$BK\9F$G$O(B $\bm{u}_i =\bm{v}_i = \bm{w}_i$
$B$H$J$kNc$7$+5s$2$J$$$,(B, 
$\mathcal{L}$ $B7?Be?t4v2?Id9f$NI|9f$G$O(B
$\bm{u}_i \neq \bm{v}_i = \bm{w}_i$
$B$H$9$k$H9b$$8m$jD{@5G=NO$r<B8=$G$-$k(B \cite{matsumoto99ldecode}.
\end{remark}
%
%\begin{remark}\normalfont 
%$BB?$/$N8&5f<T$,(B Feng-Rao $B8B3&$*$h$SI|9f%"%k%4%j%:%`$r(B
%$BDj<02=$7$J$*$7$F$$$k(B. $BB>$NDj<02=$O(B
%\cite{hoholdt97,hoholdt98,kirfel95,miurathesis,miura98a,pellikaan96,shen95decomp,shen95matrix}
%$B$K$"$k(B. 
%\end{remark}

\section{$B@~7ABe?t$@$1$r;H$C$?Be?t4v2?Id9f$NDj5A(B}
H\o holdt, van Lint, Pellikaan $B$O(B \cite{hoholdt97,hoholdt98}
$B$K$*$$$FBe?t4v2?Id9f$r@~7ABe?t$H=E$_4X?t$H$$$&35G0$r;H$C$F(B
$BDj<02=$7D>$7$?(B. $BK\@a$G$OH`$i$NDj<02=$r>R2p$9$k(B.

\subsection{$B=E$_4X?t$HId9f(B}\label{weightconstruction}
\begin{definition}\normalfont \cite[Definition 3.5]{hoholdt98}
$R$ $B$r(B $\fq$ $B$r4^$`2D494D$H$7(B, $\n$ $B$rHsIi@0?t$N=89g$H$9$k(B. 
$B$3$N$H$-<LA|(B $\rho : R \rightarrow \n\cup\{-\infty\}$ $B$,$9$Y$F$N(B $f,g\in R$
$B$K$D$$$F0J2<$N>r7o$rK~$?$9(B
$B$H$-(B\keyword{$B=E$_4X?t(B}{$B$*$b$_$+$s$9$&(B}$B$H$$$&(B. 
\begin{enumerate}
\setlength{\itemsep}{0mm}
\setlength{\parsep}{0mm}
\item $\rho(f) = -\infty \Leftrightarrow f = 0$.
\item $B$9$Y$F$N(B $0 \neq c \in \fq$ $B$K$D$$$F(B $\rho(cf) = \rho(f)$.
\item $\rho(f+g) \leq \max\{\rho(f),\rho(g)\}$.
$BEy9f$O(B $\rho(f) \neq \rho(g)$ $B$N$H$-@.N)(B. 
\item $\rho(f) = \rho(g) \neq -\infty$ $B$J$i(B, 
$\rho(f - cg)<\rho(g)$ $B$H$J$k(B $c\in \fq$ $B$,B8:_$9$k(B.
\item $\rho(fg) = \rho(f) + \rho(g)$.
\end{enumerate}
\end{definition}

$d$ $B$,(B $\rho(R \setminus \{0\})$ $B$N:GBg8xLs?t$J$i(B $\rho / d$ $B$b(B
$B=E$_4X?t$G$"$k(B. $B4JC1$N$?$a0J9_$G$O=E$_4X?t(B $\rho$ $B$N:GBg8xLs?t$O(B $1$
$B$H$9$k(B. $B$^$?(B $R \neq \fq$ $B$r>o$K2>Dj$9$k(B. 

