Códigos lineares disjuntos e corpos de funções algébricas

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-* #

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-* #

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L D M G ! ! + K M ( + * #

L M ) D ) ! D ( + * + #

6 ) 7 : ! + K ! ) * #

• Fq K q q - ? N O

• ⌈x_⌉ + @ - ⌈x_⌉:=min_{n_∈Z; x_≤n_}O

• [ ] + ? O

• v , . . . , vn ! {v , . . . , vn}O

• A_≤B : - ! ( O

• ⌊x_⌋ + @ - ⌊x_⌋:=max_{n_∈Z; n_≤x_}O

• |C_| * O

• (m , . . . , mn) m , . . . , mnO

• deg (p) ; <@=O

• [F : K] F K− ( F - @

KO

• F/K F K K F - + , !

PO

• degP ( P.

*

!

8

1 % & ' ( 1

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/ & % & //

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4 + 3 !< =1

8

>

9 E: F $D ! 6S>&

H K

# - H

K ( ( + (

A #

: ! H K + ,

D - + ! ( " ( (

# : T ! ! K - +

D + #

@ H

K !- A ! , # @

-+ , @ ? B *

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+ , -!

! K ! * N

* #

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5

- " " #

" 1 ! ( H ,

: -! ! # :

-K + , -! ( 1 (

+ , Q Q D#

" 2 + A ! ( ! @ , : -!

, : -! * H J R #

" 3 + ! H K #

K @ @ #

" 4 ! U * B : !

U B * ! K # : - (

A ! + A #

! + " * @ ? - ( #

%

1

%

&

' (

K , ! + , -! ( A ,

? + , + -!

- ! D # , !

( ( ! H Q #

1,1

&

* D (" K - +

K , ( @ , # 9

D 283#

$ * K # K - F - K#

F ( ! K# : F ! K

-F ( ! K# 1 [F :K]# $ [F :K] - K A F

-K# $ K _⊆F _⊆L - ( K

[L:K] = [L:F][F :K].

: - [L:K]<_∞, [L:F]<_∞ [F :K]<_∞.

$ * F @ K# a _∈ F - D ! K @

f _∈K[X] f(a) = 0# ; f _∈K[X]

f(a) = 0 @ N ; ; "( p p(a) = 0

p= irr(a, K).

- -! - D # $ F + -!

! K A F - K# I K @ K

-! * F -! ! K - ! F.

% 1,1 K f _∈K[x]

F = K[x]

(f) ={g + (f) :g ∈K[x]}

f K degf =n

F =_{r+ (f) :r_∈K[x], r = 0 degr < n_}.

F ( ) K n

{1 + (f), x+ (f), . . . , xn− + (f)_}

F

$ * F @ K a ∈F# : ! F

-K a < ! F - K a = K(a)#

(

K(a) = p(a)

q(a) :p, q ∈K[X] q(a) = 0 ,

a_∈F - -! ! K,

[K(a) :K]<_∞.

! " # ! K - @ F K

F - @ K K(x) x ∈ F - ! K#

F/K + , -! ( ( ! K# $ K

-K A F/K - ! " # !* (

( + F/K + , #

% 1,/ $ F K %

: z _∈F K F K(z)

# K @ x∈F ! K

[F :K(x)]<∞.

B F - @ -! ! K(x)# : z ∈F @

p(X) = irr(z, K(x))∈K(x)[X]

p(z) = 0#

zn+rn− zn− + +r = 0, ri =

ai(x)

bi(x) ∈

K(x), bi(x) = 0, ∀ i= 0, . . . , n−1.

b(x) =b (x) bn− (x)∈K(x)∗ !

b(x)p(z) =b(x)zn+b(x)an− (x)zn− + +b(x)a (x) = 0. <6#6=

@ ; K K(z)

x# : - ; -

-b(0)zn+b(0)an− (0)zn− + +b(0)a (0)

% - ! K# : <6#6= x - -! ! K(z)#

[K(z)(x) :K(z)]<_∞,

F - @ -! ! K(z, x)

K(x)_⊆K(x, z)_⊆F.

[F :K(z)] = [F :K(x, z)][K(x, z) :K(z)]<_∞.

Q D ! % * -! ! K#

[F :K] = [F :K(z)][K(z) :K]<∞,

- #

9 *

K ={z ∈F :z - -! ! K}

- ! F# 9 K - D F/K#

% 1,9 F ! " K K ! &

F F −K K

# $ * z _∈F -! ! K# @ p_∈K[X] p(z) = 0

p= irr(z, K) =c +c X + +cn− Xn− +Xn.

B z - -! ! K(c , . . . , cn− )# :

[K(c , . . . , cn− , z) :K(c , . . . , cn− )]<∞

[K(c , . . . , cn− ) :K]<∞,

c , . . . , cn− -! ! K#

[K(c , . . . , cn− , z) :K] = [K(c , . . . , cn− , z) :K(c , . . . , cn− )][K(c , . . . , cn− ) :K]<∞.

z - -! ! K * z _∈K#

? 1,: z _∈ F K z K '

K _⊆K _⊆F

F ! " K K = F ( ! x ∈ F K

) * +

[F :K(x)]<∞.

K(x)⊆K(x)⊆F.

,

[F :K(x)]<_∞.

A K- ! & F K- K =K#

+ , -! F/K - ! " F = K(x)

x_∈F - ! K#

% 1,; F/K ! " K =K. ( !

z = f

g ∈K(x), mdc(f, g) = 1,

K a , . . . , an ∈K

a +a z+ +anzn= 0 ⇒a gn+a fgn− + +anfn = 0.

'

f_|a g_|an.

) f, g _∈K z _∈K

% ! + , F/K - ! O ⊆F + A

1 6# K ⊂ O ⊂F

4# z _∈F z /_{∈ O} z− _{∈ O#}

K - ( ! ( 1 ; ; "(

p_∈K[x] *

Op =

f

g :f, g∈K[x], p∤g g = 0

- ( A K(x)/K# + - K _{⊂ O}p ⊂K(x)# :

-z = f

g ∈K(x), mdc(f, g) = 1,

p∤ f p∤g,

z /∈ Op z− ∈ Op# p q ; "( !

K Op =Oq#

% 1,= _O % ! " F/K

" ! :

* _{O O} - P = _{O − O}• _O•

O

+ x _∈F∗ _{x∈P x}− _{∈ O/}

. ) K F/K K ⊆ O K ∩P = {0}

[F :K(x)]<∞ x∈P

# K 265 #4 6#6#%3#

@ 1,A O % ! " F/K P

-:

* P

+ P =t_O z _∈F∗ _{- !}

z =tnu,

n_∈Z u_{∈ O}•

# K 265 #8 HD 6#6#'3#

< = P + , F/K - @

( A O F/K# t ∈ P P = tO D

P# I PF * F/K

" !

PF ={P :P - F/K}.

@ ? ! "( O ↔P =tO# (

? 1,B _O % F/K P _O

P ( ! 2 ) 1.6

O =_{z _∈F :z− _∈/ P_{} ∪ {}0_}.

O =_OP & / ( A P

( " Z *

∞ Z - " Z - * Z_{∪ {∞}}

Z Z∪ {∞}

∞+∞=∞+m=m+∞=∞ ∞> n, ∀ m, n∈Z.

% ! F/K - + v : F → Z ∪ {∞} + A

1

6# v(x) =_∞ x = 0#

4# v(xy) = v(x) +v(y) x, y _∈F#

8# v(x+y)_≥min_{v(x), v(y)_} x, y _∈F# <# @ = ># @ z ∈F v(z) = 1#

%# v(a) = 0 a_∈K∗_#

2 4 v - ! * # :

-2 ( v - D K ( F∗

( Z#

? 1,C v % F/K c_∈R 0< c <1

! _{| |v} :F _→R

|x|v =

cv x_{, x= 0}

0, x= 0

! % :

* _|x|v >0 |x|v = 0 x= 0

+ _|xy_|v =_|x_|v|y_|v x, y _∈F

. _|x+y|v ≤ |x|v+|y|v x, y ∈F

! _{| |v} ! % & '

! _{| |v} ! % 1 2 3′

.0 _|x+y_{|v ≤}max{|x|v,|y|v} x, y ∈F

, ! % 3 ) % F/K 1

% ! % 1 2 3′ _{& / :}

" 1,10 D# @ E F/K ! " v

% F/K x, y ∈F v(x) =v(y)

v(x+y) = min{v(x), v(y)}.

# 2 5 K ( A !

v(ay) =v(a) +v(y) = 0 +v(y) =v(y), _∀ a _∈K∗.

v(−y) = v(y)# $ v(x) = v(y) v(x) < v(y)# $

D !

v(x+y) = min{v(x), v(y)}.

: 3 K ( A !

v(x+y)> v(x).

v(x) =v((x+y)₋y)_≥min_{v(x+y), v(₋y)_}= min_{v(x+y), v(y)_}> v(x),

- !

P ∈ PF + vP : F → Z∪ {∞} K 1

D ( t P# z ∈ F∗

N ! +

z =tnu,

n∈Z u∈ O•

P# : K

vP(z) =

n, z = 0

∞, z = 0.

9! ( vP - +

P D t# + t - ( P

P =t_OP =t OP.

