Сидельников В.М. Теория кодирования. Справочник по принципам и методам кодирования
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m
X
k=0
v
k
Q
j
(k)Q
s
(k) =
½
0, j 6= s
t
s
|X|, j = s
m + 1 × m + 1− U
k
= kr
k
i,j
k
i,j=0,...,m
, k = 0, . . . , m, r
k
i,j
v
k
X
j=0
r
k
i,j
= v
i
X
j=0
r
i
k,j
m
X
k=0
r
k
i,j
r
l
k,s
=
m
X
h=0
r
l
i,h
r
h
j,s
.
m+1×m+1− U = {U
k
|k = 0, . . . , m}
A
0
A
U
i
U
j
=
m
X
t=0
r
t
i,j
U
t
,
¤
P
k
(s), s = 0, . . . , m,
U
k
A
k
P
k
(s), s = 0, . . . , m, det(U
k
− xI
m+1
) = 0
m + 1 |X|
A |X|×|X|−
A
0
m + 1 × m + 1−
|X|×|X| P
k
(s)
P
k
(j)
S
H
(G) H = Inn(G)
X G
|G|×|G|− A
k
R
k
G = {g
1
, . . . , g
N
}, N = |G|
A
k
= ka
k
g,g
0
k
g,g
0
∈G
a
k
g,g
0
= ϕ
k
(g, g
0
),
ϕ
k
R
k
S
H
(G)
ϕ
k
(g, g
0
) =
½
1, (g, g
0
) ∈ R
k
0, (g, g
0
) 6∈ R
k
E
s
A
k
E
s
= P
k
(s)E
s
.
a ∈ V
E
s
a
P
k
(s) A
k
P
k
(s)
a ∈ V
E
s
h
s
C
s
χ
j
(x)
Γ
j
G
m
X
j=0
χ
j
(h
s
)χ
j
(g) =
0, g h
s
|G|
|C
s
|
, g h
s
C
s
A
k
ϕ
k
(g, g
0
) = ψ
k
(g
0
g
−1
) =
|C
k
|
|G|
m
X
j=0
χ
j
(h
k
)χ
j
(g
0
g
−1
) =
1
|G|
m
X
j=0
X
h∈C
k
χ
j
(h)χ
j
(g
0
g
−1
),
ψ
k
C
k
a
s
(h) = (χ
s
(hg
1
), . . . , χ
s
(hg
N
)), N = |G|
h ∈ G a
s
(h) A
k
P
s
(k) =
|C
k
|
f
s
χ
s
(h
−1
k
),
h
k
C
k
ha
s
(h), a
j
(h
0
)i =
(
χ
s
(h
0
h)
f
s
, j = s
0, j 6= s
V
E
s
, s = 0, . . . , m, C
|G|
{a
s
(h)|h ∈ G}
A
k
A
k
= ka
k
g,g
0
k
g,g
0
∈G
= kψ
k
(g
0
g
−1
)k
g
0
,g∈G
.
b
g
0
a
s
(h)A
k
g
0
b
g
0
=
P
g∈G
a
k
g,g
0
χ
s
(hg) ψ
k
b
g
0
=
|C
k
|
|G|
X
g∈G
m
X
j=0
χ
j
(h
k
)χ
j
(g
0
g
−1
)χ
s
(hg).
f = hg
b
g
0
= |C
k
|
m
X
j=0
χ
j
(h
k
)
1
|G|
X
f∈G
χ
j
(g
0
hf
−1
)χ
s
(f).