\begin{proposition}\normalfont \cite[Proposition 3.12]{hoholdt98}
$B$*8_$$$K=E$_4X?t(B $\rho$ $B$NCM$,0[$J$k(B $R$ $B$N(B $\fq$ $B4pDl$,B8:_$9$k(B.
\end{proposition}
\startproof
$B3F!9$N(B $i \in \rho(R\setminus\{0\})$ $B$K$D$$$F(B
$\rho(h_i) = i$ $B$H$J$k(B $h_i$ $B$rG$0U$KA*$V(B. 
$B$3$N$H$-=E$_4X?t$NDj5A$h$j(B $\{ h_i\mid i\in
\rho(R\setminus\{0\})\}$ $B$O(B $\fq$ $B>e@~7AFHN)$G$"$k(B.
$B$J$<$J$i@~7A7k9g(B $d_1 h_1 + \cdots + d_t h_t$ $B$KHsNm(B
$B$J78?t$,$"$C$?$H$9$k$H(B,
$\rho(d_1 h_1 + \cdots + d_t h_t) = \max\{ \rho(h_i) \mid
d_i \neq 0 \} \neq -\infty$ $B$h$j(B
$d_1 h_1 + \cdots + d_t h_t \neq 0$ $B$@$+$i$G$"$k(B.

$g \in R \setminus \{0\}$ $B$H$9$k$H(B,
$B$"$k(B $c \in \fq$ $B$,B8:_$7(B, $\rho(g - c h_{\rho(g)}) < \rho(g)$ $B$H(B
$B$G$-$k(B. $B$3$N5DO@$r9b!9(B $(\rho(g)+1)$ $B2s7+$jJV$9$3$H$K$h$j(B
\[
g - \sum_{i \in \rho(R\setminus\{0\})} c_i h_i
\]
$B$N=E$_4X?t(B $\rho$ $B$NCM$r(B $-\infty$ $B$K$9$k(B $\{ c_i \mid i \in \rho(R\setminus
\{0\}) \}$ $B$,B8:_$9$k$3$H$,J,$+$k(B. 
$B$D$^$j(B $g$ $B$O(B $\{ h_i \}$ $B$N@~7A7k9g$G=q$1$k(B.
\qed

$\rho(f_i) < \rho(f_{i+1})$ $B$rK~$?$9(B
$R$ $B$N4pDl(B $\{ f_i \mid i=1,2,\ldots \}$ $B$r$3$l0J9_8GDj$9$k(B. 
$B$^$?(B $\fq^n$ $B$K:BI8$4$H$N@Q$rF3F~$7$F4D$H8+$J$7(B, 
$B4D$NA4<M=`F17?(B $\varphi : R \rightarrow \fq^n$ $B$,M?$($i$l$F$$$k$H$9$k(B.
$\{ g_1, \ldots, g_n\} := \{ f_i \mid \varphi(f_i)$ $B$O(B $\varphi(f_1),
\ldots,\varphi(f_{i-1})$ $B$GD%$i$l$k@~7A6u4V$KB0$5$J$$(B $\}$ $B$H$7(B
$\rho(g_1) < \cdots < \rho(g_n)$ $B$HE:$(;zIU$1$5$l$F$$$k$H$9$k(B. 
$BBh(B1$B@a$K=R$Y$?(B $\fq^n$ $B$N4pDl(B $\{\bm{u}_1, \ldots, \bm{u}_n\}$ $B$J$I$r(B
$\bm{u}_i = \bm{v}_i = \bm{w}_i = \varphi(g_i)$
$B$HDj5A$9$k(B.
$B@-G=$NNI$$@~7AId9f$r:n$k$?$a$K$O(B $\sharp M(s)$ $B$OBg$-$$$[$INI$$$,(B, 
$B0J>e$N$h$&$K(B $\bm{u}_i$ $B$J$I$rDj$a$?$H$-(B $\sharp M(s)$ $B$O(B
$B$"$^$j>.$5$/$J$i$J$$$3$H$r0J2<$K<($9(B. 