B @ w ∈ O•

P t=wt #

tnu= (wt )nu=tn(uwn),

uwn_{∈ O}•

P#

@ 1,11 F/K ! "

* ) P _∈PF ! vP % F/K '

OP ={z ∈F :vP(z)≥0}, O•P ={z ∈F :vP(z) = 0} P ={z ∈F :vP(z)>0}.

+ 1 t_∈F P vP(t) = 1

. , % _O F/K 2 F

3 v % F/K

P =_{z _∈F :v(z)>0_}

F/K

OP ={z ∈F :v(z)≥0}

%

# I ( 3 K 265 #% HD 6#6#683#

$ * M ! F O ⊆ M ⊆ F M = F# $ D !

M =_O# @ z _∈M z /_{∈ O}#

'< # F =_O[z]#

+ z /∈ O z− ∈P# B vP(z− )>0# : y ∈F @

B : k_∈N

kvP(z− )>−vP(y).

vP(yz−k) =kvP(z− ) +vP(y)>−vP(y) +vP(y) = 0,

- w=yz−k _{∈ O y=wz}k _{∈ O[z] F ⊆ O[z] F =M}

-#

$ * F/K + , P _∈ PF OP ( A #

FP = OP

P ={x+P :x∈ OP}={x(P) :x∈ OP}

- # FP FP ∪ {∞} K x∈F − OP x(P) =∞#

+ πP : OP → FP K πP(x) = x(P) - K

-ϕ=πP|K :K →FP

- *

x_∈kerϕ_⇔x(P) =P _⇔x_∈P _⇔x_∈P _∩K =_{0_}.

: K <K= ! FP# + F

FP ∪ {∞} K x→x(P) D !

P# 9 N

degP = [FP :K]

D P# 1 - D F/K#

% 1,1/ F/K ! " P _∈PF x∈P x= 0

degP _≤[F :K(x)]<_∞.

# $ 1.6 [F : K(x)] < _∞# : ! (

z , . . . , zn∈ OP * z (P), . . . , zn(P)∈FP * LI ! K * LI !

K(x)# $ D ! @ ϕ_i(x)_∈K(x)

n

i

ϕ_i(x)zi = 0, <6#4=

ϕi(x) ; x

* ( "( x

-ϕ_i(x) = ai+xgi(x), ai ∈K gi(x)∈K[x],

ai # x∈P gi(x)∈ OP

ϕ_i(x)(P) =ai(P) =ai, i = 1, . . . , n.

: + <6#4= ! 1

0 = 0(P) =

n

i

ϕ_i(x)(P)zi(P) = n

i

aizi(P),

A ? z (P), . . . , zn(P) ! K#

! 1,19 F/K ! "

[K :K]<_∞.

, I PF = ∅ K 6#6'# : @ P ∈ PF

[FP :K] = [FP :K][K :K]⇒[K :K]≤[FP :K]<∞,

- * #

? 1,1: F/K ! " P _∈PF degP = 1 FP =K

1 = degP = [FP :K].

$ * F/K + , z _∈ F P _∈ PF# A P - % z

vp(z)>0 P - z vP(z)<0# $ * m, n∈N# $

vP(z) =m >0,

P - A z m

vP(z) =−n <0,

P - z n

@ 1,1; F/K ! " R F K _⊆ R_⊆F

{0_}=I _⊂R 2 R P _∈PF I ⊆ P K ⊆P

# K 265 #7 HD 6#6#6S3#

! 1,1= F/K ! " z _∈ F K

P, Q_∈PF P % z Q z PF =∅

# R = K[z] I = zK[z]# z -! K

K[z]_≃K[X].

: + I = zK[z] XK[X]# B H 6#6% @ P ∈ PF

I ⊆ P# z ∈P P - A z# ( z−

A Q∈PF# z Q#

% 1,1A F/K ! " P , . . . , Pr % x ∈ F r

i

vP(x) degPi ≤[F :K(x)].

# K 265 #68 6#8#83#

! 1,1B F/K ! " x _∈ F x = 0

%

# $ x - -! ! K x∈ K ⊂ O•

P# : vP(x) = 0 - x

A # $ x - ! K 6#67

r

i

vP(x) degPi ≤[F :K(x)].

P , . . . , Pr A x vP (x)≥1#

r_≤[F :K(x)],

degPi = dimK(FP )≥1.

N A x− _{< N x=}

- [F :K(x− _{)]# x K A #}

1,/

?

%

&

4

I ( ( A + ,

F =K(x) x - ! K#

; ; "( p_∈K[x] ( A

Op =

f

g :f, g∈K[x], p∤g g = 0

! Op ! ( "( #

+

f g

−

= g

f

( Op p ( f @

Op " ! Pp

-Pp =

f

g :f, g ∈K[x], p|f, p∤g g = 0 .

O•p =

f

g :f, g ∈K[x], p∤f, p∤g g = 0 .

p=x₋α ( Pα =Px−α ∈PF#

@ ( A F/K !

O∞= f

g :f, g ∈K[x], g= 0 degf ≤ degg .

9 ; - −∞

−∞+_−∞=_−∞, _−∞+n=_{−∞ − ∞}< n, _∀ n_∈Z.

B

0 = 0

1 ∈ O∞. 9 ( "( O∞

f

g, degf <degg,

O∞ D

-P∞= f

g :f, g ∈K[x], g = 0 degf <degg .

: ( D

" K(x)# * (

-K(x) =K 1

x .

" 1,1C A & A ! - a∈ A∗

p A - np ∈Z

pn _|a, pn ∤a.

# $ D ! * + # n=np ∈

Z K@ @ bn ∈A a=pnbn# B

bn =pbn ,

a=pn _b

n

(b )_⊂(b )_⊂(b )_⊂

- A - #

$ * A " + N a∈A p A# : /

p a " ! vp(a) - N np ∈Z

pn _|a, pn ∤a,

vp(a) = ∞ a = 0# V vp(a) = 1 <vp(a) > 1= A p - !

< - = a#

% 1,/0 F/K ! "

* P = Pp ∈ PF p ∈K[x] K

p P %

vP(z) =

n, z =pn f_g

∞, z = 0,

n_∈Z f_{g ∈ O}•

P '

Fp = OP

P ≃

K[x] (p) ,

f(x) + (p)_→f(x)(P) degP = degp

+ p=x₋α degP = 1 !

z(P) = z(α), _∀ z= f

g ∈F

z(α)

z(α) =

f α

g α , g(α) = 0

∞, g(α) = 0,

. P =P∞ degP∞= 1 1 P∞

t= 1

x

% v∞

v∞ f

g = degg−degf.

' ! P∞

z(P∞) =z(∞),

z(∞)

z(_∞) =

    

a

b , m=n

0, n < m

∞, m < n,

z= f

g =

a +a x+ +anxn

b +b x+ +bmxm

, an, bm = 0,

# I ( 1 3 + , 265 #S

6#4#631 1 Pp - p

z= f

g ∈Pp,

@ u∈K[x] f =up# :

z = f

u u

g =p

u

g ⇒z ∈pOP ⇒Pp =pOP.

B p - ( P# K[x] - " + N F

-z _∈F N !

+

z =pnf g,

n_∈Z f_{g ∈ O}•

P# : K

vP(z) =

n, z =pn f

g

∞, z = 0,

* # 3 $ !

1

x ∈P∞ K(x) =K

1

x ⇒ K(x) :K

1

x = 1,

B 6#64

degP∞_≤ K(x) :K 1

x = 1⇒degP∞ = 1.

x - ( P∞

z = f

g ∈P∞,

z= 1

x xf

g ⇒z ∈

1

xO∞⇒P∞ =

1

xO∞.

9 K #

@ 1,/1 F/K ! " - F/K

Pp P∞ p∈K[x]

#$ * OP ( A F/K#

'*? 1

: x∈ OP# K[x]⊆ OP# ( * I :=K[x]∩P

-K[x]#

H !- A D K 1

γ :K[x] _→ K(x)P

f(x) _→ f(x)(P)

kerγ =I# : H L K 1 K[x]/I ≃Imγ≤K(x)P.

9! ( I = _{0_}, I = _{0_} K[x]/I = K[x] Imγ

K ! # L K @ ; "( ; <

= p(x) _∈ K[x] I = p(x)K[x]. H g(x) _∈ K[x] p(x) ∤ g(x)

I 6#'

g(x)_∈/ P 1

g(x) ∈ OP. B "

Op =

f(x)

g(x) :p(x)∤g(x) ⊆ OP

$ H 6#66 ( A - ! @ K(x)

OP =Op.

'*? /

$ D x /_{∈ O}P 6#' x− ∈P,

K[x− ]⊆ OP x− ∈P ∩K[x− ].

: P ∩K[x− ] =p(x− )K[x− ] p(x− )∈ K[x− ]- ; ; "( # p(x− ) =x− P∩K[x− ] =

x− K[x− ] ( A '*? 1 ! 1

OP ⊇

f

g :f, g∈K[x

− _{], g = 0 x}− _∤g(x− _{) =}

a + ... +anx−n

b + ... +bmx−m

, b = 0 =

a xm n _{+ ... +a}

nxm

b xm n _{+ ... +b}

mxn

, b = 0 =

u

v :u, v ∈K[x], v= 0 degu≤degv = O∞.

: OP =O∞.

? 1,// K q

K q+ 1 ' q ! Pα P∞.

