Сидельников В.М. Теория кодирования. Справочник по принципам и методам кодирования
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M G
M
M S
n−1
G SO(n)
n− H SO(n)
a S
n−1
M F
n
q
K G
n
+ g : (x, y) → R, x, y ∈ G
n
,
K ⊆ G
n
T (K) =
1
|K|
2
X
x,y∈K
g(x, y) ≥
1
|G
n
|
2
X
x,y∈G
n
g(x, y).
g(x, y) g
0
(x, y) a, b ≥
0 ag(x, y) + bg
0
(x, y)
g(x, y) = <
X
$∈
e
G
n
a
$
$(x −y)
, a
$
∈ R,
e
G
n
G
n
<(z)
z a
$
,
$ 1
K ⊆ F
n
q
a
$
∈ R T
0
(K) =
P
$∈
e
F
n
q
a
$
$(x − y)
T
0
(K) =
X
x,y∈G
n
X
$∈
e
G
n
a
$
$(x −y) =
X
x,y∈G
n
X
$∈
e
G
n
a
$
$(y −x) = T
0
(K).
<
T
$
(K) =
1
|K|
2
X
x,y∈K
$(x −y) =
1
|K|
2
¯
¯
¯
¯
¯
X
x∈K
$(x)
¯
¯
¯
¯
¯
2
≥ 0,
$ T
$
0
(K) = 1 $
0
(x)
1 x ∈ F
n
q
.
T
$
(G
n
) = 0 $
T
$
0
(G
n
) = 1 a
$
, $ ∈
f
G
n
\ {$
0
}, ¤
G = F
q
F
q
2
F
q
q = p, p ≥ 3 G
n
= F
n
p
g(x, y) = exp
µ
2π iha, x − yi
p
¶
− a exp
µ
−2π iha, x − yi
p
¶
=
= (1 − a) cos
µ
2π ha, x − yi
p
¶
, 1 > a > 0, a ∈ F
n
p
,
g(x, y)
g(x, y)
g(x, y)
K ⊆ G
n
1
|K|
2
X
x,y∈K
g(x, y) ≥ a
$
0
,
$
0
G
n
$
0
1 G
n
1
|G
n
|
2
X
x,y∈G
n
$(x −y) =
(
0, $ 6= $
0
a
$
0
, $ = $
0
.
¤
$
1 K ⊆ G
n
g(x, y) =
Φ
$
(x−y)
|St($)|
=
e
P
e
H
(w, s) $ ∈
e
A
w
x − y ∈ A
s
Ψ
$
(K) =
1
|K|
2
X
x,y∈K
Φ
$
(x − y)
|St($)|
=
1
|K|
2
X
x,y∈K
e
P
e
H
(w, s
H
(x − y)) =
1
|K|
2
m
X
s=0
D
s
e
P
e
H
(w, s) ≥
1
|G
n
|
2
X
x,y∈G
n
Φ
$
(x − y)
|St($)|
=
1
|G
n
|
2
m
X
s=0
|A
s
|
e
P
e
H
(w, s) = 0,
D
s
x, y ∈ K s
H
(x −y) = s D
s
x, y ∈ K x −y ∈ A
s
$ 1
Ψ
$
(K) =
1
|K|
2
X
x,y∈K
Φ
$
(x − y)
|St($)|
=
1
|K|
2
X
x,y∈K
e
P
e
H
(w, s
H
(x − y)) =
1
|K|
2
m
X
s=0
D
s
e
P
e
H
(w, s) = 1
G
n
= F
n
q
H M
n
(F
n
q
) D
s
x, y ∈ K d(x, y) = s
1
|K|
2
X
x,y∈K
K
(q,n)
w
(d(x, y)) ≥ 0, w 6= 0
1
|K|
2
X
x,y∈K
K
(q,n)
0
(d(x, y)) = 1,
K ⊂ F
n
p
d(x, y) = wt(x−y)
x, y ∈ F
n
q
V {A
0
, . . . , A
m
}
G
n
H
K ⊆ G
n
V − x, y ∈ K x − y
A
s
V V − K x, y ∈ K A
s
∈ V
x − y 6∈ A
s
A
0
6∈ V
f
K
(s) =
X
x,y∈K;x−y∈A
s
g(x, y).
g(x, y), x, y ∈ G
n
,
f
K
(s) ≤ 0, A
s
6∈ V s 6= 0
a
$
, $ ∈
e
G
n
, g(x, y)
a
0
= a
$
0
> 0 $
0
G
n
K V − g(x, y)
|K| ≤
f
K
(0)
a
0
.
g(x, y)
X
x,y∈K
g(x, y) =
n
X
s=0
D
s
f
K
(s) = |K|f
K
(0) +
X
s∈v
D
s
f
K
(s) ≤ |K|f(0),
D
s
x, y ∈ K (x, y) ∈ A
s
v
j, j 6= 0, j 6∈ V
X
x,y∈K
g(x, y) ≥ |K|
2
a
$
0
= |K|
2
a
0
.
¤
G
n
= F
n
q
H M
n
(F
p
)
F
n
q
F
n
q
f(x)
{0, 1, . . . , n}
f(x) ≤ 0, x ∈ {d, . . . , n}
a
w
, w = 0, 1, . . . , n,
f(x) =
n
X
w=0
a
k
K
(p,n)
w
(x)
K
(q,n)
w
(x) a
0
> 0
K d f(x)
|K| ≤
f(0)
a
0
,
a
0
f(x)
X
x,y∈K
f(d(x, y)) =
n
X
s=0
D
s
f(s) ≤ |K|f(0),
D
s
x, y ∈ K d(x, y) = s
f(d(x, y))
X
x,y∈K
f(d(x, y)) =
n
X
s=0
f(s)D
s
=
n
X
s=0
n
X
w=0
a
w
K
(p,n)
w
(s)D
s
≥ a
0
|K|
2
.
