Сидельников В.М. Теория кодирования. Справочник по принципам и методам кодирования
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K ∈ F
n
q
η
j
= η
j
(K), j = 0, . . . , n,
K j η
0
= 1 η
j
= 0, j = 1, . . . , d − 1, d
K η(K) = (η
0
, . . . , η
n
) K
K K
⊥
η(K
⊥
)
η(K) K
η(K)
η(K
⊥
)
W
K
(x) K
W
K
(x, y) =
n
X
j=0
η
j
x
j
y
n−j
,
x, y w(a)
a ∈ F
n
q
W
K
(x, y) =
X
a∈K
x
w(a)
y
n−w(a)
.
ψ
K
(a) =
(
1, a ∈ K
0, a 6∈ K.
K
ψ
K
(x)
W
K
(x, y) =
X
a∈F
n
q
ψ
K
(a)x
w(a)
y
n−w(a)
.
ψ
K
(x)
K
⊥
K K
⊥
ψ
K
(x)
K
⊥
W
K
(x, y) =
1
|K
⊥
|
W
K
⊥
(y − x, y + (q − 1)x) =
=
1
|K
⊥
|
n
X
s=0
ν
s
(y − x)
s
(y + (q − 1)x)
n−s
,
ν
s
s K
⊥
l(x) F
p
F
q
F
p
ψ
K
(x) K
ψ
K
(
x
) =
1
|K
⊥
|
X
y∈K
⊥
exp
µ
2πi l(hx, yi)
p
¶
,
x
∈
F
n
q
.
x
K hx, yi = 0 y ∈ K
⊥
hx, yi = x
1
y
1
+ ···+ x
n
y
n
F
q
x 6∈ K
y
0
∈ K
⊥
hx, y
0
i 6= 0 b ∈ F
q
by
0
K
⊥
b ∈ F
q
X
y∈K
⊥
exp
µ
2πi l(hx, yi)
p
¶
=
X
y∈K
⊥
exp
µ
2πi l(hx, y + by
0
i)
p
¶
= exp
µ
2πi l(bhx, y
0
i)
p
¶
X
y∈K
⊥
exp
µ
2πi l(ahx, yi)
p
¶
.
l(·) hx, y
0
i 6= 0 b
0
∈ F
q
l(b
0
hx, y
0
i) 6= 0 exp
³
2πi l(b
0
hx,y
0
i)
p
´
1 ψ
K
(x) = exp
³
2πi l(b
0
hx,y
0
i)
p
´
ψ
K
(x)
ψ
K
(x) = 0
x ∈ K ψ
K
(x) = 1 ¤
F
q
q = p
ψ
K
(x) K
ψ
K
(x) =
1
|K
⊥
|
X
y∈K
⊥
exp
µ
2πi hx, yi
p
¶
, x ∈ F
n
q
. ¤
ψ
K
(x)
W
K
(x, y) =
1
|K
⊥
|
X
y∈K
⊥
X
x∈F
n
q
x
w(x)
y
n−w(x)
exp
µ
2πi l(hx, yi)
p
¶
.
w(x) = w(x
1
) + ··· + w(x
1
), x = (x
1
, . . . , x
n
) ∈ F
n
q
w(x) =
(
1, x 6= 0
0, x = 0
X
x∈F
n
q
x
w(x)
y
n−w(x)
exp
µ
2πi l(hx, yi)
p
¶
=
n
Y
s=1
X
x
s
∈F
q
x
w(x
s
)
y
1−w(x
s
)
exp
µ
2πi l(x
s
y
s
)
p
¶
.
X
x
s
∈F
q
x
w(x
s
)
y
1−w(x
s
)
exp
µ
2πi l(x
s
y
s
)
p
¶
=
(
(y − x), y
s
6= 0
y + (q − 1)x, y
s
= 0
.
a 6= 0
X
x∈F
n
q
x
w(x)
exp
µ
2πi l(ahx, yi)
p
¶
= ((1 − x))
w(y)
(1 + (q − 1)x)
n−w(y)
.