1,9

#

9 K + , -! F/K - @ K

K# F + , ! K (

@ " + , F/K - K

-F/K#

9 F/K - K ! ( < ( =

- F/K " ! div(F)# 9 div(F) D

F/K# ( ( - +

D =

P∈P

nPP,

nP ∈ Z nP = 0 @ K P ∈ PF# 9 D

-K 1

suppD=_{P _∈PF :nP = 0}.

( + ?

D=

P∈S

nPP,

S ⊆PF - * K suppD ⊆S# ( + D=P P ∈PF

- D # (

D= nPP D′ = n′PP

+

D+D′ = (nP +n′P)P.

9 div(F) - (

0 =

P∈P

nPP

nP = 0 P ∈PF# Q∈PF

D =

P∈P

nPP,

K vQ(D) =nQ # :

suppD=_{P _∈PF :vP(D) = 0} D=

P∈ D

vP(D)P.

! div(F) - K

D _≤D _⇔vP(D )≤vP(D ), ∀ P ∈ PF.

$ D _≤ D D =D !- ( D < D # ( D _≥0 - D

%# - (

D =P + 2P D = 2P +P

( #

9 ( D∈div(F)- K

degD=

P∈P

vP(D) degP.

@ D K ϕ: div(F)_−→Z K ϕ(D) = degD#

$ ! 6#6& x _∈ F x = 0 ! K

K A PF +

K , 1

$ * x_∈F x= 0 _Z < ( _N= * A < ( = x PF# K

(x) =

P∈Z

vP(x)P

% x

(x)∞=

P∈N

(₋vP)(x)P

x#

(x) := (x) ₋(x)∞

x# (x) ≥0 (x)∞ ≥0

(x) =

P∈P

vP(x)P.

9 x _∈ F x = 0 -! < = ! K A

x_∈K _⇔(x) = 0.

(x)∞ = (x) = 0#

9 * (

PF ={(x) :x∈F∗}

- D F/K#

Cl(F) := div(F)

PF

F/K# ( D _∈ div(F)

[D] ( D# : !

( ? 1

D_∼D′ _⇔[D] = [D′],

- D=D′+ (x) x∈F∗#

" 1,/9 K V K W V

dim V

W = dimV −dimW.

# : +

π :V → V

W

K π( ) = +W - ! * # B H N L

dim V

W = dimV −dimW,

kerπ =W#

( A _∈ div(F) K@ K 4 /4 & A

L(A) =_{x_∈F∗ : (x)_{≥ −}A_{} ∪ {}0_}.

9! (

x_{∈ L}(A)_⇔vP(x)≥ −vP(A), ∀ P ∈PF.

: - L(A) = {0} @ D ∈ div(F) D ∼ A D ≥ 0# +

L(A) =_{0_{} ⇔ ∃} x_∈F∗; (x)_{≥ −}A_⇔A+ (x)_≥0_⇔D=A+ (x)_≥0_⇔D_∼A.

" 1,/: F/K ! " A_∈div(F)

* _L(A) K

+ D A _L(A)_{≃ L}(D)

. _L(0) =K _L(A) =_{0_} A <0

3 B A_≤B 2 _L(A)_{⊆ L}(B)

dim L(B)

L(A) ≤degB −deg A.

# K 265 #67 6& B 6#>#' 6#>#7 6#>#&3#

% 1,/; F/K ! " A ∈div(F) L(A)

K 5 A=A ₋A−

A A−

dimL(A)≤degA + 1.

# K 265 #6S 6#>#S3#

( A_∈div(F)

ℓ(A) = dim_L(A)

- D ( A#

@ 1,/= F/K ! " ) x∈F −K

deg(x) = deg(x)∞= [F :K(x)].

%

# $ *

n= [F :K(x)] B = (x)∞ =

r

i

−vP(x)Pi,

P , . . . , Pr x# 6#67

degB =

r

i

vP (x− ) degPi ≤[F :K(x− )] = [F :K(x)] =n,

x - A x− _{( ( degB ≥ n# D}

!

u , . . . , un

F/K(x)# A + * K

*

T ={vP(ui) :i= 1, . . . , n P ∈S}

- K

S =

n

i

supp(ui).

: @ n ∈N −n <minT# (

C =n

P∈S

P ≥0,

! (ui)≥ −C i= 1, . . . , n#

: * *

A=_{xjui : 0≤j ≤t 1≤i≤n}

t_∈ Z t_≥0#9 A LI ! K A_{⊆ L}(tB+C)

vP(xjui) =jvP(x) +vP(ui)≥ −jvP(B)−vP(C)≥ −tvP(B)−vP(C) =−vP(tB+C).

: A - ! L(tB+C)

dim A ≤ℓ(tB+C)⇒(t+ 1)n≤ℓ(tB+C). <6#8=

C ≥0 t≥0 (tB+C)≥0# B 6#4%

ℓ(tB+C)_≤deg(tB+C) + 1 =tdegB + degC+ 1,

-(t+ 1)n≤tdegB+ degC+ 1.

H c= degC !

(t+ 1)n_≤tdegB+c+ 1_⇒n₋c₋1_≤t(degB₋n).

degB−n≥0#

! 1,/A F/K ! "

* A A′ A∼A′

ℓ(A) =ℓ(A′) degA= degA′.

+ degA <0 ℓ(A) = 0

. ) A % " :

A ;

ℓ(A)≥1;

ℓ(A) = 1

# K 265 #6S ) 6#>#643#

% 1,/B F/K ! " γ ∈Z

deg(A)₋ℓ(A)_≤γ, _∀ A_∈div(F).

γ A ! " F/K

# K 265 #46 6#>#6>3# *

X ={deg(A)−ℓ(A) + 1 :A∈div(F)}.

6#4& @ γ _∈Z

deg(A)−ℓ(A) + 1≤γ+ 1, ∀ A∈div(F).

: X - # 6 @ # 9 7

F/K - K

g = maxX.

! 1,/C F/K ! " 7 F/K

. H A= 0 ! B 6#4>

g _≥deg 0₋ℓ(0) + 1 = 0,

- * #

@ 1,90 D@ 4 E F/K ! " 7 g

* ℓ(A)_≥degA+ 1₋g A_∈div(F)

+ c F/K

ℓ(A) = degA+ 1₋g,

degA_≥c

# 1 K ? !

g ≥degA−ℓ(A) + 1⇒ℓ(A)≥degA+ 1−g,

A ∈div(F)#

2 @ ( A

g = degA −ℓ(A ) + 1.

: 1

ℓ(A−A )≥deg(A−A ) + 1−g. c= deg A +g degA _≥c

deg(A₋A ) + 1₋g = degA₋degA + 1₋g ≥ c₋degA + 1₋g

= degA +g−degA + 1−g

= 1.

B ℓ(A₋A )_≥ 1 " L(A₋A ) = _{0_}.: @ z _{∈ L}(A₋A )

z = 0# ( A′ _{:=A+ (z) A}′ _≥A

z _{∈ L}(A₋A )_⇒(z)_{≥ −}A+A _⇒A′ = (z) +A_≥A .

B 6#4> 6#47

degA₋ℓ(A) = degA′₋ℓ(A′)_≥deg A ₋ℓ(A ) = g₋1.

ℓ(A)_≤ degA+ 1₋g

1 #

% 1,91 F/K ! " F/K 7 g= 0

* # $ ! H Q @ c∈Z

ℓ(A) = degA+ 1−g,

degA≥c# $ * r >0 r > c P∞ = (x)∞ L(rP∞)#

H 6#4'

deg(rP∞) =rdegP∞ =r[F :K(x)] = r

B

ℓ(rP∞) = deg(rP∞) + 1₋g =r+ 1₋g.

'< # 1, x, . . . , xr _{∈ L(rP∞)#}

+

−vP(rP∞)≤0, ∀ P ∈PF,

vP(rP∞) =

r >0, P =P∞

0, P =P∞

B

0 = vP(1)≥ −vP(rP∞), ∀ P ∈PF.

: - P∞ x !

vP∞(x

i_{) =iv}

P∞(x)<0⇒ivP∞(x)> rvP∞(x) =−rvP∞(P∞).

vP(x)≥0, ∀ P ∈PF − {P∞}.

H

vP(xi)≥ −vP(rP∞), ∀ P ∈PF,

* xi _{∈ L(rP∞) i= 0, . . . , r. ℓ(rP∞) = r+ 1 g = 0#}

@ 1,9/ F/K ! " A degA≥2g−1

ℓ(A) = degA+ 1−g.

# K 265 #86 HD 6#%#673#

% 1,99 P ∈ PF n ∈ N n ≥ 2g x ∈ F

(x)∞ =nP.

# K 265 #8> 6#'#'3#

%

/

&

%

&

: + , F/K - "( N

#I ! ( ! @ ,

-! @ , " K H J R # I

+ , -! F/K F′_/K′ _K

F K′ _F′_{# F}

q K q

q - ? N # 9 D

2653#

/,1

&

' (

+ , -! F′/K′ - D F/K F′/F

- @ -! K ⊆ K′# : @ -! F′/K′ F/K - D

F′ =F K′ - F K′ F K′ - ! F′

F K′

F K′ =F(K′) =K′(F).

@ -! F′_/K′ _{F/K- D [F}′ _:F]<∞.