1
|G|
X
f∈G
χ
j
(g
0
hf
−1
)χ
s
(f) =
1
|G|
X
f∈G
χ
j
(g
0
hf)χ
s
(f
−1
) =
hχ
s
(f), χ
j
(g
0
hf)i =
(
χ
s
(g
0
h)
f
s
, j = s
0, j 6= s
χ
s
(g
0
h) = χ
s
(hg
0
hh
−1
) = χ
s
(hg
0
)
b
g
0
= |C
k
|
χ
s
(h
k
)
f
s
a
g
0
=
|C
k
|
f
s
χ
s
(h
−1
k
)a
g
0
,
a
g
0
a
s
(h) g
0
¤
C
|G|
a
s
(h), h ∈ G |C
s
|
G (F
p
, +)
Γ
a
, a ∈ F
p
, G
χ
a
(x) Γ
a
χ
a
(x) = exp
µ
2π iax
p
¶
, x ∈ 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
F
p
ψ
k
(x −
y), k 6= 0, C
k
Φ
d
F
p
ψ
k
(x) =
1
|C
k
|p
X
a,z∈F
p
exp
Ã
2π ia(x − τ
k
z
p−1
d
)
p
!
=
1
p
X
a∈F
p
exp
µ
2π iax
p
¶
ϑ
k
(−a),
ϑ
k
(b) =
1
|C
k
|
X
z∈F
p
exp
Ã
2π ibτ
k
z
p−1
d
p
!
, k = 0, . . . , d − 1.
A
k
= ka
x,y
k
x,y∈F
p
= kψ
k
(y − x)k
x,y∈F
p
a
k
(b) = (ϑ
k
(a
1
b), . . . , ϑ
k
(a
p
b)), { a
1
, . . . , a
p
} = F
p
,
A
k
|C
k
| = d
α
y
y ∈ F
p
a(b)A
k
= a(b)kψ
k
(y − x)k
α
y
=
X
x∈F
p
ϑ
k
(xb)ψ
k
(y − x) =
1
p
X
x∈F
p
X
a∈F
p
exp
µ
2π ia(y − x)
p
¶
ϑ
k
(−a)ϑ
k
(xb).
1
p
X
x∈F
p
exp
µ
−2π iax
p
¶
ϑ
k
(bx) =
(
exp
³
−2π iax
p
´
, a = bτ
k
z
p−1
d
0,
α
y
= |C
k
|β
y
,
β
y
a(b) y ∈ F
p
¤
a
k
(b) a
s
(b
0
), b, b
0
6= 0,
d
{a
1
bτ
k
, . . . , a
p
bτ
k
} {a
1
b
0
τ
s
, . . . , a
p
b
0
τ
s
}
ϑ
k
(a) = ϑ
k
(ay
p−1
d
) y 6= 0 ϑ
k
(a) = ϑ
k
(b) a b
a
k
(b) d = p − 1
p−1
2
p
k
i,j
H
n
q
P
k
(x)
C
H
(G)
λ(g, g
0
) C
H
(G
n
)
λ(g, g
0
) = λ(h, h
0
) (g, g
0
) (h, h
0
)
C
H
(G
n
)
λ(g, g
0
)
C
H
(G
n
) λ(g, g
0
) =
λ(h, h
0
) g
0
g
−1
h
0
h
−1
G λ
C
H
(G
n
)
G
λ G
G
G mod 4
X = {0, . . . , r −1}
mod r
l(a, b) a, b ∈ X
b − a ∈ X a − b ∈ X l(a, b) = min(b − a mo d r, a − b mod r)
r = 5 l(0, 1) = 1 l(4, 1) = 2
l(a, b)
λ
q−
λ(g, g
0
), g, g
0
∈
G, G
λ(g, g
0
) = λ(gh, g
0
h) h ∈ G.