\begin{lemma}\normalfont 
$\rho(g_i) + \rho(g_j) = \rho(g_s)$ $B$J$i$P(B
$(i,j) \in M(s)$ $B$G$"$k(B. 
\end{lemma}
\startproof
$\rho(h) \leq \rho(g_s)$ $B$G$"$k(B $h\in R$ $B$K$D$$$F(B
\begin{eqnarray*}
\varphi(h) &\in& \mathcal{W}_s \setminus \mathcal{W}_{s-1}
\mbox{ if } \rho(h) = \rho(g_s),\\
\varphi(h) &\in& \mathcal{W}_{s-1} \mbox{ if }
\rho(h) < \rho(g_s)
\end{eqnarray*}
$B$G$"$k(B. 
$B$9$Y$F$N(B $1 \leq u \leq i$,
$1 \leq v \leq j$,
$(u,v) \neq (i,j)$ $B$K$D$$$F(B
$\rho(g_u g_v) < \rho(g_s)$ $B$G$"$k$3$H$+$i<gD%$OF3$+$l$k(B.
\qed

$g := \sharp (\n \setminus \rho(R \setminus \{0\}))$ $B$H$9$k(B.
\begin{proposition}\label{msg}\normalfont \cite[Theorem 5.24]{hoholdt98}
$\sharp M(s) \geq s-g$.
\end{proposition}
\startproof
$B$^$:(B $\rho(g_i f_j) = \rho(g_s)$ $B$rK~$?$9(B $(g_i, f_j)$ $B$N?t$r?t$($k(B.
$B$"$k(B $g_i$ $B$KBP$7$F(B $\rho(g_i f_j) = \rho(g_s)$ $B$H$J$k(B
$f_j$ $B$,B8:_$7$J$$$N$O(B $\rho(g_s) - \rho(g_i) \in
\n \setminus \rho(R \setminus \{0\})$ $B$H$J$k(B
$B$H$-$K8B$k(B. $B=>$C$F(B
$\rho(g_i f_j) = \rho(g_s)$ $B$rK~$?$9(B $(g_i, f_j)$ $B$N?t$O(B
$s-g$ $B0J>e$G$"$k(B. 

$B$D$.$K(B $\rho(g_i f_j) = \rho(g_s)$ $B$J$i(B $g_{j'} = f_j$ $B$H$J$kE:$(;z(B
$j'$ $B$,B8:_$9$k$3$H$r<($9(B. 
$B$3$l$,<($5$l$l$PA0JdBj$h$j<gD%$N>ZL@$,=*N;$9$k(B. 
$B$b$7$=$N$h$&$JE:$(;z(B $j'$ $B$,B8:_$7$J$$$H$9$k$H(B
\[
\varphi(f_j) = \sum_{\ell=1}^{j-1} c_\ell \varphi(f_\ell)
\]
$B$H=q$1$k(B. $B=>$C$F(B $\varphi(g_s)$ $B$,(B
\begin{eqnarray*}
\varphi(g_s) &=& d \varphi(g_i f_j) + \sum_{k=1}^{s-1} e_k \varphi(g_k)\\
&=& \sum_{\ell=1}^{j-1} d c_\ell \varphi(g_i f_\ell) + \sum_{k=1}^{s-1} e_k \varphi(g_k)
\end{eqnarray*}
$B$N$h$&$K(B $g_s$ $B$h$j??$K=E$_4X?t$NCM$,>.$5$$MWAG$K(B
$B<LA|(B $\varphi$ $B$r:nMQ$5$;$FF@$i$l$k%Y%/%H%k$N@~7AOB$G=q$1$k$3$H$K$J$j(B, 
$g_s$ $B$NDj5A$KL7=b$9$k(B.
\qed