" /,1 F′/K′ F/K / 8

* K′_{/K F ∩K}′ _=K

+ [F′ _{:F]<∞ [K}′ _:K]<∞

# K 265 #'S B 8#6#43#

$ * F′_/K′ _{@ -! F/K# A P}′ _{∈ P}

F′ #

< = P _∈PF P # < =P′ " ! P′ |P P ⊆P′#

H !- A P′ _{- P#}

% /,/ F′/K′ F/K & P (P′)

F/K (F′/K′) OP ⊆F (OP′ ⊆F′) v_P (v_P′) %

" :

* P′ _|P;

+ _OP ⊆ OP′;

. e_≥1 vP′(x) =ev_P(x) x∈F

' P′ _|P

P =P′∩F OP =OP′ ∩F.

) % P & P′ _F

# K 265 #'S 8#6>3#

? /,9 ) ) 2.2 &

FP FP′′

x(P)_→x(P′), _∀ x_{∈ O}P,

# FP FP′′ )

FP FP′′ F_P′′

FP

$ * F′_/K′ _{@ -! F/K P}′ _{∈ P}

F′ F′/K′

P ∈PF# A e =e(P′ |P)

vP′(x) = ev_P(x), ∀ x∈F,

- P′ _{! P# 9}

f =f(P′ _|P) = [FP′ :F_P]

- D P′ ! P# f K K e

-N #

% /,: F′/K′ F/K P′ ∈PF′ F′/K′

P _∈PF.

f =f(P′ _|P)<_{∞ ⇔}[F′ :F]<_∞.

# K 265 #76 8#6#'3#

% /,; F′/K′ F/K

* ) P′ ∈ PF′ P ∈ P_F P′ | P P =

P′∩F

+ 9 P _∈PF " P′ ∈PF′

# 1 I ( @ z ∈ F z = 0 vP′(z) = 0#

+ D ! * + # D t∈ F′ _v

P′(t)>0#

F′/F - @ -! @ c , . . . , cn ∈F c = 0 cn = 0

cntn+cn− tn− + +c t+c = 0.

vP′(c_i) = 0 i= 0, . . . , n v_P′(c_iti) =v_P′(c_i) +iv_P′(t)>0, i= 1, . . . , n.

:

vP′(c ) = 0 v_P′(c_ntn+ +c t)≥min{v_P′(c_ntn), . . . , v_P′(c t)}>0 H

vP′((c_ntn+c_n− tn− + +c t) +c ) = min{v_P′(c_ntn+c_n− tn− + +c t), v_P′(c )}= 0.

∞=vP′(0) =v_P′(c_ntn+c_n− tn− + +c t+c ) = 0,

- # @ ω ∈F vP′(ω)>0# B P′∩F =∅

OP′ ∩F =∅#

'< # OP =OP′ ∩F P =P′∩F#

+ K ⊆ K′ ⊂ OP′ K ⊂ O_P. O_P = F F ⊆ O•

P′

vP′(z) = 0 z ∈ F - "( # : @ z ∈ F z /∈ O_P# B

z /∈ OP′ z− ∈ O_P′ - z− ∈F ∩ O_P′# z− ∈ O_P# O_P

-( A P - N @ P =_OP − OP•

: 4#4#

2 $ * P F/K# 6#88 @ x_∈F * N

A - P#

'< , P′ _|P vP′(x)>0# + P′ _|P

vP′(x) =e(P′ |P)v_P(x)>0, e(P′ |P)≥1.

vP′(x) > 0 Q - F/K ! @ P′ v_Q(x) >0#

: Q=P P - N A x F/K#

x " @ K A F′/F

#

$ * F′_/K′ _{@ -! F/K# P ∈ P}

F K@ K

F′/F

ConF′_/F(P) =

P′_|P

e(P′ |P)P′,

P′ ∈PF′ P#

@ /,= F′_/K′ _{F/K P F/K P , . . . , P}

m

F′/K′ P ei =e(Pi|P)

fi =f(Pi|P) Pi|P.

m

i

eifi = [F′ :F].

# K 265 # 7> HD 8#6#663#

! /,A F′/K′ F/K P ∈PF

|{P′ ∈PF′ :P′ |P}| ≤[F′ :F].

# i 1≤i≤m !

1_≤eifi ⇒m≤ m

i

eifi = [F′ :F],

- * #

% /,B F′ =F K′ F/K F′/K′

7 F/K

# K 265 # 66> HD 8#'#83#

/,/

@

>

F

>

F

F - + , -! * - Fq# 9 @

- + T ( ( #

" /,C ) n_≥0 n

, ( *

S =_{P _∈PF : degP ≤n}

- K ( ( - ( # D x _∈ F ₋Fq

*

S =_{P _∈PF x : degP ≤n}

@ -! K F/Fq Fq(x)/Fq#

P _∩Fq(x)∈S , ∀ P ∈S.

OP∩F x

P ∩Fq(x)

- ! (

OP

P

deg(P _∩Fq(x)) ≤ n# : - 4#% P ∈ S

K @ , F# : ( S - K # +

Fq(x) @ x, ; ; "(

Fq[x] Fq - K S - K #

@ /,10 F Fq F qr

r= [F :Fq] dim(F) = r= logq|F|

# F - ( ! Fq r F - K #

$ *

β ={ , . . . , r}

! F# ∈F @ N x , . . . , xr ∈Fq

=

r

i

xi i.

I K + Tβ :Frq→F

Tβ(x , . . . .xr) = .

W + ( K Tβ ! K - * # F - + Frq#

xi @ q ! |F|=qr

dim(F) = r= log_q|F_{| |}F_|=q F ,

- * #

? /,11 ' ! Tβ : Frq → F & A F

β T_β− & K ! ; F

β

I K F H 4#65 Fq # : r ∈ N @

@ @ Fq /Fq #

Fq ⊆Fq ⇔r|s.

: - ( + D -! Fq Fq

-Fq = r∈N

Fq .

9 D ! K 263 283#

@ < = F =FFq F/Fq

Fr=FFq ⊆F .

% /,1/ F/Fq ! " : / 8

* Fq Fr

+ P _∈PF m

ConF /F(P) = P + +Pd,

d= mdc(m, r) Pi ∈ PF i= 1, . . . , d m_d

# K 265 #6S5 B %#6#S3#

$ * F/Fq + , K

An =|{A∈div(F) :A≥0 degA=n}|.

@ A = 1 A - N 1# : - ?

Z(t) :=ZF(t) =

∞

n

Antn ∈C

- D ! ; F/Fq# 9! ( t - ( ( @ Z(t)

-- ? ! @ # : (

( A D A #

% /,19 ' 7 Z(t) |t|< q−

# K 265 #6&& %#6#'3#

: Z(t) ( ! @ N 1

F/Fq Fr# |t|< q− * #

% /,1: F/Fq ! "

* F/Fq 7 0 F/Fq ! " ! ;

Z(t) = 1

(1₋t)(1₋qt).

+ F/Fq 7 g ≥ 1 ! ; ! Z(t) =

F(t) +G(t)

F(t) = 1

q₋1

≤ C≤ g−

qℓ C t C

G(t) = h

q₋1 q

g_t g− 1

1₋qt−

1 1₋t .

$ h= _|{[A]_∈Cl(F) : deg[A] = 0_}|

# K 265 #6S4 ) %#6#643#

$ * F/Fq + , # 9 ;

L(t) =LF(t) = (1−t)(1−qt)Z(t)

- D L/ F/Fq# 4#6>

L(t) = (1₋t)(1₋qt)F(t) + h

q−1(q

g_t g− ₍₁

−t)₋(1₋qt)) L(t) = 1

B L(t) - + ; # : - L(t) - + , ! N An

L(t) = (1₋t)(1₋qt)

∞

n

Antn.

@ /,1; F/Fq ! " / 8

* deg(L(t)) = 2g

+ )

L(t) =a +a t+ +ant g,

a = 1 a =N ₋(q+ 1) N - P _∈PF 1

. L(t) ! C[t] !

L(t) =

g

i

(1−αit). <4#6=

3 Lr(t) := (1−t)(1−qrt)Zr(t) L/ Fr

Lr(t) = g

i

(1−αr_it),

αi (2.1)

# K 265 #6S8 HD %#6#6%3#

$ * F/Fq + , K 1

N(F) =N =_|{P _∈PF : degP = 1}|

r_≥1 N

Nr =N(Fr) =|{P ∈PF : degP = 1}|

Fr - @ F/Fq#

! /,1= F/Fq ! " r∈N

Nr=qr+ 1− g

i

αri,

α , . . . , α g ∈ C % L(t) N = N(F)

N(F) =q+ 1−

g

i

αi.

# H 4#6%

Nr−(qr+ 1)

- K ; Lr(t)# ( H 4#6% !

Lr(t) = g

i

(1−αr_it).

K

-−

g

i

αri

#

Sr =−

g

i

αr_i,

4#6'

Nr =qr+ 1 +Sr, <4#4=

Sr = 0 g = 0 9! ( 6#44#

$ * F/Fq + , ? g K 1

Bτ =Bτ(F) = |{P ∈PF : degP =τ}|.

9! ( B =N(F)#

% /,1A F/Fq ! " 7 g

Nr =

i|r

iBi.

, $ * P _∈PFOdegP =i i| r# 4#64

ConF /F(P) =P + +Pi,

Pj ∈ PF deg Pj = 1 j = 1, . . . , i# :

degP ( r degQ >1 Q P.

Q _∈PF N P ∈ PF, PF

PF * ( r.

Nr =

i|r

iBi,

P ∈PF deg P =i @ @ Q P#

: @ ( ! K + X!

