Сидельников В.М. Теория кодирования. Справочник по принципам и методам кодирования
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f(x) H
f(x) =
m
X
w=0
b
w
e
P
e
H
(w, s), b
w
∈ C, x ∈ A
s
.
Y
w
(x)
H (f(x), f
0
(x))
f(x) = Y
w
(x) f
0
(x) = Y
w
0
(x) G = ∪
m
s=0
A
s
G
(Y
w
(x), Y
w
0
(x)) =
1
|G|
X
x∈G
Y
w
(x)Y
w
0
(x) =
1
|G|
m
X
s=0
X
x∈A
s
Y
w
(x)Y
w
0
(x) =
1
|G|
m
X
s=0
|A
s
|
e
P
e
H
(w, s)
e
P
e
H
(w
0
, s).
w 6= w
0
e
A
w
∩
e
A
ew
0
= ∅
(Y
w
(x), Y
w
0
(x)) = 0
(Y
w
(x), Y
w
(x)) = (
X
$∈
e
A
w
$(x),
X
$∈
e
A
w
$(x)) = |
e
A
w
|.
¤
P
e
H
(w, s)
H G
H Aut(
e
G) P
e
H
(w, s)
e
A
w
P
e
H
(w, s)
w
G
l
= (F
l
p
, +) l−
a = (a
1
, . . . , a
l
) ∈ F
l
p
mod p p
Aut(G) (F
l
p
, +)
$ ∈ Aut(G)
$ : a → aA, A ∈ M
l
(F
p
) M
l
(F
p
)
l ×l− F
p
Aut(G
l
)
M
l
(F
p
)
l > 1 Aut(G
l
)
Aut(G
l
)
F(G
l
) a ∈ F
l
p
F
p
l
F
l
p
F
p
l
F
p
l
a = (a
1
, . . . , a
l
) ∈ F
l
p
b
a F
q
, q = p
l
,
b
a =
P
l
i=1
a
i
ω
i
Ω = {ω
1
, . . . , ω
l
} F
q
F
p
α
b
a, α ∈ F
∗
q
aA
α
α
b
a Ω A
α
F(G
l
) A
α
, α ∈ F
q
, α 6= 0 F(G
l
) ∼ F
∗
p
l
F
p
l
|F(G
l
)| = p
l
− 1.
F(G
l
)
G
l
= F
l
p
a, b
F(G
l
) σ a b σ : a → b
G
l
A
0
= {0}
A
1
= G
l
r {0} F(G
l
)
G
n
l
n G
l
x = (x
1
, . . . , x
n
), x
s
∈ G
l
, ln
F
p
G
n
l
n F
q
, q = p
l
G
n
l
σ : (x
1
, . . . , x
n
) → (x
i
1
α
1
, . . . , x
i
n
α
n
), α
s
∈ F
q
r {0},
λ =
µ
1 2 ··· n − 1 n
i
1
i
2
··· i
n−1
i
n
¶
F
n
q
λ i
s
∈ {1, 2, . . . , n}
S
n
n!
S
n
{1, 2, . . . , n} S
n
r−
i
1
, . . . , i
r
j
1
, . . . , j
r
S
n
σ
σ : x → xΛ,
Λ = Γ ·D D
α
s
F
q
Γ n × n−
λ x
Λ
F
q
Λ
M
n
(F
q
)
n!(q − 1)
n
M
n
(F
q
) Aut(G
n
l
)
F(G
l
)
G
l
= F
l
p
r− r, 0 < r ≤ n
A
s
a ∈ F
n
q
s
M
n
(F
q
)
A
s
|A
s
| =
¡
n
s
¢
(q − 1)
s
G
n
l
G
l
G
l
l− F
p
x = (x
1
, . . . , x
l
)
f
G
l
$
a
(x) = exp
µ
2π i ha, xi
p
¶
, a ∈ G
l
,
ha, xi =
P
l
k=1
a
k
x
k
F
p
G
l
G
l
f
G
l
ρ a ∈ G
l
$
a
(x)
σ F(G
l
) ⊂ Aut(G
l
)
a ∈ G
l
ϕ ∈ Aut(G
l
)
$
a
(x
ϕ
) = exp
µ
2π i ha, x
ϕ
i
p
¶
= exp
Ã
2π i ha
ϕ
T
, xi
p
!