$BL?Bj(B\ref{msg}$B$h$j(B, $W = \{ \varphi(g_1), \ldots, \varphi(g_{n-k}) \}$
$B$H<h$k$H(B, $C(W)$ $B$O(B $[n,k,d]$ $B@~7AId9f(B($d \geq n-k+1-g$)$B$K$J$k$3$H$,J,$+$k(B.
$B>pJs%l!<%H(B $k/n$ $B$KBP$7AjBP:G>.5wN%(B $d/n$ $B$rBg$-$/$9$k$?$a$K$O(B
$g$ $B$KHf3S$7$F(B $n$ $B$rBg$-$/<h$kI,MW$,$"$k(B.
\cite{matsumoto99weight}$B$N7k2L$H(B Hasse-Weil-Serre $B$NDjM}(B \cite[Theorem V.3.1]{bn:stichtenoth}$B$h$j(B $n$ $B$N:GBgCM$O(B $q+1+g \lfloor 2 \sqrt{q} \rfloor$ $B$G$"$k$3$H$,(B
$B$o$+$k(B.
$BId9fD9(B $n$ $B$r(B $g$ $B$KBP$7$FBg$-$/<h$l$k4D(B $R$ $B$N9=@.K!$O<!@a$G=R$Y$k(B.

\subsection{$B2D494D(B $R$ $B$H=E$_4X?t(B $\rho$ $B$N6qBNNc(B}
$B<B:]$KId9f$r9=@.$9$k$?$a$K$O(B $g_1, \ldots, g_n$ $B$*$h$S(B $\varphi$ $B$,(B
$B7W;;2DG=$G$"$kI,MW$,$"$k(B. $B0J2<Id9f9=@.$,2DG=$J2D494D(B $R$ $B$H(B
$B=E$_4X?t(B $\rho$ $B$NNc$r5s$2$k(B. 
$2 \leq a < b$ $B$r8_$$$KAG$J@0?t$H$7(B, $F(X,Y)$ $B$r(B
\[
Y^a + X^b + \sum_{ai+bj<ab} \alpha_{i,j} X^i Y^j
\]
$B$H$$$&7A$NFsJQ?tB?9`<0$H$*$/(B.
$I$ $B$r(B $F(X,Y)$ $B$G@8@.$5$l$kFsJQ?tB?9`<04D(B $\fq[X,Y]$
$B$N%$%G%"%k$H$7(B, $R := \fq[X,Y]/I$ $B$H$9$k(B.
$B$3$N$H$-(B
\[
B := \{ X^i Y^j \mid 0\leq i,\; 0\leq j \leq a-1 \},
\]
$B$H$9$k$H(B $\{ X^i Y^j \bmod I \mid X^i Y^j \in B \}$ $B$O>jM>N`4D(B $R$ $B$N4pDl$K$J$k(B. 
$BC19`<0(B $X^i Y^j$ $B$N=E$_IU$-<!?t$r(B $ai+bj$ $B$HDj$a(B, 
$BHsNm$JB?9`<0$N=E$_IU$-<!?t$r:GBg$N=E$_IU$-<!?t$r;}$D9`$N(B
$B=E$_IU$-<!?t$H$9$k(B. 
$G(X,Y)$ $B$,(B $B$ $B$NMWAG$NHsNm$J@~7A7k9g$GI=$5$l$kB?9`<0$G$"$k$H$-(B, 
$G(X,Y) \bmod I \in R$ $B$KBP$7=E$_4X?t(B $\rho$ $B$NCM$r(B
$G(X,Y)$ $B$N=E$_IU$-<!?t$HDj$a(B, $\rho(0) = -\infty$
$B$HDj$a$k(B. 
$B$ $B$K4^$^$l$kC19`<0$O8_$$$K0[$J$k=E$_IU$-<!?t$r;}$D$?$a(B, 
$B>e$KDj$a$?(B $\rho$ $B$,=E$_4X?t$NDj5A$rK~$?$9$3$H$r3NG'$9$k$K$O(B, 
$f,g\in R$ $B$KBP$7(B $\rho(fg) = \rho(f) + \rho(g)$ $B$r3NG'$9$l$P$h$$(B. 