C
H
(G) C
H
(G)
λ(g, g
0
) =
b
λ(hg, hg
0
)
λ N
g,g
0
(a, b)
h ∈ G λ(g, h) = a, λ(h, g
0
) = b
c = λ(g, g
0
) N
g,g
0
(a, b) = N
c
(a, b)
λ(g, g
0
) = 1 g 6= g
0
λ(g, g) = 0
G
λ(g, g
0
), g , g
0
∈ G,
G N
g,g
0
(a, b) h ∈ G
λ(g, h) = a, λ(h, g
0
) = b c = λ(g, g
0
)
λ G
N
g,g
0
(a, b) 0, 1, 2
C
H
(G) C
H
(G)
G
Γ = {Γ(g)|g ∈ G} G
C
f
H Aut(G) G a
U
f−1
h·, ·i
H,a
G
hg, g
0
i
H,a
=: hg, g
0
i =:
1
|H|
X
σ∈H
(aΓ(g
σ
), aΓ(g
0
σ
)) =
1
|C
H
j
|
X
h∈C
H
j
(a, aΓ(h)),
(·, ·) U
f
C
H
j
H
g
0
g
−1
Γ(g) f × f (aΓ(g
σ
), aΓ(g
0
σ
)) = (a, aΓ(g
0
σ
)Γ
−1
(g
σ
)) =
(a, aΓ((g
0
g
−1
)
σ
))
hg, gi
H,a
= 1
hg, g
0
i
H,a
C
f(t+1)
b
g
=
1
√
t+1
(aΓ(g
σ
0
), . . . , aΓ(g
σ
t
)) {σ
0
, . . . , σ
t
} = H, b
g
0
=
1
√
t+1
(aΓ(g
0 σ
0
), . . . ,
aΓ(g
0 σ
t
)) 1
hg, g
0
i
H,a
G H
G F
p
Φ
d
H ⊆ Aut(F
p
) d, d|p − 1,
kexp
³
2πiax
p
´
k, a ∈ F
∗
p
, G
ha, bi
Φ
d
,a
= ha, bi
Φ
d
, a, b ∈ F
p
, a 1
ha, bi
Φ
d
=
1
d
X
x∈F
∗
p,d
exp
µ
2πi(b −a)x
p
¶
=
1
p − 1
X
x∈F
∗
p
exp
Ã
2πi(a −b)x
p−1
d
p
!
.
d = p − 1
ha, bi
Φ
p−1
=
(
1, a = b,
−1
p−1
, a 6= b
.
d =
p−1
2
b − a 6= 0
ha, bi
Φ
p−1
2
=
1
p − 1
X
x∈F
∗
p
exp
µ
2πi(b −a)x
2
p
¶
=
1
p − 1
³
b−a
p
´
(−1 + p
1
2
), p = 4t + 1
³
b−a
p
´
(−1 + ip
1
2
), p = 4t −1
,
i =
√
−1.
X
x∈F
p
exp
µ
2πix
2
p
¶
=
(
√
p, p = 4t + 1
i
√
p, p = 4t −1
H = Inn(G) a ∈ U
f−1
hg, g
0
i
H,a
=
tr Γ(g
0
g
−1
)
f
.
Γ(g) g G
hg, g
0
i
H,a
=
1
|G|
X
h∈G
(aΓ(h)Γ(g)Γ(h
−1
), aΓ(h)Γ(g)Γ(h
−1
)) =
1
|G|
(a, a(
X
h∈G
Γ(h)Γ(g
0
g
−1
)))Γ(h
−1
))i
H,a
=
1
f
tr Γ(g
0
g
−1
)(a, a) =
1
f
tr Γ(g
0
g
−1
).
¤
H = Inn(G) hg, g
0
i
H,a
a U
f−1
Γ(G) G
G
λ
H,a
(g
0
, g) = G
λ
H,a
(g
0
, g) =: {hg
0
, g
0
i
H,a
+ hg, gi
H,a
− hg
0
, gi
H,a
− hg, g
0
i
H,a
}
1
2
=
q
2 − 2<hg
0
, gi
H,a
,
<z z
λ
0
U
s−1
C
s
x, y ∈ U
s−1
λ
0
(x, y) =
p
(y − x, y − x) = ((x, x) + (y, y) − (x, y) −(y, x)) =
p
2 − 2<(x, y).