<:N→ {−1,0,1}

<(n) =

  

 

1, n= 1

0, @ k >1 k _|n,

(₋1)k_{, n - k #}

+ ( X!

Nr =

i|r

iBi ⇔rBr=

i|r

< r

i Ni. <4#8=

+ , ! + X! 2'3#

@ /,1B D@ > F E $ % LF(t) ! % /

:

|αi|=q , i= 1, . . . ,2g.

# K 265 #6S7 HD %#4#63#

! /,1C D > F E $ - N =N(F) 1 F/Fq

! %

|N −(q+ 1)| ≤2gq .

, $ 4#6'

N ₋(q+ 1) =₋

g

i

αi ⇒ |N −(q−1)|= − g

i

αi ≤

g

i

|αi|.

: H J R !

|N ₋(q₋1)_|= ₋

g

i

αi ≤

g

i

|αi| ≤2gq ,

- * #

Fr/Fq +

|Nr−(qr+ 1)| ≤2gq ,

r _≥1#

$ * F/Fq + , ? g# A F/Fq

-N =q+ 1 + 2gq .

% /,/0 F/Fq ! "

αi =−q , i= 1, . . . ,2g.

, 4#6' H 4#6& !

N =q+ 1−

g

i

αi |αi|=q , i= 1, . . . ,2g.

F/Fq - @

q+ 1₋

g

i

αi=q+ 1 + 2gq ⇒ g

i

αi =−2gq .

B

g

i

αi =|α |+ +|α g| ⇒αk =bklαl, bkl∈R 1≤k, l≤2g.

: αt K@

g

i

αi = 1 +

g

i

bit αt=−2gq ⇒αt∈R⇒αi ∈R, 2≤i≤2g,

|αi|=q ⇒αi =±q , i= 1, . . . ,2g.

g

i

αi =−2gq ,

! α = =α g =−q #

%

9

"

" ( + A B +

@ A " ! # 9

D 2&3 273#

9,1

! - * ( A A# 2 C ! + ! A

-! * ( A * AI _?

={ci :i∈I}= (ci)i∈I,

ci ∈A I⊆N# V

I={i: 1≤i≤n}

An_=AI_#

2 C n ! + ! A - ! *

( A * An _{( < ? =}

=_{ci :i∈I}.

( ! D + ! K # @

+ ! K - A =Fq Fq - K q q = pm

N p m_∈N#

$ * = (x , . . . , xn), = (y , . . . , yn) ∈ An# : < =

-K

d( , ) =_|{i_∈I:xi =yi}|.

K ! + ! K - A

+ ! '# $ _|A_|= q, A

- q # q = 2 ! q = 3 q = 4

# : - |C|= 1 C =An _{A - 2 #}

: < C _|C_|= 1 - K

d=d(C) = min_{d( , ) : , _∈C, = _}.

9 ∈C ωt(x),- K

ωH( ) =d( , )

C - K

min{ωH( ) : ∈C, = }.

2 - ! C Fn

q (

C# D n C dim(C) C# 2 [n, k]

-n k# $ T " d - D

[n, k, d] n k T " d# H

!-[n, k, _≥ d] n k T " " d#

( T " - " #

$ * C [n, k] ! Fq

{ , . . . , k}

! C# + Φ :Fk

q →Fnq K

Φ(x , . . . , xk) =x + +xk k

- * C = Im Φ# C

+ * Φ :Fk

q →Fnq C = Im Φ ( ( #

$ * C [n, k] ! Fq % C - Ak×n

* D ! C#

$ * = (x , . . . , xn), = (y , . . . , yn)∈Fnq# 9 , - K

n

i

xiyi.

C # 9 ! Fn

q

C⊥:={ ∈Fn_q : , = 0, ∀ ∈C}

D 2 C# : A C⊥ _{D %}

C# ( C⊥ - [n, n−k]

t _=O.

C ={ ∈Fn_q : t= }.

: A ( K A ( u_∈Fn

q - (

#

$ ( @ "

# K A #

! ! - ! + ! K@ Fq

* T " * #

@ , # :

-< = T " - # :

- 1

% 9,1 D * E ) 2 q/# [n, k, d]

k+d≤n+ 1.

# ! ( E ⊆Fn

q K

E =_{(x , . . . , xd− , xd, . . . , xn)∈Fnq :xi = 0, ∀ i≥d}

" C - d E _∩C =_{{ }}#

k+d₋1 = dim(C) + dim(E) = dim(E+C) + dim(E_∩C) = dim(E+C)_≤n,

- * #

9 k +d =n+ 1 D

2 M DS < 2 # # < =#

% 9,/ D > E C 2 q/# [n, k, d]

qk

t

i

n

i (q−1)

i

≤qn t= d−1 2 ,

⌊x_⌋= max_{n_∈Z:n_≤x_}.

, $ * , _∈C = t_∈N K

St( ) ={ ∈Fnq :d( , )< t}

St( )∩St( ) = ∅ ∈St( ) ∈St( )

d_≤d( , )_≤d( , ) +d( , )<2t =d₋1,

- "( # : - qk + St( ) c∈C -t

i

n

i (q−1)

i

( Fn

q @ i ∈Fnq - (

D

n

1

- F∗

q# @

n

1 (q−1)

( # @ qn _{( F}n

q

qk

t

i

n

i (q−1)

i

≤qn,

- * #

9,/

- - ! (

( ! #

W ( ! @ Fq Fq# !

H 4#65

[Fq :Fq] =t.

% 9,9 D E C 2 [n, k, d] Fq Fq

Fq 2 A[m, t, s] Fq 2 C′[nm, kt,≥

ds] Fq

, A Fq ( ! Fq @ K Φ : Fq −→ A#

Φ∗ _{: F}n

q → Fmnq

( , . . . , n) → (Φ( ), . . . ,Φ( n))

Φ∗ _{- * Φ - * # C - !}

Fn

q C′ = Φ∗(C) - ! Fnmq # : H 4#65 !

dim(C′) = log_q|C′_|= log_q|C_|= log_q((qt₎k_{) = log}

qqkt=kt.

*

= ( , . . . , n)

( # $ i = Φ( i) - (

A# :

ωH(Φ( i))≥s.

" d , N ,

(Φ( ), . . . ,Φ( n))

- " # C′ _{- [nm, kt,≥ds] ! F}

q#

9 C′ _{D 2 - ! ? - D /}

# 9 C - 2 '

-2 #

% 9,:

F ={0,1}

p=x +x+ 1_∈F [x]

F '

F = F [x]

(p) ={0,1, α,1 +α},

0 = (p),1 = 1 + (p) α = x+ (p) _∈ F % p F .

$ _{1, α_} F F ' 2 [2,1,2]

C = (1, α) =_{(0,0),(1, α),(α,1 +α),(1 +α,1)_}

F & 2 [3,2,2]

A = (1,1,0),(1,0,1) =_{(0,0,0),(1,1,0),(1,0,1),(0,1,1)_}

F 1 Φ F A Φ(1) Φ(α)

Φ(1) = (1,1,0) Φ(α) = (1,0,1).

Φ(a+bα) = (a+b, a, b).

>

C′ = Φ∗(C) ={(Φ( ),Φ( )) : ( , )∈C}

= _{(0,0,0,0,0,0),(1,1,0,1,0,1),(1,0,1,0,1,1),(0,1,1,1,1,0)_} = (1,1,0,1,0,1),(1,0,1,0,1,1) .

) C′ _{2 [6,2,4] F}

9,9

' (

(

9 : -! - < AG= + A I# # #

Y ( A D - # (

( Q $ < RS=

! Fq# - ! D H D !

# $ ! , : -! - A

RS#

@ 9,; D 4 * E Fq n≤ q+ 1

2 [n, k, d] MDS Fq 1≤k≤n

# L n=q+ 1.

Lk ={f ∈Fq[X] : degf ≤k−1}

ϕ:_Lk −→Fnq + < ( = K

ϕ(f) = (f(α ), f(α ), . . . , f(αq), ak− ), <8#6=

α , α , . . . , αq q Fq

f =a +a x+ +ak− xk− ∈Fq[X].

9! ( Lk - ( ! Fq k ( K ϕ -

* # : C =ϕ(Lk) - n k# $

ak− = 0 H C ! f @ k −1 "A #

: J (

-wH( )≥n−(k−1).

: 8#6

wH( ) =n−k+ 1.

$ ak− = 0 H C ! f @ k−2 "A #

B

wH( )≥n−[(k−2) + 1] = n−k+ 1.

( 8#6

wH( ) =n−k+ 1.

C - MDS# $ n < q+1 !

#

n < q ! ( RS

D + ! .

? 9,= 2 ( ) [4,2,3] F ( ! &

[4,2,3]/ 2 C F

d(C) =d(C+ )

∈ F (0,0,0,0) _∈ C C

2 MDS " ! '

2 (0,1) " (0,1,1,1)

# d(C)<3 C

(1,0,1,1),(1,1,0,1) (1,1,1,0).

5 < = 2

) 2 [4,2,3] F , 3.5

2 [4,2,3] Fq q≥3 '

< 2

: -! - ( A

, 1

F/Fq + , -! ? g#

P , . . . , Pn F/Fq 1#

D=P + +Pn#

G ( F/Fq

suppG_∩suppD=_∅.