$G_1(X,Y), G_2(X,Y)$ $B$,HsNm$J(B $B$ $B$NC19`<0$N@~7A7k9g$GI=$5$l$k(B
$BB?9`<0$H$7(B, $B3F!9$N=E$_IU$-<!?t$,:G9b$N9`$r(B $X^{i_1} Y^{j_1},
X^{i_2} Y^{j_2}$ $B$H$9$k$H(B,
$G_1(X,Y) G_2(X,Y)$ $B$N=E$_IU$-<!?t:G9b$N9`$O(B $X^{i_1+i_2} Y^{j_1+j_2}$
$B$G$"$k(B. 
$G_1(X,Y) G_2(X,Y)$
$B$r(B $Y$ $B$rJQ?t$H$9$k0lJQ?tB?9`<0$H8+$F(B $F(X,Y)$ $B$G3d$j;;$r$9$k$3$H$K$h$j(B
$BF@$i$l$kM>$j$r(B $H(X,Y)$ $B$H$9$k$H(B
$G_1(X,Y) G_2(X,Y) \bmod I =  H(X,Y) \bmod I$
$B$G$"$k$,(B,  $F(X,Y)$ $B$N7A$+$i(B $H(X,Y)$ $B$N=E$_IU$-<!?t$O(B
$a(i_1+i_2) + b (j_1 +j_2)$ $B$G(B $G_1(X,Y)$ $B$H(B $G_2(X,Y)$ $B$N=E$_IU$-<!?t$NOB$K(B
$BEy$7$$(B. $B=>$C$F>e$KDj$a$?(B $\rho$ $B$,=E$_4X?t$NDj5A$rK~$?$9$3$H$,(B
$B3NG'$G$-$?(B. $B=>$C$F(B, $\{f_1, f_2, \ldots \}$ $B$O(B $B$ $B$K4^$^$l$kC19`<0$G(B
$BBeI=$5$l$k(B $R$ $B$N>jM>N`$K<h$k$3$H$,$G$-$k(B. 

$\sharp (\n \setminus \rho(R\setminus \{0\}))$ $B$,(B $\sharp M(s)$ $B$N2<3&$r(B
$B7h$a$?$,$3$N>l9g(B
\[
\n \setminus \rho(R\setminus \{0\})
= \n \setminus \{ ia+bj \mid 0\leq i, \; 0\leq j \leq a-1 \}
\]
$B$G$"$k$N$G(B, $g = \sharp (\n \setminus \rho(R\setminus \{0\}))
= (a-1)(b-1)/2$ $B$G$"$k(B.