λ
H,a
λ
H,a
(g
0
, g) = λ
H,a
(g, g
0
) λ
H,a
(g, g) = 0 λ
H,a
(g
0
, g) =
λ
H,a
(h
0
, h) (g
0
, g), (h
0
, h)
S
H
(G) λ
H,a
S
H
(G)
λ
H,a
(g
0
, g)
λ
H,a
(g, g
0
) = λ
H,a
(g
σ
, g
0
σ
), σ ∈ H.
w(g) = λ(e, g) g
λ
Φ
p−1
(a, b) =
(
0, a = b,
q
2p
p−1
, a 6= b
,
H = Φ
p−1
λ
Φ
p−1
H = Φ
p−1
2
p = 4t − 1
λ
Φ
p−1
2
(a, b) =
s
2
p − 1
(p +
µ
b − a
p
¶
(1 −
√
p)), b −a 6≡ 0 mod p.
p = 4t + 1 λ
Φ
p−1
2
(a, b)
λ
H,a
(g
0
, g)
λ
H,a
(g
0
, g)
G H λ
H,a
(g
0
, g)
λ
H,a
(g
0
, g)
λ(g
0
, g) G
a
0
= 0, a
1
, . . . , a
m
S = S(G, R
0
, . . . , R
m
)
(g
0
, g) ∈ R
j
λ(g
0
, g) = a
j
S
R
T
j
R
T
j
= R
j
λ(g
0
, g) = λ(g, g
0
) (g
0
, g) (g, g
0
)
H
n
q
n S
H
((F
p
, +)) H = Φ
p−1
H = Φ
p−1
2
λ
H,a
(g
0
, g) G
λ
H,a
(g
0
, g) G S
H
(G)
λ
H,a
(g
0
, g) 6= λ
H,a
(h
0
, h) (g
0
, g) (h
0
, h)
λ
H,a
(g
0
, g)
G
Γ(h), h ∈ G (aΓ(g)Γ(h), aΓ(g
0
)Γ(h)) =
= (aΓ(g), aΓ(g
0
)) λ
H,a
λ
H,a
(g, g
0
) = λ
H,a
(e, g
0
g
−1
) = w(g
0
g
−1
)
λ
H,a
g
0
= e e, g g
0
, g
c
(g, g
0
) λ
H,a
(g, g) = c N
c
a,b
(g, h) (h, g
0
)
λ
H,a
(g, h) = a λ
H,a
(g
0
, h) = b c
a, b, c R
i
, R
j
, R
k
(g, h), (g
0
, h) (g, g
0
)
N
c
a,b
= r
k
i,j
¤
G H
G
λ
H,a
(g, g
0
) = λ
H,a
(g
−1
, g
0
−1
).
Γ
G H = Inn(G)
λ
H,a
(g
0
, g) = λ
H,a
(g
0
, g) =
s
2 −
2<tr g
0
g
−1
f
,
λ
H,a
a U
f−1
w(g) g ∈ G
w(g) = w(g
0
) = w
j
g, g
0
C
j
G
n
λ
H,a
G
n
λ
H,a
(g, g
0
) =
v
u
u
t
n
X
j=1
λ
2
H,a
(g
j
, g
0
j
), g = (g
1
, . . . , g
n
), g
0
∈ G
n
.
λ
H,a
(g
0
, g), g, g
0
∈ G
n
,
b
g
= (b
g
1
, . . . , b
g
n
) b
g
0
b
g
j
=
1
√
t+1
(aΓ(g
j
σ
0
), . . . , aΓ(g
j
σ
t
)),
{σ
0
, . . . , σ
t
} = H, U
ftn−1
√
n
C
ftn
λ
H,a
G
n
λ
H,a
(g
0
, g)
(g
0
g) C
H
(G
n
)
λ
H,a
(g, g
0
) =
b
λ
c
(g, g
0
) ∈ R
c
λ
H,a
(g
0
, g) =
v
u
u
t
m
X
s=1
c
s
λ
2
s
, c = (c
0
, . . . , c
m
),
(g
0
, g) ∈ R
c
C
H
(G
n
)
λ
H,a
λ
H,a
λ
H,a
G
n
λ
H,a
G
n
R
c
w
1
, . . . , w
m
G
C
1
, . . . , C
m
(c
1
, . . . , c
m
) 6= (c
0
1
, . . . , c
0
m
)
c
1
w
1
+ ··· + c
m
w
k
6= c
0
1
w
1
+ ··· + c
0
m
w
m
¤