9 -! - < AG= CL(D, G) ( D G -K

CL(D, G) =_{(x(P ), . . . , x(Pn)) :x∈ L(G)} ⊆Fnq,

K + A

suppG∩suppD=∅

vP(x)≥ −vP (G) = 0, ∀ x∈ L(G) i= 1, . . . , n.

B

x_{∈ O}P, ∀ i= 1, . . . , n.

: - degPi = 1 K

Fq x(Pi)∈ OP

Pi

=Fq, ∀ i= 1, . . . , n,

K 9! ( 6#6>#

<8#6= + ϕ:L(G)−→Fnq K

ϕ(x) = (x(P ), . . . , x(Pn)),

( K ϕ- -CL(D, G) L(G)

-( ! Fq# B CL(D, G) =ϕ(L(G)) - ! Fq n#

@ 9,A CL(D, G) [n, k, d]/ 2 Fq <

k =ℓ(G)₋ℓ(G₋D) d_≥n₋degG.

# ( (

kerϕ=_{x_{∈ L}(G) :vP(x)>0, ∀ i= 1, . . . , n}=L(G−D).

x_∈kerϕ !

(x(P ), . . . , x(Pn)) = (0, . . . ,0).

B x(Pi) = 0 i= 1, . . . , n# vP(x)>0 i= 1, . . . , n#

vQ(x)≥ −vQ(G) Q∈PF

suppG∩suppD=∅

vQ(x)≥ −vQ(G) =−vQ(G) +vQ(D), ∀ Q /∈ {P , . . . , Pn},

vQ(x)≥1 =−vQ(G) +vQ(D), ∀ Q∈ {P , . . . , Pn}.

kerϕ⊆ L(G−D)# x∈ L(G−D) !

vQ(x)≥ −vQ(G) +vQ(D)≥ −vQ(G), ∀ Q∈PF.

B x∈ L(G)

-L(G−D)⊆ L(G).

vQ(x)≥1 =−vQ(G) +vQ(D)>0, ∀ Q∈ {P , . . . , Pn}.

x∈ L(G) x∈Pi, ∀ i= 1, . . . , n,

- x_∈kerϕ# _L(G₋D)_⊆kerϕ# : H L K

!

L(G)

kerϕ ≃Imϕ=CL(G, D) ℓ(G)−dim kerϕ= dimCL(G, D) =k.

kerϕ=L(G−D),

k =ℓ(G)−ℓ(G−D).

$ D CL(G, D) = {0} + +

" # D 0 = x∈ L(G)

ωH(ϕ(x)) =d.

B

|{i:x(Pi) = 0}|=n−d,

* @ @ n−d Pi , . . . , Pi − suppD A #

( A suppG∩suppD=∅,

vQ(x)≥ −vQ(G) =−vQ(G) +vQ(D), ∀ Q /∈ {Pi , . . . , Pi − },

vQ(x)≥1 =−vQ(G) +vQ(D), ∀ Q∈ {Pi , . . . , Pi − }. :

0 =x∈ L(G−(Pi + +Pi − )) ℓ(G−(Pi + +Pi − )) = 0

B 6#47 !

degG₋n+d= deg(G₋(Pi + +Pi − ))≥0.

d≥n−degG#

! 9,B & degG < n ϕ : L(G) −→ Fnq

CL(G, D) [n, k, d]/ 2

d ≥n−degG, k =ℓ(G)≥degG+ 1−g

%

=

  

x (P ) x (P ) . . . x (Pn)

xk(P ) xk(P ) . . . xk(Pn)

  ,

{x , . . . xk} L(G)

, $

deg(G−D) = degG−n <0

6#47= L(G−D) = {0}# : kerϕ = {0} * ϕ - * #

H 8#7

ℓ(G)−ℓ(G−D) =k d≥n−degG.

ℓ(G−D) = 0 H Q

k=ℓ(G)_≥degG+ 1₋g.

: - {x , . . . , xk}- ! L(G)

{(xi(P ), . . . , xi(Pn))}, i = 1, . . . , k,

- ! CL(G, D) ϕ- * # - A CL(G, D)#

9,:

' (

(

4

( ( AG ( + ,

# + ( !

A Q $ Q

$ A #

-! - CL(D, G) ( G D

+ , Fq(z)/Fq - D 2 2

AG #

9! ( AG - @ q+1 H

6#46Fq(z) q+ 1 ! P∞ % α∈Fq

Pα A z−α# B ! @ 6#86 + ,

? g = 0#

% 9,C C =CL(D, G) 2 AG [n, k, d] Fq :

* k =n_⇔degG > n₋2

+ 0_≤degG_≤n₋2

k= 1 + degG d=n₋degG.

. n=q+ 1 C %

=

       

v v . . . vn− 0

α v α v . . . αn− vn− 0

α v α v . . . αn− vn− 0

α k− _{v α} k− _{v . . . α}

n− k− vn− 1        

Fq={α , . . . , αn− } v , . . . , vn− ∈F∗q

# 1 $ k=n 8#6

k+d≤n+ 1 ⇒d≤1.

: H 8#7 !

1_≥d_≥n₋degG_⇒1₋n_{≥ −}degG_⇒degG_≥n₋1_⇒degG > n₋2.

Q n_≥1 H 6#84

degG > n₋2_≥1₋2 =₋1 = 2g₋1_⇒ℓ(G) = degG+ 1.

:

degG−n > −2⇒deg(G−D) = degG−degD= degG−n >−2

⇒ deg(G₋D)_{≥ −}1_⇒ℓ(G₋D) =deg G₋n+ 1.

H 8#7

k =ℓ(G)₋ℓ(G₋D) = degG+ 1₋degG+n₋1 =n.

2

0 ≤ degG≤n−2⇒ −2<0≤degG≤n−2< n

⇒ −2 + 2g <degG < n.

: 8#& k =ℓ(G) H 6#84 k =ℓ(G) = degG+ 1. : 8#6

k+d≤n+ 1 ⇒degG+ 1 +d≤n+ 1⇒d≤n−degG.

H 8#7

d ≥n−degG⇒d=n−degG.

3 D 1< k < n k=n ! ;

Fn

q - ! k= 1 ( !

! A - D #

z Fq(z)

D=P + +Pn Pn =P∞.

<1= <2= (

k = degG+ 1,

0_≤degG_≤n₋2,

degG <0 _L(G) =_{0_} 6#47# (

(k₋1)P∞₋G

+ A , 1

deg((k₋1)P∞₋G) = deg((k₋1)P∞)₋degG=k₋1₋k+ 1 = 0_{≥ −}1 = 2g₋1

H 6#84

ℓ((k−1)P∞−G) = deg((k−1)P∞−G) + 1 = 1.

: 6#47

(k₋1)P∞₋G= (u), 0 = u_∈Fq(z).

: *

{u, zu, . . . , zk− u_}.

ziu∈ L(G) i= 0, . . . , k−1

(ziu) =i(P ₋P∞) + (k₋1)P∞₋G=iP + (k₋1₋i)P∞₋G.

iP + (k₋1₋i)P∞_≥0_⇒iP + (k₋1₋i)P∞₋G_{≥ −}G

(ziu)_{≥ −}G_⇒ziu_{∈ L}(G).

'< # {u, zu, . . . , zk− _{u} - LI ! #F}

q

+ *

k−

i

ai(ziu) = 0.

k−

i

ai(ziu) =u k−

i

aizi ⇒ k−

i

aizi = 0 ⇒ai = 0, i,

z ! Fq#

{u, zu, . . . , zk− u}

- ! L(G) H 6#84k =ℓ(G)# : - αj =z(Pj)∈Fq

αi =αj Pi =Pj# 9! ( u(Pj)∈F∗q j = 1, . . . , n−1

( A Pj ∈/ supp(u).

( Pn = P∞# + i =

0, . . . , k−2

vP (uzi) =vP (u) +ivP (z) =k−1−i≥1⇒(uzi)(Pn) = 0(Pn).

$ i=k₋1

vP (uzi) = 0⇒(uzi)(Pn) =γ ∈F∗q.

((uzi_{)(P ), . . . ,(uz}i_)(P

n)) = (αiu(P ), . . . , αin− u(Pn− ),0), i= 1, . . . , k−2

((uzi)(P ), . . . ,(uzi)(Pn)) = (αiu(P ), . . . , αin− u(Pn− ), γ), i=k−1

: ! u γ− u vi = γ− u(Pi) ! 8#&

A * #

? 9,10

L(G) =_{uf(z) :f _∈Fq[z] degf ≤k−1}

C =_{(v f(α ), . . . , vn− f(αn− ), θ) : degf ≤k−1 θ ∈Fq},

2 RS %

! @ ! @ : !

+ , #

% 9,11

F =_{0,1,2,3,4_}

%

< 2 CL(G, D)

D=P +P +P +P +P +P∞.

p=z ₋3 F )

G=Pz − ,

) 1.20

degG= 2<6−2 = 4.

) k = 3 d= 4

1

z ₋3

% P∞ Pz −

v∞ 1

z ₋3 = 2 vP −

1

z ₋3 =−1,

1

z −3 = 2P∞−G.

'

1

z ₋3,

z

(z ₋3),

z

(z ₋3)

L(G) ? ) 1.20

1

z −3(P ) =

1

1 −3 =− 1 2,

z

z −3(P∞) = 1.

'

" D ) % CL(G, D)

  

− − 1 0

0 − 2 0 0 − 4 1

  =   

− − 1 1 0

0 − 2 0 0 − 4 1

  .