$B:G8e$KId9f9=@.$r9T$&$?$a$KA4<M=`F17?(B $\varphi$ $B$r6qBNE*$K7W;;$9$k(B
$BI,MW$,$"$k(B. 
$x := X \bmod I$, $y := Y \bmod I$ $B$H$7(B, 
$(\alpha_1, \beta_1), \ldots, (\alpha_n, \beta_n)$ $B$r(B
$BJ}Dx<0(B $F(X,Y) = 0$ $B$N(B $\fq^2$ $B$K$*$1$kAj0[$J$k2r$H$9$k(B. 
$M_i$ $B$r(B $R$ $B$N(B $x-\alpha_i, y-\beta_i$ $B$G@8@.$5$l$k6KBg%$%G%"%k$H(B
$B$9$k$H(B,
$G(X,Y) \bmod I \in R$ $B$K(B $G(\alpha_i, \beta_i)$ $B$rBP1~$5$;$k(B
$B<LA|$O(B $G(X,Y) \bmod I \in R$ $B$K(B
$G(X,Y) \bmod I$ $B$GBeI=$5$l$k(B
$B>jM>N`4D(B $R/M_i$ $B$NMWAG$rBP1~$5$;$F$$$k$H$_$k$3$H$,$G$-$k(B.
$B<LA|(B
\begin{eqnarray*}
R &\longrightarrow&\fq^n,\\
G(X,Y) \bmod I&\longmapsto& (G(\alpha_1,\beta_1), \ldots, G(\alpha_n,\beta_n))
\end{eqnarray*}
$B$O<LA|(B
\begin{eqnarray*}
R &\longrightarrow&\fq^n,\\
g&\longmapsto& (g \bmod M_1, \ldots, g \bmod M_n)
\end{eqnarray*}
$B$KEy$7$/(B, $M_1, \ldots, M_n$ $B$OAj0[$J$k6KBg%$%G%"%k$J$N$G(B, 
$BCf9q$N>jM>DjM}(B\footnote{$B=iEyBe?t$N652J=q$r8+$h(B.}$B$h$j$3$N<LA|$OA4<M$K$J$k(B. 
$B=>$C$F(B $\varphi$ $B$H$7$F(B $F(X,Y)=0$ $B$N0[$J$k2r$rBeF~$9$k<LA|$rA*$s$GNI$$(B.
$\{\varphi(g_1), \ldots, \varphi(g_n)\}$ $B$O(B $\varphi(f_1),
\varphi(f_2), \ldots$ $B$r7W;;$7$?$"$H@~7A=>B0$J%Y%/%H%k$r(B
$B=|$/$3$H$GF@$i$l$k(B. 

\begin{remark}\normalfont 
$(\alpha, \beta)$ $B$,(B $F(X,Y) = 0$ $B$N2r$G$J$1$l$P(B, 
$G(X,Y) \bmod I \in R$ $B$K(B $(\alpha, \beta)$ $B$r!VBeF~!W$9$kA`:n$O(B
$B$=$b$=$b<LA|$K$J$i$J$$(B. 
$G(X,Y) \bmod I$ $B$H(B $G(X,Y) + F(X,Y) \bmod I$ $B$O(B $R$ $B$KB0$9$k(B
$BF1$8MWAG$@$,(B $G(\alpha,\beta) \neq G(\alpha, \beta)+F(\alpha,\beta)$
$B$@$+$i$G$"$k(B. 
\end{remark}

\begin{remark}\normalfont 
$BK\>.@a$G=R$Y$?7A$NB?9`<0(B $F(X,Y)$ $B$O(B1992$BG/$K;01:$K$h$j(B
$BBe?t4v2?Id9f$N9=@.$KI,MW$J>pJs$,$9$Y$F$o$+$kBe?t6J@~B2(B
\keyword{$C_a^b$$B6J@~(B}{Cab$B$-$g$/$;$s(B}$B$H$7$FDs0F$5$l$?(B \cite{miura92}.
\end{remark}

\begin{example}\normalfont 
$B>pJs%l!<%H(B $k/n$ $B$KBP$7AjBP:G>.5wN%(B $d/n$ $B$,Bg$-$$Id9f$r:n$k$K$O(B
$g = \sharp (\n \setminus \rho(R\setminus \{0\}))
= (a-1)(b-1)/2$ $B$KBP$72r$N?t$,B?$$B?9`<0(B $F(X,Y)$ $B$,I,MW$@$,(B, $BNc$($P(B
\[
F(X,Y) = Y^q + Y - X^{q+1} = 0
\]
$B$O(B $\mathbf{F}_{q^2}$ $BFb$K(B $q^3$ $B8D$N2r$r;}$A(B,
$g=q(q-1)/2$ $B$H$J$kB?9`<0$NCf$G(B $\mathbf{F}_{q^2}^2$ $BFb$N(B
$B2r$N?t$,:GBg$NB?9`<0$G$"$k(B.
$g$ $B$KBP$72r$N?t$,B?$$B?9`<0$O%3%s%T%e!<%?$K$h$kA4?tC5:w0J30$K(B
$B8+$D$1$kJ}K!$OL5$$$H;W$o$l$k(B.
$BB?$/$N2r$r;}$DB?9`<0$NI=$,(B \cite{miura92} $B$K$"$k(B.
\end{example}

