$C$
1
$C$
$P_{1}$
$C’$
,
$\cdot\cdot \mathrm{c}$
,
$P_{k}$
$C$
.
$2g-2$
$W$
$C$
(canonical
divisor) linear series)
$C$
$C$
$g-1$
$C$
(complete
(=E
$D$
$|$
W|
$D$
$D$
)
1
$i$
(special) (D)
$b_{1}$
2.2
$\mathrm{V}.\mathrm{D}$
${\rm Max}$
Noether
1983
”
.Goppa codes, Math. USSR. Izvestia 21
,
: “AlgebraicO-geometric
A.Brill -M.Noether
(linear series)
$\mathrm{F}_{q^{m}}$
$n$
$g(z)\in \mathrm{F}_{g^{m}}[z]$
$g(\gamma i)\neq 0,0\leq i\leq n-1$
$\Gamma(L,g):=\{(c_{0}, c_{1}, \cdot\cdot \mathrm{c}, c_{n-1})\in \mathrm{F}_{q}^{n}$
Goppa
1980
2
Riemann-Roch
144
2.1
Abel
G\"opel
‘
Riemann
Jacobi
$|^{\backslash }\S^{\mathfrak{n}}\mathrm{x}^{4}$
Rosenhein
Riemann
$g\geq 2$
Riemann-Roch
Riemann Riemann
$z=\gamma i$
1
3
$c$
$0\leq i\leq n-1$
$d_{Z}=- \frac{1}{u}\tau^{du}$
.
$z=\infty$
:
$\infty$
$u.= \frac{1}{z}$
$\omega=-\sum_{i=0}^{n-1}\frac{c_{i}}{1-\gamma_{i}u}$
${\rm Im}(\varphi_{\Omega})$
$\Gamma_{\Omega}(D, G)$
$(D, G)$
$2g-2<\deg G<n$
$\Gamma_{\Omega}(D, G)$
$k$
,
$d$
$k=$
n-deg $G+g-1$
$d\geq\deg G-2g+2$
Riemann-Roch
$P_{k}$
$r_{1},$
$\cdot\cdot\not\subset,$
$rk$
$C$
$g= \frac{1}{2}(m-1)(m-2)-.\sum_{1=1}^{k}\frac{1}{2}r_{\dot{l}}(r_{i}-1)$
145
$g$
$C$
(4v2?
)
$r_{i}(1\leq i\leq k)$
$C$
$(\omega)\geq G-D$
$\mathrm{F}_{q}$
$\Omega(G-D)$
$\Psi\Omega$
:
$\Omega$
(G-D)
$arrow$
F;
$\varphi_{\Omega}(\omega)=({\rm Res}_{P_{1}}(\omega), \cdots, {\rm Res}_{P_{\hslash}}(\omega))$
:
$\sum_{i=0}^{n-1}\frac{c_{i}}{z-\gamma_{i}}\equiv 0\mathrm{m}\mathrm{o}\mathrm{d} g(z)\}$
$\Gamma(L, g)$
$g$
(z)
Goppa
Goppa
$\omega=\sum_{i=0}^{n-1}\frac{c_{i}}{z-\gamma_{i}}dz$
‘
$\frac{1}{u}$
du
$c_{n}=$
$\omega$
1
$c_{n}$
$- \sum_{i=0}^{n-1}c$
:
$\mathrm{c}=(c_{0},c_{1}, \cdot\cdot. , c_{n-1}., c_{n})$
$\varphi_{\Omega}$
:
$\omega\mapsto({\rm Res}_{\gamma_{\mathrm{O}}}\omega,{\rm Res}_{\gamma_{1}}\omega, \cdot\circ\cdot ,{\rm Res}_{\gamma_{n}-1}\omega,{\rm Res}_{\infty}\omega)$
$\mathrm{T}\mathrm{s}\mathrm{f}\mathrm{a}\mathrm{s}\mathrm{m}\mathrm{a}\mathrm{n},\mathrm{M}.\mathrm{A}.,$
$G= \sum m$ QQ
$G$
$\mathrm{F}_{q}$
$D$
$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}D\cap \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}G=\phi$
.
$\omega$
$(\omega)$
$\mathrm{F}_{q}$
$D$
(speciality index)
$C$
$L(D)=\{f :
(f)+D\geq 0\}\cup\{0\}$
$l$
(D)
Riemann-Roch
$l(D)=\deg D-g+1+i(D)$
$i(D)=l(W-D)$ .
2.3
Riemam Dedekind
Kronecker
Kronecker
:
$C$
Riemann
$P_{i}$
$r_{i}-1$
$C$
(adjoint curve) (special adjoint curve)
$g$
$m-3$
$\text{ }$
$g\gg 1$
.
$C$
$C’$
: (X, )
$\emptyset$
$\mathrm{Y}$
$=0$
1
$\omega=\frac{\phi(X,\mathrm{Y})}{f_{\mathrm{Y}}(X,\mathrm{Y})}dX$
function fields and codes, Springer, I993
Goppa
$\mathrm{G}\mathrm{o}\mathrm{p}\mathrm{p}\mathrm{a},\mathrm{V}.\mathrm{D}.$
:AlgebraicO-geometric codes, Math. USSR Izvestia, 1983
, A Mathematical Theory of Communication, Bell SysHamming 1950 Reed-Solomon
tem Tech. J. (1948)
1960
BCH
1970
Goppa Goppa, V.D., A new class of linear error-correcting codes
$g= \frac{1}{2}b$
1
Weierstrass 2 Jacobi “Theorie der abelischen Ehnctionen” 1857 Jacobi Riemann Part Roch Riemann Riemann 1 Riemann abel Betti 1 Riemann
$\Gamma(D, G)$
$\mathrm{D}\mathrm{o}\mathrm{k}1.,1981$
148
$C$ $C$
$\mathrm{F}_{q}$
$g$
$P_{1},$
$\cdot\cdot 1,$
$P_{n}$
$\mathrm{F}_{q}$
$D= \sum_{i}P_{\dot{8}},$
Stichtenoth
:
$\mathrm{S}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{h}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{h},\mathrm{H}.:\mathrm{A}\mathrm{l}\mathrm{g}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{c}$
. :Theorie der algebraischen Funktionen einer
$\mathrm{S}\mathrm{c}\mathrm{h}\mathrm{m}\mathrm{i}\mathrm{d},\mathrm{F}.\mathrm{K}$
Veriuderlichen, Crelle 92,1882 . , Landsberg, G., to the theory of algebraic functions of one variable, AMS. Math. Surveys, 1951
:” F.Severi :Vorlesungen \"uber alge-