= $
a
0
(x),
ϕ
T
ϕ a
0
= a
ϕ
T
ϕ : x → xA A
l × l− F
p
ϕ
T
: x → xA
T
A
T
A
eρ G
l
f
G
l
ϕ ∈ Aut(G
l
)
eϕ : $
a
(x) → $
a
ϕ
T
(x) = $
a
0
(x) = $
eϕ
a
(x)
e
G a
0
= a
ϕ
T
eϕ $
a
(x) → $
a
(x
ϕ
)
G
n
l
G
n
l
$
a
(x) = exp
µ
2π i ha, xi
p
¶
, a ∈ G
n
l
,
ha, xi, a, x ∈ G
n
l
, a = (a
1
, . . . , a
n
), F
p
ha, xi = ha
1
, x
1
i + ··· + ha
n
, x
n
i
G
l
Λ = Γ ·D M
n
(F
p
) G
n
l
σ σ : x → xΛ
$
a
(x
σ
) = $
a
0
(x) σ
a
0
= aΛ
e
A
w
$
a
(x)
wt(a) = w
f
M
n
(F
q
) Aut(G
l
)
f
M
n
(F
q
)
M
n
(F
q
)
f
G
n
e
A
w
|
e
A
w
| =
¡
n
w
¢
(q − 1)
w
p
l
P
l
Z
p
l
mod
p
l
p p
l
P
l
= (Z
p
l
, +)
l−1 p
s
, s = 1, . . . , l−
1 n− P
l
P
n
l
== (Z
n
p
l
, +)
Aut(P
l
) P
l
σ : x → ax, a ∈ P
l
(a, p) = 1 .
|Aut(P
l
)| = ϕ(p
l
) = p
l−1
(p − 1) ϕ
P
l
1 + l A
s
, s = 0, . . . , l
Aut(P
l
) A
s
a ∈ Z
p
l
p
l−s
p
l−s+1
|A
s
| = ϕ(p
s
)
Aut(P
n
l
), n > 1, P
n
l
= (Z
n
p
l
, +)
M
n
(P
l
) M
n
(G
l
)
M
n
(P
l
)
σ : (x
1
, . . . , x
n
) → (x
σ
1
i
1
, . . . , x
σ
n
i
n
), σ
s
∈ Aut(P
l
), x
j
∈ P
l
,
λ =
µ
1 2 ··· n − 1 n
i
1
i
2
··· i
n−1
i
n
¶
P
l
|M
n
(P
l
)| = (p − 1)p
l−1
n!
c
s
(x), x ∈ P
n
l
, x
A
s
c(x) l + 1− c(x) =
(c
0
, c
1
(x), . . . , c
l
(x))
c = (c
0
, . . . , c
l
), c
0
+ ···+ c
l
= n, l + 1−
A
c
a ∈ P
n
l
c(x) =
c M
n
(P
n
l
)
A
c
|A
c
| =
¡
n
c
0
,...,c
l
¢
|A
0
|
c
0
···|A
l
|
c
l
¡
n
c
0
,...,c
l
¢
=
n!
c
0
!···c
l
!
|A
0
| = 1 |A
c
| =
¡
n
c
0
,...,c
l
¢
|A
1
|
c
1
···|A
l
|
c
l
$ P
l
$
a
(x) = exp
µ
2π i a · x
p
l
¶
, a ∈ P
l
.
P
l
f
P
l
$
a
(x) = exp
µ
2π i ha, xi
p
¶
, a ∈ P
n
l
,
ha, xi Z
p
l
P
n
l
f
P
l
P
l
f
P
l
e
A
w
, w = (w
0
, . . . , w
l
)
H
e
P
e
H
(w, s)
H Aut(G
n
), G = F
q
,
M
n
(F
q
)
f(x) = wt(x) x, x ∈ F
n
q
.
M
n
(F
q
)
e
P
e
H
(w, s) b
w
M
n
(F
p
)
e
P
e
H
(w, x)
e
P
e
H
(w, s) =
w
X
j=0
µ
s
j
¶µ
n − s
w − j
¶
(−1)
s
(q − 1)
w−j
.
e
P
e
H
(w, s) =
P
$∈
e
A
w
$(x) x ∈ A
s
,
X
a∈F
l
p
,a6=0
exp
µ
2π i ha, bi
p
¶
=
(
−1, b 6= 0
q − 1, b 6= 0
.
e
P
e
H
(w, x) = coeff
x
s
y
n−s
(y − x)
w
(x + (q − 1)y)
n−w
.
