Сидельников В.М. Теория кодирования. Справочник по принципам и методам кодирования
Подождите немного. Документ загружается.
G Γ
C
f
Γ = {Γ(g)|g ∈ G}
G
Γ(g)Γ(g
0
) = Γ(h), gg
0
= h.
σ G G
g
σ
g
0
σ
= (gg
0
)
σ
,
g
σ
σ g
G
Aut(G)
H = {σ
0
, . . . , σ
t
} Aut(G)
h ∈ G C
H
h
G C
H
h
=
{h
σ
|σ ∈ H} C
H
h
h ∈ G
H H h
H h C
H
e
e G e
h
0
6∈ C
H
h
C
H
h
T
C
H
h
0
= ∅
G 1 + m
G =
[
j=0
m
C
H
j
,
C
H
j
= C
H
h
j
, j = 0, . . . , m,
H h
j
C
H
j
H C
H
j
S
H
(G)
H ⊆ Aut(G) S
H
(G)
S
H
(G)
X S
H
(G) G X = G
G ×G R
j
, j = 0, . . . , m G ×G =
∪
m
j=0
R
j
R
j
= {(g, hg)|h ∈ C
j
, g ∈ G}.
R
j
(g, g
0
) g
0
g
−1
∈ C
j
|R
j
| =
|G||C
j
| R
j
(e, h
j
) h
j
C
j
G S
H
(G)
S
H
S
H
(G)
(g, g
0
) (g
0
−1
, g
−1
)
S
H
(G)
R
T
j
= R
j
0
S
H
(G)
C
j
0
h
j
0
= h
−1
j
r
i,j
(g, g
0
) h ∈ G
(g, h) ∈ R
j
, (h, g
0
) ∈ R
i
(g, g
0
) ∈ R
k
r
i,j
(g, g
0
) = r
k
i,j
R
k
(g, g
0
)
(g, h) ∈ R
j
, (h, g
0
) ∈ R
i
(gh
0
, hh
0
) ∈ R
j
, (hh
0
, g
0
h
0
) ∈ R
i
h
0
∈ G
r
i,j
(g, g
0
) r
i,j
(gh
0
, g
0
h
0
) h
0
∈ G
r
i,j
(g, g
0
) = r
i,j
(e, g
0
g
−1
) h
0
= g
−1
r
i,j
(g, g
0
) = r
i,j
(g
σ
, g
0
σ
) σ ∈ H h
k
C
k
(g, g
0
) ∈ R
k
σ ∈ H (g
0
g
−1
)
σ
= h
k
r
i,j
(g, g
0
) = r
i,j
(e, g
0
g
−1
) = r
i,j
(e, h
k
) r
i,j
(g, g
0
)
R
k
(g, g
0
)
S
H
(G)
(g, g
0
) (g
0
−1
, g
−1
)
r
i,j
(g, g
0
) = r
j,i
(g, g
0
) (g, g
0
) ∈ G × G
(e, h) ∈ R
i
(h, h
k
) ∈ R
j
(e, h
k
h
−1
) ∈ R
j
(h
k
h
−1
, h
k
) ∈ R
i
(h
−1
k
, (h
k
h
−1
)
−1
) (e, h) R
i
h −→ h
k
h
−1
R
i
R
j
(e, h) ∈ R
i
(h, h
k
) ∈ R
j
r
i,j
(g, g
0
) =
r
j,i
(g, g
0
)
{(hg, g)|h ∈ C
j
, g ∈ G}
R
j
0
= {(g, hg)|g ∈ C
j
0
, g ∈ G}
C
j
0
h
−1
j
h
j
C
j
iii. S
H
(G) ¤
S
H
G
H
S
H
j = j
0
g, g
0
∈ G
g, g
−1
G
R
j
= {(x
σ
, y
σ
)|σ ∈
e
H} R
j
e
H
X (x, y) ∈ X ×X
S
H
(G)
G X H G
e
H
S
H
(G) G H
S
H
G = (F
p
, +) F
p
Aut(G) (F
p
, +)
σ
a
: x → ax, a ∈ F
∗
p
= F
p
r {0} |Aut(G)| = p − 1.