¤
l(x)
F
q
F
p
K K
⊥
x
s
y
n−s
q = p
η
w
=
1
|K
⊥
|
n
X
s=0
P
w
(s)ν
s
,
P
w
(s) = K
(p,n)
w
(x)
η(K) η(K
⊥
) = (ν
0
, . . . , ν
n
)
W (x, y) η(K)
K K
⊥
K
wt(x)
H ⊂ Aut(F
n
p
) H
F
n
p
F
n
p
H F
n
p
F
n
q
H n + 1 A
0
, . . . , A
n
A
w
x ∈ F
n
p
w wt(x) A
w
w
η
w
=
X
wt(x)=w
ψ
K
(x),
ψ
K
(x) K
P
wt(x)=w
x F
n
p
w
Φ
w
(y) =
X
wt(x)=w
exp
µ
2πi hx, yi)
p
¶
wt(y) = s Φ
x
(y) y
H
Φ
H
(x, $) = S
H
(s)P
H
(s, w) =
e
S
e
H
(w)
e
P
e
H
(w, s), x ∈ A
s
, $ ∈
e
A
w
,
Φ
w
(x) Y
w
(y) =
P
$∈
e
A
w
$(y) y = y
Φ
w
(x) = P
H
(s, w) = P
w
(s),
wt(x) = s
ψ
K
(x)
F
n
p
H
H
c
k
(x) x
k ∈ F
p
c(x) = (c
0
(x), . . . , c
p−1
(x)) x
x
c(x)
F
n
p
S
n
F
n
p
κ
c
= κ
c
(K) x K c(x)
c
κ = κ(K) = {κ
c
(K) |c = (c
0
, . . . , c
p−1
) c
0
+ ··· + c
p−1
= n
K
κ
c
=
X
c(x)=c
ψ
K
(x) =
1
|K
⊥
|
X
y∈K
⊥
X
c(x)=c
exp
µ
2πi hx, yi
p
¶
,
ψ
K
(x) K
P
c(x)=c
x ∈ F
p
c(x) = c
c
k
(y) = w
k
, k = 0, . . . , p−1, w
0
+···w
p−1
= n w = (w
0
, . . . , w
p−1
)
Φ
c
(y) =
X
c(x)=c
exp
µ
2πi hx, yi
p
¶
,
c(y) = w Φ
c
(y) = Φ
c
(y
0
)
c(y) = c(y
0
)
c(y) = w
p 0, . . . , p −1 c
0
, . . . , c
p−1
p w
0
, . . . , w
p−1
Φ
c
(y) := p
c
(w) =
g
X
µ
w
0
c
0,0
, . . . , c
0,p−1
¶
···
µ
w
p−1
c
p−1,0
, . . . , c
p−1,p−1
¶
exp
Ã
2πi
P
p−1
t=0
stc
t,s
p
!
,
c
t,s
t x s y
f
P
c
(j)
= (c
j,0
, . . . , c
j,p−1
), j =
0, . . . , p −1,
P
p−1
s=0
c
j,s
= c
j
, j = 0, . . . , p − 1,
P
p−1
t=0
c
t,s
= w
s
, s = 0, . . . , p − 1.
z
0
, . . . , z
m
p
c
(w) = coeff
z
c
0
0
···z
c
p−1
p−1
l
Y
j=0
Ã
m
X
s=0
exp
µ
2πi s
p
¶
z
s
!
w
t
.
κ
c
=
X
w
p
c
(w)ν
w
(K
⊥
),
ν
w
(K
⊥
) K
⊥
w
X
c
κ
c
(K)z
c
0
0
z
c
1
1
···z
c
p−1
p−1
=
1
|K
⊥
|
X
w
ν
w
(K
⊥
)
p−1
Y
j=0
µ
z
0
+ exp
µ
2πij
p
¶
z
1
+ ··· + exp
µ
2πi(p −1)j
p
¶
z
p−1
¶
w
j
,
P
c
P
w
c = (c
0
, . . . , c
p−1
)
w = (w
0
, . . . , w
p−1
) c
0
+ ··· + c
p−1
= w
0
+ ··· + w
p−1
= n
p
c
(w)
w
0
, . . . , w
p−1
w
0
+ ··· + w
p−1
= n p
c
(w) p − 1
p
c
(w) p = 2
p
c
(w)
C
w
=
¡
n
w
0
,...,w
p−1
¢
1
p
n
X
w
C
w
p
c
(w)p
c
0
(w) =
(
0, c 6= c
0
C
c
, c = c
0
.
K F
p
p n
k η
s
= η
s
(K)
K s η(K) = (η
0
, η
1
, . . . , η
n
) K
W
K
(x, y) =
n
X
j=0
η
j
x
j
y
n−j
=
X
a∈K
x
wt(a)
y
n−wt(a)
,
wt(x) x K
K
⊥
F
n
p
K K
⊥
b
∈
F
n
p
(
b
,
a
) =
0 a
∈
K (
b
,
a
) =
P
n
j=1
a
j
b
j
F
n
p
η(K) η(K
⊥
) = (ν
0
, ν
1
, . . . , ν
n
)
η(K) K
K
⊥
K
K = R
m−2
m−2−
n = 2
m
k = 2
m
− m − 1
B =
1 1 ··· 1 1
a
1,1
a
1,2
··· a
1,2
m
−1
a
1,2
m
a
2,1
a
2,2
··· a
2,2
m
−1
a
2,2
m
a
m,1
a
m,2
··· a
m,2
m
−1
a
m,2
m
, a
i,j
∈ F
2
,
a
1,j
a
2,j
a
m,j
, j = 1, . . . , 2
m
, F
2
B
F
2
B
K = R
m−2
4
R
m−1
4
R
⊥
m−2
= R
1
m + 1
B
η(R
⊥
m−2
) = (1, 0, . . . , 0, 2(n −1), 0, . . . , 0, 1),
2(n − 1)
n
2
= 2
m−1
R
m−2
W
R
m−2
(x, y) =
1
2n
¡
(x + y)
n
+ 2(n − 1)(x + y)
n/2
(y − x)
n/2
+ (y − x)
n
¢
x
w
y
n−w
η
w
=
1
2n
µµ
n
w
¶
(K
(2,n)
w
(0) + K
(2,n)
w
(n)) + 2(n − 1)K
(2,n)
w
³
n
2
´
¶
,
K
(2,n)
w
(c) = coeff
x
c
y
n−c
(y −x)
w
(x+y)
n−w
K
(2,n)
w
(0) =
¡
n
w
¢
K
(2,n)
w
(n) = (−1)
w
¡
n
w
¢
(y−x)
n
2
(y+x)
n
2
= (y
2
−x
2
)
n
2
K
(2,n)
w
(
n
2
) = 0 w K
(2)
2t
(
n
2
) = (−1)
t
¡
n/2
t
¢
w = 2t
η
w
=
(
0, w
1
n
³
¡
n
2t
¢
+ (−1)
t
(n − 1)
¡
n
2
t
¢
´
, w = 2t
.