P
H
(s, w) =
P
y∈A
s
$(y),
$ ∈
e
A
w
P
H
(w, s) =
s
X
j=0
µ
w
j
¶µ
n − w
s − j
¶
(−1)
j
(q − 1)
s−j
=
e
P
e
H
(s, w).
e
P
e
H
(w, x)
w x
e
P
e
H
(w, x)
e
P
e
H
(w, x) K
(q,n)
w
(x)
n
X
x=0
µ
n
x
¶
(q − 1)
u
K
(q,n)
w
(x)K
(q,n)
w
0
(x) =
(
0, w 6= w
0
¡
n
w
¢
(q − 1)
w
, w = w
0
.
K
(q,n)
w
(s)
K
(q,n)
s
(w)
Φ
H
(x, $) = (q −1)
s
µ
n
s
¶
K
(q,n)
w
(s) =
e
S
e
H
(w)K
(q,n)
s
(w) = (q − 1)
w
µ
n
w
¶
K
(q,n)
s
(w),
K
(p,n)
w−1
(x), K
(p,n)
w
(x), K
(p,n)
w+1
(x)
K
(p,n)
w
(x) n → ∞ p = 2
K
(q,n)
w
(x)
H Aut(F
n
q
)
H < Aut(F
q
)
M
1
(F
p
) (F
q
, +) F
q
Φ
q−1
2
F
q
F
q
H
E, |E| − 1, F
∗
q
F
q
q
Φ
E F
∗
q
K
(p,n)
w
(x)
K
(q,n)
1
(x) = (q − 1)n − qx
K
(q,n)
2
(x) =
1
2
((q − 1)
2
(n
2
− n) − (2nq(q −1) −q(q − 2))x + q
2
x
2
)
P
H
(w, s), c, w ∈ {0, . . . , m}
H
G
n
l
= F
n
q
S
n
H Aut(G
n
l
)
A
w
⊂ F
n
q
H
n = 1 H
F
q
H b ∈ F
q
A
b
c
b
(x), x ∈ G
n
l
, b ∈ G
l
,
x ∈ F
n
q
A
b
b c(x) = (c
b
0
(x), . . . , c
b
q−1
(x)) F
q
= {b
0
, . . . , b
q−1
}
x x
c = (c
b
0
, . . . , c
b
q−1
), c
b
0
+···+ c
b
q−1
= n,
a ∈ F
q
c(a) = c
A
c
x ∈ F
n
q
c(x) = c
F
n
q
S
n
S
n
a c(a) = c
a A
c
e
A
c
$
b
(x)
c(b) = c
S
n
$ = $
a
(x)
e
A
c
¤
|A
c
| = |
e
A
c
| =
¡
n
c
0
,...,c
q−1
¢
=
n!
c
0
!···c
q−1
!
c
j
= c
b
j
q = p l =
1 A
c
e
A
w
G
n
1
f
G
n
1
c = (c
0
, . . . , c
p−1
), c
0
+ ··· + c
p−1
= n,
w = (w
0
, . . . , w
p−1
), w
0
+ ··· + w
p−1
= n,
x ∈ A
c
e
P
e
H
(w, c) =
X
$∈
e
A
w
$(x) =
X
µ
c
0
w
0,0
, . . . , w
0,p−1
¶
···
µ
c
p−1
w
p−1,0
, . . . , w
p−1,p−1
¶
exp
Ã
2π i (
P
p−1
k=0
P
p−1
s=0
c
k
w
k,s
)
p
!
,
P
(w
k,0
, . . . , w
k,p−1
),
k = 0, . . . , p−1,
P
p−1
k=0
w
k,s
= c
s
, s = 0, . . . , p−1,
P
p−1
s=0
w
k,s
= w
k
, k = 0, . . . , p−1.
e
P
e
H
(w, x) n
¡
n
c
0
,...,c
p−1
¢
X
c
0
+···+c
p−1
=n
µ
n
c
0
, . . . , c
p−1
¶
e
P
e
H
(w, c)
e
P
e
H
(w
0
, c) =
(
0, w 6= w
0
¡
n
w
0
,...,w
p−1
¢
, w = w
0
.