Φ
(p)
d
= Φ
d
, d|p − 1, Aut(G) σ
a
a F
∗
p,d
d F
∗
p
F
p
σ
a
∈ Φ
d
a a = y
p−1
d
, y ∈ F
∗
p
Γ
a
G
kexp
³
2πiax
p
´
k, a ∈ F
∗
p
x ∈ F
p
Γ
a
(x) = kexp
³
2πiax
p
´
k
F
p
Γ
a
Γ
a
(x)Γ
a
(y) =
kexp
³
2πiax
p
´
k · kexp
³
2πiay
p
´
k = kexp
³
2πia(x+y)
p
´
k = Γ
a
(x + y)
x → exp
³
2πiax
p
´
F
p
exp
³
2πiax
p
´
F
p
Φ
d
G d
0
+1 = 1 +
p−1
d
C
0
, C
1
, . . . , C
d
0
, C
0
= {0}, C
j
= {τ
j
x
p−1
d
|x ∈ F
∗
p
}, j = 1, . . . , d
0
, dd
0
= p − 1 τ
F
∗
p
R
j
R
j
= {(g, h + g)|h ∈
C
j
, g ∈ F
p
}
S
Φ
d
−1 ∈ F
∗
p,d
(F
p
, +)
d = p−1 G×G G×G = R
0
∪R
1
R
0
= {(g, g)|g ∈ G}
R
1
= {(g, h + g)|g ∈ F
p
, h ∈ F
∗
p
} C
0
= {0} C
1
= F
∗
p
d =
p−1
2
G×G = R
0
∪R
+
∪R
−
R
0
= {(g, g)|g ∈ G} R
+
= {(g, h+g)|g ∈
F
p
, h ∈ F
∗
p
,
³
h
p
´
= 1} R
−
= {(g, h + g)|g ∈ F
p
, h ∈ F
∗
p
,
³
h
p
´
= −1}
³
h
p
´
G
F
q
, q = p
l
l > 1
F
q
σ
a
: x → ax, a ∈ F
∗
q
G H = Inn(G)
g → h
−1
gh, h ∈ G
S
H
(G)
G
Aut(G)
G
C
H
(G)
C
H
(G
n
)
C
H
(G)
H
n
|G|
G = (F
p
l
, +)
H < Aut((F
p
l
, +))
S
H
(G)
G
n
S
H
(G) G
G
n
= G ×···×G S
H
(G) 1 + m
R
j
S
H
(G
n
)
H
n
2
G
n
n− n−
n−
λ
λ x = (x
1
, . . . , x
n
) y = ( y
1
, . . . , y
n
)
x y
λ(x, y) =
v
u
u
t
n
X
j=1
λ
2
(x
j
, y
j
),
λ(x, y) =
n
X
j=1
λ(x
j
, y
j
) = d,
λ d
x y
H
n
2
(x, y) ∈ R
d
λ(x, y) = d
G
n
x, y
G
n
G
n
ii.
S
H
(G) 1 + m R
j
H G
n
H
n
H
n
(x
1
, . . . , x
n
) → (x
σ
1
1
, . . . , x
σ
n
n
), σ
j
∈ H.