\begin{remark}\normalfont 
$B=E$_4X?t$r;}$D4D$N9=@.K!$OB>$K(B
\cite[Proposition 3.17, Example 3.22]{hoholdt98},
\cite[$BDjM}(B 5.17]{miurathesis}, \cite[p.1401 $B<gDjM}(B]{miura98b}
$B$K<($5$l$F$$$k(B. 
$BFC$K(B \cite[$BDjM}(B 5.17]{miurathesis}, \cite[p.1401 $B<gDjM}(B]{miura98b}
$B$O=E$_4X?t$r;}$D$9$Y$F$N4D$rM?$($k9=@.K!$K$J$C$F$$$k(B. 
\end{remark}

\subsection{$B=>Mh$NBe?t4v2?Id9f$H$N4X78(B}\label{relation}
$BK\@a$G=R$Y$?(B H\o holdt $B$i$N@~7AId9f$N9=@.K!$H=>Mh$NBe?t4v2?Id9f$N(B
$B4X78$r=R$Y$k(B. $BFI<T$,Be?t4v2?$rMQ$$$FBe?t4v2?Id9f$rM}2r$7$F$$$k$3$H$rA0Ds$H$7$F(B
$B5-=R$9$k(B.
$BFC$KCG$i$J$$8B$jI=5-K!$O(BStichtenoth$B$N652J=q(B \cite{bn:stichtenoth}
$B$K=>$&(B.
$F/\fq$ $B$r0lJQ?tBe?t4X?tBN(B,
$P_1, \ldots, P_n, Q$ $B$rAj0[$J$k0l<!$N:B(B,
$D := P_1 + \cdots + P_n$ $B$H$9$k(B.
$B$3$N$H$-Be?t4v2?Id9f(B $C_\Omega(D, mQ)$ $B$OId9f(B
\[
\{ (f(P_1), \ldots, f(P_n)) \mid f \in \mathcal{L}(mQ) \}
\]
$B$NAPBPId9f$H$7$FM?$($i$l$k(B. $B$3$l$O(B\ref{weightconstruction}$B@a$N9=@.K!$r;H$$0J2<$N$h$&$K(B
$B9=@.$G$-$k(B. $\mathcal{L}(\infty Q) := \bigcup_{i=1}^\infty \mathcal{L}(iQ)$
$B$H$*$/$H(B $\mathcal{L}(\infty Q)$ $B$O4X?tBN(B $F$ $B$NItJ,4D$K$J$k(B.
$B$^$?(B $\rho$ $B$r(B $Q$ $B$K$*$1$kN%;6IUCM$r(B $-1$ $BG\$7$?$b$N$H$9$k$H(B
$\rho$ $B$O(B $\mathcal{L}(\infty Q)$ $B$N=E$_4X?t$K$J$k(B.
$f \in \mathcal{L}(\infty Q)$ $B$K(B $(f(P_1), \ldots, f(P_n))$ $B$rBP1~$5$;$k(B
$B<LA|(B $\varphi$ $B$O(B $\mathcal{L}(\infty Q)$ $B$+$i(B $\fq^n$ $B$X$NA4<M=`F17?(B
$B$G$"$k(B. $g_1, \ldots, g_n$ $B$r(B\ref{weightconstruction}$B@a$N$h$&$KDj$a(B,
$W := \{ \varphi(g_i) \mid \rho(g_i) \leq m\}$ $B$H$9$l$P(B $C(W)$ $B$O(B $C_\Omega(D, mQ)$
$B$KEy$7$$(B.
$B$3$N$H$-L?Bj(B\ref{msg}$B$K$*$1$k(B $g$ $B$O4X?tBN(B $F$ $B$N<o?t$KEy$7$/$J$j(B,
$d$ $B$r(B $C_\Omega(D, mQ)$ $B$N:G>.5wN%$H$9$k$HL?Bj(B\ref{msg}$B$+$iD>$A$K(B
Goppa$B8B3&(B
$d \geq \sharp W + 1 - g$ $B$rF3$/$3$H$,$G$-$k(B.