R
m−2
R
m−2
R
⊥
m − 2 = R
1
η
0
= 1, η
1
= η
2
= η
3
= 0
η
4
=
1
n
µµ
n
4
¶
+ 2(n − 1)
µ
n
2
2
¶¶
.
R
m−2
4
X n S(t, w, n) = {B
1
, . . . , B
N
}
w− B
j
X
t− X S(t, w, n)
S(2, 3, n)
{B
1
, . . . , B
N
} B
j
X = {1, . . . , 2
m
}
4 R
m−2
S(3, 4, n) N = η
4
η
4
=
(
0, w ,
1
n
¡
n
w
¢
(1 + ²
4,n
), w ,
²
4,n
∼
1
n
, n → ∞
K
(p,n)
w
(x)
K
η
w
K
⊥
η
w
x
K
K
⊥
|K
(p,n)
w
(x)|
x = 0 x = n K
(p,n)
w
(0) =
¡
n
w
¢
(p − 1)
w
K
(p,n)
w
(n) =
¡
n
w
¢
(−1)
w
|K
(p,n)
w
(x)|
x =
(p−1)n
p
(1 + ²), n → ∞
|K
(p)
w
(x)| ¿ K
(p)
w
(0) = (p −1)
j
¡
n
w
¢
(p−1)n
p
w = 0 w = n
η
w
≈
1
|K
⊥
|
µ
n
w
¶
((p − 1)
w
+ (−1)
w
ν
n
),
ν
n
n K
⊥
K
η
w
n = 2
m
− 1 n → ∞ t = o(
√
n)
P
n−1
s=1
ν
s
K
(2,n)
s
(x) ¿
¡
n
j
¢
η
w
K
χ
2
K
K F
p
K n
Ξ(K) =
1
|K|
n
X
w=1
(η
w
− ω(w)|K|)
2
ω(j)
,
ω(w) = ω
p
(w) :=
(p−1)
w
(
n
w
)
p
n
ω(w) w
F
n
p
ω(w)
C
√
n
w ∼
(p−1)n
p
w |w −
(p−1)n
p
| ≥ c ·n, c > 0, ω(w) n
ω(
(p−1)n
p
)
Ξ(K)
η
w
K ω(w)|K|
1
ω(w)
η
w
w
p−1
p
n
ω(w) = (p−1)
w
¡
n
w
¢
p
−n
η
w
w ω(w)
|η
w
−ω(w)|K|
p
ω(w)|K| ≈
√
η
w
n k
n − k
Ξ(K)
ν K
⊥
K
Ξ(K)
K
⊥
Ξ(K)
P (x) ≤ n
P (x) =
n
X
t=0
α
t
K
(p,n)
t
(x)
{K
(p,n)
t
(x)|t = 0, . . . , n}
η
j
K ν
j
K
⊥
1
|K
⊥
|
n
X
t=0
(ν
t
− ω(t)P (t))
2
ω(t)
=
1
|K|
n
X
s=0
(η
s
− ω(s)|K|α
s
)
2
ω(s)
.
ω(j) =
(p−1)
j
(
n
j
)
p
n
K K
⊥
ν
j
=
1
|K|
n
X
x=0
η
x
K
(p,n)
j
(x).
K
(p,n)
j
(s)ω(s) = K
(p,n)
s
(j)ω(j).
ν
j
ω(j)
=
1
|K|
n
X
s=0
η
s
K
(p,n)
s
(j)
ω(s)
.
ν
j
ω(j)
− P (j) =
n
X
s=0
µ
η
s
ω(s)|K|
− α
s
¶
K
(p,n)
s
(j).
ω(j)
j 0 n
K
(p)
j
(s)
n
X
t=0
(ν
j
− ω(j)P (j))
2
ω(j)
=
p
n
|K|
2
n
X
s=0
(η
s
− ω(s)|K|α
s
)
2
ω(s)
,
|K| · |K
⊥
| = p
n
¤
P (x)