C
j
j = (j
1
, . . . , j
n
), j
s
∈
{0, . . . , m} g = (g
1
, . . . , g
n
) g
s
∈ C
j
s
S
H
n
(G
n
)
X S
H
n
(G
n
) G
n
G
n
× G
n
(1 + m)
n
{R
j
|j = (j
1
, . . . , j
n
); 0 ≤ j
s
≤
m} (g, g
0
) ∈ R
j
g
0
g
−1
∈ C
j
S
H
n
(G
n
) n− S
H
(G)
S
H
n
(G
n
)
C
H
(G
n
) S
H
n
(G
n
)
R
j
n = 1 C
H
(G
n
) S
H
n
(G
n
)
c
j
(g) g
s
g = (g
1
, . . . , g
n
)
g
s
∈ C
j
c(g) = (c
0
(g), . . . , c
m
(g)) 1 + m
G H
g
C
H
(G
n
)
X
n
C
H
(G
n
) G
n
G
n
× G
n
{R
c
|c = (c
0
, . . . , c
m
); c
0
+ ··· + c
n
= n}
(g, g
0
) ∈ R
c
c(g
0
g
−1
) = c
R
c
(g, g
0
)
c(g
0
g
−1
) = c C
H
(G) = S
H
(G)
C
H
(G
n
)
C
H
(G)
C
H
(G
n
)
H
n
G
n
c
C
H
(G
n
)
C
H
(G
n
)
τ = (i
1
, . . . , i
n
) {1, . . . , n} σ = (σ
1
, . . . , σ
n
)
G
n
G
n
H
n
oS
n
(σ, τ) : g = (g
1
, . . . , g
n
) → (g
σ
1
i
1
, . . . , g
σ
n
i
n
), σ ∈ H
n
, τ ∈ S
n
.
S
H
n
oS
n
(G
n
)
S
H
n
oS
n
(G
n
)
C
H
(G
n
)
C
H
(G
n
)
(g, g
0
) ∈ R
c
g, g
0
∈ G
n
(h, h
0
) R
c
C
H
(G
n
)
Γ = Γ(G) G
f−
ϕ G Γ(G) = {Γ(g)|g ∈ G}
f × f−
ϕ : g → Γ(g) Γ(g)Γ(h) = Γ(gh) g, h ∈ G.
Γ(G)
Γ(G) L C
f
R
f
L 6= C
f
L 6= {)} L
LΓ(g) = L g ∈ G L f
0
, 0 < f
0
< f,
Γ(g) C
f
Γ(g) =
µ
Γ
1
(g) 0
0 Γ
2
(g)
¶
,
Γ
1
(g) f
0
× f
0
− Γ
2
(g) (f − f
0
) × (f − f
0
)−
C
f
C
f
= L ⊕L
0
, dim L
0
= f −f
0
.
Γ
u(x)
G u(x) u
0
(x)
hu(x), u
0
(x)i =
1
|G|
X
g∈G
u(g)u
0
(g).
Γ(g) = kγ
i,j
(g)k
i,j=1,...,f
Γ g
hγ
i,j
(g), γ
i
0
,j
0
(g)i =
½
1
f
, i, j = i
0
, j
0
0, i, j 6= i
0
, j
0
Γ
G 1 + m
1 + m
G Inn(G)
Γ
k
(G), k = 0, . . . , m Γ
k
(G)
C
f
k
R
f
k
f
k
f
k
|G| G
f
2
0
+ f
2
1
+ ··· + f
2
m
= |G|.
hγ
k
i,j
(x), γ
s
i
0
,j
0
(x)i =
0, k 6= s
hγ
k
i,j
(xg), γ
k
i
0
,j
0
(x)i, k = s
tr A A
A = Γ(g) tr Γ(g) g
Γ χ
Γ
(g) Γ = Γ
k
(G) tr Γ
k
(g)
χ
k
(g)
hχ
k
(x), χ
s
(g
0
x)i =
(
0, k 6= s
χ
k
(g
0
)
f
k
, k = s
k = s g
0
= e G
hχ
k
(x), χ
k
(x)i = 1.
Γ
s
(g
0
g) = Γ
s
(g
0
)Γ
s
(g)
χ
s
(g
0
g) = tr Γ
s
(g
0
g) =
f
s
X
j=1
f
s
X
i=1
γ
s
i,j
(g
0
)γ
s
j,i
(g).
k 6= s htr Γ
k
(g), tr Γ
s
(g
0
g)i = 0 k = s
hχ
k
(x), χ
s
(g
0
x)i = htr Γ
k
(x), tr Γ
k
(g
0
x)i =
1
|G|
X
g∈G
Ã
f
k
X
i=1
γ
k
i,j
(g
0
)γ
k
j,i
(g)
!Ã
f
k
X
i=1
γ
k
i,i
(g)
!