P
n
l
= (Z
n
p
l
, +) M(P
n
l
) H
Aut(P
n
l
) M(P
n
l
)
n = 1 r
t,k
M(P
n
l
)
r
w,s
=
X
x∈A
w
exp
µ
2π i ax
p
l
¶
, a ∈ A
s
.
a A
t
l = 2
R
H
= kr
w,s
k
R
H
=
°
°
°
°
°
°
1 1 1
p − 1 p − 1 −1
p(p − 1) −p 0
°
°
°
°
°
°
,
P
2
s=0
|A
s
|r
w,s
r
w
0
,s
= 0, w 6= w
0
|A
0
| = 1, |A
1
| = p −
1, |A
2
| = p(p − 1)
n = 1 l ≥ 1
e
P
e
H
(w, s) = r
w,s
.
x ∈ A
c
, c = (c
0
, . . . , c
l
), c
0
+
··· + c
l
= n
e
P
e
H
(w, c) =
X
$∈
e
A
w
$(x) =
X
µ
c
0
w
0,0
, . . . , w
0,l
¶
···
µ
c
l
w
l,0
, . . . , w
l,l
¶
Y
0≤j,s≤l
r
w
j,s
j,s
,
P
(w
k,0
, . . . , w
k,l
),
k = 0, . . . , l,
P
p−1
s=0
w
k,s
= c
k
, s = 0, . . . , l ,
P
l−1
k=0
w
k,s
= w
s
, k = 0, . . . , l.
e
P
e
H
(w, x) l
|A
0
|
c
0
···|A
l
|
c
l
X
c
0
+···+c
p−1
=n
µ
n
c
0
, . . . , c
l
¶
|A
0
|
c
0
···|A
l
|
c
l
e
P
e
H
(w, c)
e
P
e
H
(w
0
, c) =
=
(
0, w 6= w
0
¡
n
w
0
,...,w
l
¢
|A
0
|
w
0
···|A
l
|
w
l
, w = w
0
= (w
0
, . . . , w
l
)
.
x = (x
0
, . . . , x
l
)
e
P
e
H
(w, x) x
0
+ ··· + x
l
= n
e
P
e
H
(w, x)
l x
1
, . . . , x
l
, x
1
+···+x
l
≤ n
e
P
e
H
(w, x) w
1
+ ··· + w
l
M
G M
R
n
F
n
q
ϕ(x), x ∈ M
H ≤ G ϕ(x) = ϕ(xh) h ∈ H
M = F
n
q
H x
F
n
q
s M = F
n
q
Y
n
q
n
− F
n
q
C
$
a
(x), a ∈ F
n
q
Y
n
ϕ ∈ Aut(F
n
q
) T
ϕ
: f → T
ϕ
(f)
T
ϕ
: $
a
(x) → $
ϕ(a)
(x).
Y
n
Y
n
T
ϕ
T
ϕ
0
= T
ϕ(ϕ
0
)
T
ϕ
, ϕ ∈ Aut(F
n
q
), T(F
n
q
)
T(F
n
q
) q
n
× q
n
− C
Aut(F
n
q
)
C
q
n
T(H) T(F
n
q
) T
ϕ
,
ϕ ∈ H Φ
H
(x, a) =
P
ϕ∈H
a
ϕ(a)
(x) ∈ Y
n
, $ = $
a
(x),
T(H)
Φ
H
(x, a)T
ϕ
= Φ
H
(x, a).
f(x) ∈ Y
n
Φ
$
(ϕ) = (f(x)T
ϕ
, Φ
H
(x, $)),
(·, ·)
H Φ
H
(x, $)
Φ
$
(ϕ)
H ϕ
0
∈ H Φ(ϕ) = Φ(ϕ
0
ϕ)
Φ
$
(ϕ) Aut(F
n
q
)
H ϕH ϕ ∈ Aut(F
n
q
)
ϕ a ∈ F
n
q
b = ϕ(a)
H ϕH b
Φ
$
(ϕ) b
a ϕ Φ
$
(ϕ)
P
$
(t) t = wt(ϕ(a))
P
$
(t) H
K
(p,n)
w
(x), w = wt(h),
h ∈ F
n
p
$ = $
a
(x) H