%

:

"

# $

2> #>SS H 43 0 D @ ?

+ , + , K Fn _Fm _{m, n ∈ N}

Y @ ? * ! F * D

( # ( ! (

" - Z

( ? > ' > > 2S3 (

!- - ! ! K

Fq # 9 !* ( "

-+ Z #

:,1

"

# $

q

( # 1

T " q *

[ V + * q * +

* #

" :,1 Fq V(k, q) k Fq

- V(k, q) :

k−

i

(qk₋qi).

, ! V(k, q) " 1 (

∈ V(k, q) = D ( ! V(k, q)#

V(k, q) ≃ Fk

q |V(k, q)| = qk @ qk−1 D #

V ∈ V(k, q)O -

D ( ! V(k, q)#

V(1, q) = {x :x∈Fq} ≃Fq

|V(1, q)_|=q# : @ qk_{−q D}

!

(qk₋1)(qk₋q) (qk₋qk− ) =

k−

i

(qk₋qi),

- * #

! :,/ Fq V(n, q) n Fq

* $ - k V(n, q) :

Nq(k, n) = k−

i

qn−i₋₁

qk−i₋₁.

+ ) l V(n, q)

-k V(n, q) V(l, q) Nq(k−l, n−l)

, <1= *

{ , . . . , k} { , . . . , k}

V(k, q) ! V(k, q) B >#6

Nq(k, n) = k−

i

(qn_−qi₎

k−

i

(qk_−qi₎

=

k−

i

qn−i₋₁

qk−i₋₁.

<2=

W(n₋l, q) = V(n, q)

V(l, q) ≃F

n−l q .

$ V(k, q)- ! V(n, q) V(l, q)

W(k₋l, q) = V(k, q)

V(l, q) ≃F

k−l q .

- ! k−l W(n−l, q)# : <1= N !

k−l W(n−l, q) - Nq(k−l, n−l)#

W(n₋l, q)

W(k−l, q) ≃

V(n, q)

V(k, q)

N ! k V(n, q) V(l, q)

-Nq(k−l, n−l)#

N ( K@ n k d S ={C ≤Fn

q :C- [n, k, ≥

d] * }# Mq(n, k, d) =|S|#

% :,9 ) - n k d Mq(n, k, d)≤

qn₋₁

qk₋₁

# $ * S * q [n, k,_≥ d] *

|S_|=Mq(n, k, d).

- * C− { } * C ∈S

C∈S

(C− { })⊆Fnq − { }.

|C_|=qk

qn₋1 = Fn_{q − { } ≥}

C∈S

(C_{− { }}) =

C∈S

(_|C_{| −}1) =_|S_|(qk₋1),

- * #

" :,: C 2 q/#

n k < d

|C| ≤Mq(n, k, d).

# C _{∈ C} N DC DC ⊆C dimDC =k#

*

D =_{DC :C ∈ C}

- |C| = |D|# ( * >#8

|D| ≤Mq(n, k, d)# |C| ≤Mq(n, k, d)#

? :,; 2 : Fq

Φ∗ _{: F}n

q → Fmnq

( , . . . , n) → (Φ( ), . . . ,Φ( n))

) 3.3 ) ! _{Ci}i∈I 2

Fq !

{Φ∗(Ci)}i∈I

2 Φ∗

" :,= S 2 qt_{/# [n, k,≥d]}

A 2 /# [m, t,_≥s]

Mq(nm, kt, ds)≥ |S|.

, $ * S :=_{Ci}i∈I# 9! ( >#%

{Φ∗(Ci)}i∈I

- + " [nm, kt,_≥ds] * # B * ( Φ∗ >#8 !

|{Ci}i∈I|=|{Φ∗(Ci)}i∈I| ≤Mq(nm, kt, ds),

- * #

$ * R δ N ( (0,1) K

Aq(R, δ) = sup

σ lim supn→∞

log_qMq(n, kn, dn)

n ,

- ! ? σ (n, kn, dn)∈N

n∈N

lim

n→∞ kn

n =R nlim→∞

dn

n =δ.

>#8

Mq(n, kn, dn)≤

qn₋₁

qk ₋₁.

Aq(R, δ)≤1−R# +

qn₋₁

qk ₋₁ ≤

qn

qk ₋₁ =

qn−k

q_{− q} k −

,

q _≥ 2 1

q

k −

≤1_⇒q₋ 1 q

k − ≥1

⇒ 1

q_{− q} k −

≤1⇒ q

n−k

q_{− q} k −

≤qn−k .

:

log_qMq(n, kn, dn)

n ≤

log_qqn−k

n = 1−

kn n + 1 n ⇒ lim sup n→∞

log_qMq(n, kn, dn)

n ≤ lim supn→∞ 1− kn

n +

1

n = 1−R.

Aq(R, δ) = sup σ

lim sup

n→∞

log_qMq(n, kn, dn)

n ≤ 1−R.

: - A q = 2

0 D 2> H 8 # %553#

% :,A ) k, n_∈N n > k _≥2

Mq(n, k, d)≥Bq(n, k, d) =

   

Nq(k, n)− S n,d_q− Nq(k−1, n−1)

(qk_−1)(N

q(k−1, n−1)−1)

   ,

⌈x⌉= min{n∈Z:x≤n} Sq(n, d) = { ∈Fnq : 1≤ωH( ) ≤d−1} .

, $ * C * q n

k +

Sd ={ ∈Fnq : 1≤ωH( ) ≤d−1}.

C∩Sd=∅ C ∈ C T " d# (

Sd ( 1 ! Fq <2= >#4

N _C ( _∈Sd - Nq(k−1, n−1)# :

@

Nq(k, n)−Nq(k−1, n−1)

Sq(n, d)

q−1 <>#6=

C Sd# B @ <>#6= - N

C Sd ( Sd# $ * C

C#

|{C_{∈ C} :C_∩C =_{{ }}| ≤}(qk₋1)(Nq(k−1, n−1)−1),

qk_{−1 ( C @ N}

q(k−1, n−1)−1 #

D ( C C (

C .: M - [n, k, d] *

(

Nq(k, n)−Nq(k−1, n−1)

Sq(n, d)

q₋1 −[(M −1)(q

k_−1)(N

q(k−1, n−1)−1)]≥0.

Mq(n, k, d)≥

   

Nq(k, n)− S n,d_q− Nq(k−1, n−1)

(qk_−1)(N

q(k−1, n−1)−1)

   ,

- * #

! >#7 K R δ K@ ( (0,1)

Bq(R, δ) = sup σ

lim sup

n→∞

log_qBq(n, k, d)

n ,

- ! ? σ (n, kn, dn)∈N

n∈N

lim

n→∞ kn

n =R nlim→∞

dn

n =δ.

Bq(R, δ)≤sup σ

lim sup

n→∞

1

nlogq

Nq(kn, n)

qk _N

q(kn−1, n−1)

.

+ ! ( n > kn≥2

qn− ₋₁

qk − ₋₁ ≥

qn−

qk − =q

n−k _≥q.

:

Nq(kn−1, n−1) =

qn− −1

qk − ₋₁

k −

i

qn− −i₋₁

qk − −i₋₁ ⇒Nq(kn−1, n−1)≥q. <>#4=

K ( N = Nq(kn, n) S = Sq(n, dn) N = Nq(kn−

1, n₋1)# B    

N − S

q− N

(qk _{−1)(N −1)}

  

 ≤ 2   N −

S q− N

(qk _{−1)(N −1)}

 

≤ 2N

(qk _{−1)(N −1)} =

q N q N

q− q

k −

q−Nq

A <>#4= !

q N q N

q_{− q} k − q_{− N}q

≤ _q2_kq N_N .

log_qBq(n, kn, dn)

n ≤

1

nlogq

2q Nq(kn, n)

qk _N

q(kn−1, n−1)

= 1

nlogq

Nq(kn, n)

qk _N

q(kn−1, n−1)

+ 2

n+

1

nlogq2

sup

σ lim supn→∞

log_qBq(n, kn, dn)

n ≤supσ lim supn→∞

1

nlogq

Nq(kn, n)

qk _N

q(kn−1, n−1)

,

-Bq(R, δ)≤sup σ

lim sup

n→∞

1

nlogq

Nq(kn, n)

qk _N

q(kn−1, n−1)

.

% :,B "

Bq(R, δ)≤1−2R.

,

Nq(kn, n)

Nq(kn−1, n−1)

= q

n₋₁

qk ₋₁

qn₋₁

qk _(qk ₋₁₎ ≤

qn

qk _(qk ₋₁₎ =

qq q

q− q

k − ≤ qqn

q k =q

n − k _.

: sup σ lim sup n→∞ 1

nlogq

Nq(kn, n)

qk _N

q(kn−1, n−1) ≤

lim

n→∞

1

nlogq(q

n − k _{) = 1}

−2R.

Bq(R, δ)≤1−2R#

% :,C 2 q/# [m, t,_≥s]

Aq Rt m, δs m ≥ t

mAq (R, δ).

, 0 (

Aq (R, δ) = sup σ

lim sup

n→∞

logq Mq(n, kn, dn)

n ,

- ! ? σ (n, kn, dn)∈N

n_∈N

lim

n→∞ kn

n =R nlim→∞

dn

n =δ.