=
tr Γ
k
(g
0
)
f
k
=
χ
k
(g
0
)
f
k
.
¤
A ∈ GL(f, C) f × f− Γ(G)
G
1
|G|
X
v∈Γ(G)
vAv
−1
=
tr A
f
I
f
.
X = {1, . . . , n} A
i
= ka
i
k,s
k
k,s∈X
, i = 0, . . . , m,
R
i
S = S(X, R
0
, . . . , R
m
) |X| ×
|X|− a
i
k,s
= 1 (k, s) ∈ R
i
a
i
k,s
= 0 (k, s) 6∈ R
i
A
i
A
j
=
m
X
t=0
r
t
i,j
A
t
,
r
t
i,j
S
H
(G)
S r
t
i,j
= r
t
j,i
A
i
A
j
= A
j
A
i
i, j A
i
A
j
A
j
A
T
j
= A
j
0
= A
j
A
i
A
j
A
t
A = {c
0
A
0
+ · + c
m
A
m
|c
i
∈ C} A
A
j
B A C
|X| × |X|− B = {A
0
, A
1
, . . . , A
m
}, A
0
= I, I
R
0
A
0
+ A
1
+ ··· + A
m
= J,
J |X| ×|X|− 1
B
0
= {E
0
, E
1
, . . . , E
m
} A |X| × |X|− E
k
E
k
E
s
=
(
0, k 6= s ,
E
k
, s = k
.
E
k
E
2
k
=
E
k
E
k
0 1
V
A
C
|X|
A A ∈ A
1 V
AA
0
= V
A
∩ V
A
0
E
k
V
E
k
V
E
k
∩ V
A
= {0}, V
E
k
∩ V
A
= V
E
k
A ∈ A.
E
k
A ∈ A AE
k
= 0 AE
k
= E
k
aE
k
, a ∈ C
|X|
, a
V
E
k
E
k
A
C
A ∈ A C
−1
AC
C
|X|
A
E
k
t
k
t
0
+ ··· + t
m
= |X|
E
k
B B
0
A A
k
∈ A
A
k
= P
k
(0)E
0
+ P
k
(1)E
1
+ ··· + P
k
(m)E
m
, k = 0, . . . , m
E
k
= Q
k
(0)A
0
+ Q
k
(1)A
1
+ ··· + Q
k
(m)A
m
, k = 0, . . . , m,
P
j
(k), Q
j
(k) ∈ C
P = kP
j
(k)k
j,k=0,...,m
B
0
B S
H
(G)
P
P
j
(k) Q
j
(k)
P
j
(k) t
k
m
X
k=0
t
k
P
j
(k)P
s
(k) =
½
0, j 6= s
v
s
|X|, j = s
A
j
A
s
= P
j
(0)P
s
(0)E
0
+ P
j
(1)P
s
(1)E
1
+ ··· + P
j
(m)P
k
(m)E
m
tr A
j
A
s
= tr
Ã
m
X
t=0
r
t
j,s
A
t
!
= r
0
j,s
tr A
0
= |X|r
0
j,s
,
tr A
s
= 0, s 6= 0, A
s
, s 6= 0,
0
r
0
j,s
= 0 s 6= j r
0
j,s
= v
s
s = j ¤
v
s
R
s
(x, y) ∈ X ×X x ∈ X
R
s
S
H
(G) v
s
= |C
s
|
C
s
P
m
k=0
t
k
P
s
j(k)P
s
(k) = |C
s
||X|
m + 1 × m + 1− P = k P
s
(k)k
s,k=0,...,m
, Q = kQ
s
(k)k
s,k=0,...,m
T = diag (t
0
, . . . , t
m
), V = diag (v
0
, . . . , v
m
)
P
T
· T ·P = |X|V
Q = |X|P
−1
= V
−1
· P
T
· T
t
s
P
k
(s) = v
k
Q
s
(k) s, k = 0, . . . , m.