$B5U$K(B\ref{weightconstruction}$B@a$G9=@.$7$?Id9f$H=>Mh$NBe?t4v2?Id9f$N4X78$O(B
$B0J2<$N$h$&$K$J$k(B.
$B=E$_4X?t(B $\rho$ $B$r;}$D4D(B $R$ $B$O(B
$BL58B1s$KM#0l$D$N(B $\fq$ $BM-M}E*$J:B(B $Q$ $B$r;}$D%"%U%#%sBe?t6J@~$N(B
$B%"%U%#%s:BI84D$K$J$j(B, 
$B=E$_4X?t(B $\rho$ $B$O(B $Q$ $B$K$*$1$kN%;6IUCM$NIi$N@0?tG\$G(B
$BM?$($i$l$k$3$H$,<($5$l$k(B \cite{matsumoto99weight}.
$R$ $B$N$b$H$K$J$k%"%U%#%sBe?t6J@~$,HsFC0[$G$"$k$H$-(B\ref{weightconstruction}$B@a(B
$B$G<($7$?(B
$BId9f$OBe?t4v2?Id9f(B $C_\Omega(D,mQ)$
$B$^$?$O$=$N2~NIBe?t4v2?Id9f$K$J$k(B. 
$BC"$7(B, $B:B(B
$P_1, \ldots, P_n$ $B$OA4<M(B $\varphi$ $B$+$i7h$^$k(B.

$B%"%U%#%sBe?t6J@~$,FC0[$J$H$-9=@.$5$l$kId9f$O(B
$B;01:$K$h$jDs0F$5$l$?FC0[6J@~>e$N@~7AId9f(B \cite{miurathesis,miura98b}
$B$KEy$7$/$J$k(B. 
$BC"$7(B, $B$3$N>l9g6J@~$r@55,2=$7$FF@$i$l$kHsFC0[6J@~>e$NId9f$KHf$Y$F(B
$B@-G=$NNI$$Id9f$OF@$i$l$J$$(B \cite{matsumoto99nonsing}.

\section{$B$^$H$a(B}
$BBe?t4v2?Id9f$N$?$a$KDs0F$5$l$?(BFeng-Rao$B8B3&$H(BFeng-Rao$BI|9f%"%k%4%j%:%`(B
$B$r@~7ABe?t$@$1$rMQ$$$FDj5A$7(B, $BD>$7G$0U$N@~7AId9f$KE,MQ$G$-$k$h$&$K$7$?(B
$B;01:$N7k2L$r>R2p$7$?(B.
$B$^$?(B, $BBe?t4v2?Id9f$N9=@.K!$r@~7ABe?t$@$1$GDj5A$7$?(B H\o holdt$B$i$N(B
$B7k2L$r>R2p$7$?(B.

\section*{$B<U<-(B}
$B869F$N4V0c$$$*$h$SJ,$+$j$K$/$$ItJ,$r;XE&$7$FD:$$$?(B
$BEl9)Bg$N:dDm9%0l65<x(B, $B%=%K!<$N;01:?8<(Gn;N(B, $B:eBg$NCS>eBg2p;a(B,
$BEl9)Bg$NIp0fM3CR;a$K46<U$9$k(B.

\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\begin{thebibliography}{10}
\small
\setlength{\topsep}{0mm}
\setlength{\partopsep}{0mm}
\setlength{\itemsep}{0mm}
\setlength{\parsep}{0mm}
\setlength{\parskip}{0mm}
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%\bibitem{kirfel95}
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%
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%\bibitem{pellikaan96}
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\end{thebibliography}
%\printindex
\end{document}