: N ǫ >0 @ ? σ

lim sup

n→∞

logq Mq (n, kn, dn)

n > Aq (R, δ)−ǫ

B >#' !

Mq(nm, knt, dns)≥Mq (n, kn, dn),

n∈N#

1

nmlogqMq(nm, knt, dns)≥ t

nmlogq Mq (n, kn, dn),

B

Aq

Rt

m,

δs

m ≥ lim supn→∞

log_qMq(mn, knt, dns)

mn

≥ _mt lim sup

n→∞

log_q Mq (n, kn, dn)

n

> t

m(Aq (R, δ)−ǫ). ǫ >0- !

Aq Rt m, δs m ≥ t

mAq (R, δ),

- * #

:,/

&

"

# $

( * (

+ , -! # : (- ( ! T R δ

Aq(R, δ)> Bq(R, δ)#

@ ( K , 1 *

D =

P∈P

mPP D =

P∈P

nPP

( F/Fq#

D _∨D =

P∈P

max_{mP, nP}P D ∧D = P∈P

min_{mP, nP}P.

% :,10 ) D D F/Fq

* _L(D ) +L(D )⊆ L(D ∨D )

+ _L(D )_{∩ L}(D ) =_L(D _∧D )

# 1 g ∈ L(D ) +L(D ) @ f ∈ L(D ) f ∈ L(D ) g =f +f # 0

(

vP(f +f )≥min{vP(f ), vP(f )}.

$ D min{vP(f ), vP(f )}=vP(f )#

vP(f ) ≥ −vP(D )≥ −max{vP(D ), vP(D )} ⇒vP(g)≥ −vP(D ∨D )

⇒ g _{∈ L}(D _∨D ),

* L(D ) +_L(D )_{⊆ L}(D _∨D )#

2 f _{∈ L}(D )_{∩ L}(D ) ! vP(f)≥ −vP(D ) vP(f)≥ −vP(D )# :

vP(f)≥ −min{vP(D ), vP(D )} ⇒vP(f)≥ −vP(D ∧D )⇒f ∈ L(D ∧D ),

* L(D )_{∩ L}(D )_{⊆ L}(D _∧D )#

Q f _{∈ L}(D _∧D ) !

vP(f) ≥ −vP(D ∧D ) =−min{vP(D ), vP(D )}

⇒ vP(f)≥ −vP(D ) vP(f)≥ −vP(D )

⇒ f _{∈ L}(D )_{∩ L}(D ),

* L(D ∧D )⊆ L(D )∩ L(D )#

I K Fq ⊆ L(D) D - ( ( F/Fq# :

(1, . . . ,1) _⊆CL(_P, D),

P =P + +PnO degP = = degPn= 1OP , . . . , Pn *

suppD_∩supp_P =_∅.

" :,11 F/Fq ! " x ∈ F

Fq F/Fq Fq(x)/Fq & P′ #

P P′ _∈P

F P ∈PF x deg P′ = 1 deg P = 1

# $ P′ P (Fq(x))_P - ! FP′# $ (F_q(x))

P FP′

( ! Fq (Fq(x))_P ! ( FP′#

:

degP′ = 1_⇔[FP′ :F_q] = 1⇒[(F_q(x))

P :Fq] = 1⇔deg P = 1,

- * #

@ :,1/ _P =P + +Pn P , . . . , Pn

F/Fq D D F/Fq

suppDj∩suppP =∅, j = 1,2.

deg(D ∨D ) < n deg(D ∧D ) < n_q C ∩C = { } Cj

CL(_P, Dj)

Cj⊕ (1, . . . ,1) =CL(P, Dj), j = 1,2.

, ( ( L(D )_{∩ L}(D ) = Fq# $ D !

L(D )_{∩ L}(D ) =Fq# @

x_{∈ L}(D )_{∩ L}(D ) =_L(D _∧D )

x /_∈Fq# : x∈F - ! Fq#

x _{∈ L}(D )_{∩ L}(D )_⇒vP(x)≥ −vP(D ∧D )⇒ −vP(x)≤vP(D ∧D )

⇒ −vP(x) degP ≤vP(D ∧D ) degP.

P ∈PF - ( x H 6#4'

deg((x)∞)≤deg(D ∧D )⇒[F :Fq(x)] = r= deg((x)∞)≤deg(D ∧D ).

@ -! K F/Fq Fq(x)/Fq# 4#7 !

@ r 1 Pα α ∈Fq. $ P∞ N A

1

x,

x∈ L(D ∧D ) !

vP i(x)≥0, 1≤i≤n, ⇒ −vP i(x)≤0⇒vP i

1

x =−vP i(x)≤0⇒Pi ∤P∞,

N ( 4#%# @ @

N₋n 1 P∞ N =N(F) N

F/Fq# 4#% PF N PF x

B >#66 ! @ 1 1#

N =





α∈F

να



+ν∞,

να N 1 Pα ν∞ N

1 P∞# : - να ≤r ν∞≤N −n

N =

 

α∈F

να



+ν∞≤qr+N −n⇒r≥ n q.

deg(D ∧D )≥ n q,

- #

* ∈C ∩C # @ fj ∈ L(Dj) j = 1,2

= (f (P ), . . . , f (Pn)) = (f (P ), . . . , f (Pn))⇒(f −f )(Pi) = 0(Pi).

vP i(f −f )≥1, 1≤i≤n,

f ₋f _∈Pi# :

-f −f ∈ L(D ) +L(D )⊆ L(D ∨D ).

:

vP(f −f )≥ −vP((D ∨D )− P)⇒f −f ∈ L((D ∨D )− P).

deg((D ∨D )− P) = deg((D ∨D )−n <0,

deg(D ∨D )< n# B B 6#4>

L((D ∨D )− P) = 0⇒f −f = 0.

f =f _{∈ L}(D )_{∩ L}(D ) =Fq,

- = (β, . . . , β) β_∈Fq# = C ∩C ={ }#

? :,19 D D F/Fq

vP(D ) = 0, ∀ P ∈suppD vP(D ) = 0, ∀ P ∈suppD .

?

min{vP(D ), vP(D )} ≤0, ∀ P ∈suppD ∪suppD .

)

deg(D ∧D )≤0.

! D _∧D =_{0_}

(

deg(D _∧D )_≤0< n q

deg(D ∧D )< n q

!

@ :,1: F/Fq ! " 7 g n

( Br - F r

max_{1, g_}< r < n

2, :

* ) r= 2 3

Mq(n, r−g, n−r)≥Br.

+ ) r≥4

Mq(n, r−g, n−r)≥Br+

⌊ ⌋

j

min_{Bj, Br−j}.

# <1= r ≥2, @ Br * r−g

T " n−r# + *

D , . . . , DB

F # - ( ( (

* # :

-deg(Di∨Dj) = deg P∈P

max{vP(Di), vP(Dj)}P ≤deg(Di+Dj)

= r+r < n

2 +

n

2 =n, ∀ i, j ∈ {1, . . . , Br}, i=j.

: P , . . . , Pn 1 >

P =P + +Pn.

Di - r

suppDi∩suppP =∅.

$ * Ci ! CL(P, Di)

Ci⊕ (1, . . . ,1) =CL(P, Di), i= 1,2, . . . , Br.

9! ( >#68 H >#64

C , . . . , CB

q * 8#&

dim(Ci) = dim (CL(P, Di))−1 =ℓ(Di)−1≥r−g,

d(Ci)≥n−r.

B >#>

Mq(n, r−g, n−r)≥Br.

<2= : - Br * ! <1= (

⌊ ⌋

j

min_{Bj, Br−j}.

* # + r _≥4 @

M =

⌊ ⌋

j

min{Bj, Br−j}.

+ E+G E j G r−j *

( # (

( + E+G P - ( A # Y (

<1= ! M q * # : B >#>

Mq(n, r−g, n−r)≥Br+

⌊ ⌋

j

min{Bj, Br−j},

- * #

% :,1; F ! " Fq q ≥ 8 Ni

-Fi =FFq F/Fq 7 0 ) 2.8 Fi/Fq

7 0 ) ) 2.14 Fi/Fq ! " )

r= 4, n=q+ 1 >8 (2.3)

B = 1

2 < 2

1 N +< 2

2 N = 1

2((−1)(q+ 1) + 1(q + 1)) =

q −q

2

B = 1

4 < 4

1 N +< 4

2 N +< 4

4 N = 1

4(0 + (−1)(q + 1) + 1(q + 1)) =

q −q

4 .

1< r < n

2

, 4.14

Mq(q+ 1,4, q−3)≥B +B =

q +q ₋2q

4 .

! +

Mq(n, r−g, n−r)

H >#6> - Br, ( ( #

@ :,1= q - q_≥16

1

√_q−1 < λ < 1

2,

Aq λ−

1

√_q−1,1₋λ _≥λ.

q_≥49 1

3 + 2

3(√q₋1) < λ < 1 2,

Aq(R,1−λ)>1−2R,

R=λ− √_q1₋₁.

# $ * {F/Fq} + " + , ? g = g(F)

lim

g→∞ N(F)

g =

√_q −1,

N(F) - N F/Fq <( * 243 + " =#

$ * n=N(F) Br N F r =⌊nλ⌋# I K

g < r < n

2,

n K # H >#6>

Mq(n, r−g, n−r)≥Br.

$ * Ni N qi FFq# J R !

qi+ 1₋2gq _≤Ni ≤qi+ 1 + 2gq .