¤
q− n
G
n
H Aut(G
n
)
G
n
F
n
q
|K|
f(x)
f(x)
Ω
q
(n, d) = min
f(0)
a
0
,
n d
Ω(n, d)
f(0)
a
0
M
q
(n, d) ≤ Ω
q
(n, d),
M
q
(n, d) q− n
d
Ω(n, d)
Ω
(r)
q
(n, d) Ω
q
(n, d)
r
a
s
K
(q,n)
w
(x)K
(q,n)
w
0
(x) =
w+w
0
X
s=0
a
s
K
(q,n)
s
(x)
K
(q,n)
w
(s)K
(q,n)
w
0
(s)
K
(q,n)
w
(s)K
(q,n)
w
0
(s) = Y
w
(x)Y
w
0
(x) =
X
wt(y)=w, wt(y
0
)=w
0
$
y+y
0
(x),
x A
s
$
y+y
0
(x) wt(y + y
0
) = r
˜
A
r
N
(r)
w,w
0
y y
0
wt(y + y
0
) = r, wt(y) = w, wt(y
0
) = w
0
.
Y
w
(x)Y
w
0
(x) =
w+w
0
X
r=|w−w
0
|
N
(r)
w,w
0
Y
r
(x).
N
(r)
w,w
0
q = 2 r = w + w
0
− 2u N
(r)
w,w
0
=
¡
n
w
¢¡
w
u
¢¡
n−w
w
0
−u
¢
N
(r)
w,w
0
≥ 0 ¤
q = 2
K
(2,n)
s
K
(2,n)
s
(x) (0, n) s
x
s−1
1
< ··· < x
s−1
s−1
K
(2,n)
s−1
(x) (0, x
s−1
1
), (x
s−1
1
, x
s−1
2
)
, . . . , (x
s−1
s−1
, n) K
(2,n)
s
(x)
K
(2,n)
s
(x) K
(2,n)
s−1
(x)
x
s
1
K
(2,n)
s
(x)
K
(2,n)
r
(x) r < s K
(2,n)
r
(x
s
1
)
r < s K
(2,n)
r
(x
s
1
)
K
(2,n)
r
(0) =
¡
n
r
¢
(x
s
1
, x
s−1
1
) c
s
−K
(2,n)
s
(c
s
) = K
(2,n)
s−1
(c
s
).
c
s
x x
s
1
x
s−1
1
K
(2,n)
s−1
(x) + K
(2,n)
s
(x) x
s
1
K
(2,n)
s−1
(x
s
1
) x
s−1
1
K
(2,n)
s
(x
s
1
)
(x
s
1
, x
s−1
1
) c
s
x
s
K
(2,n)
s
(x)
F (x) =
K
(2,n)
s
(x)K
(2,n)
s−1
(c
s
) − K
(2,n)
s−1
(x)K
(2,n)
s
(c
s
)
x
s
− x
=
s−1
X
r=0
b
r
K
(2,n)
r
(x)
F (x) b
r
K
(2,n)
s
(x)K
(2,n)
s−1
(y) −K
(q,n)
s
(y)K
(2,n)
s−1
(x)
y − x
=
2
s
µ
n
s − 1
¶
s−1
X
r=0
K
(2,n)
r
(x)K
(2,n)
r
(y)
¡
n
r
¢
.
K
(2,n)
s+1
(x), K
(2,n)
s
(x), K
(2,n)
s−11
(x)
y = c
s
F (x) =
2
s
µ
n
s − 1
¶
s−1
X
r=0
K
(2,n)
r
(x)K
(2,n)
r
(c
s
)
¡
n
r
¢
.
x
s
1
K
(2,n)
r
(x) r < s K
(2,n)
r
(x
s
1
)
r < s K
(2,n)
r
(x
s
1
)
K
(2,n)
r
(0) =
¡
n
r
¢
¤
c
s
≤ d
f(x) =
³
K
(2,n)
s
(x)K
(2,n)
s−1
(c
s
) − K
(2,n)
s−1
(x)K
(2,n)
s
(c
s
)
´
2
c
s
− x
f(x) c
s
− x
c
s
f(x)
f(x) = F (x)
³
K
(2,n)
s
(x)K
(2,n)
s−1
(c
s
) − K
(2,n)
s−1
(x)K
(2,n)
s
(c
s
)
´
K
(2,n)
s−1
(c
s
) > 0, K
(2,n)
s
(c
s
) < 0
¤
x
s
1
K
(2,n)
s
(x)
s
n
→ λ, n → ∞ ξ = lim
n→∞
x
s
1
n
ξ =
1
2
−
p
λ(1 − λ).
d
n
→ δ, n →
∞, R (δ) = lim
n→∞
log
2
M(n,d)
n
R(δ) ≤ H
2
µ
1
2
−
p
δ(1 −δ)
¶
.
f(x) =
P
2s−1
j=0
α
j
K
(2,n)
j
(x)
α
0
=
1
2
n
n
X
j=0
µ
n
j
¶
f(j).
α
0
=
1
2
n
n
X
j=0
2
s
µ
n
s − 1
¶
³
K
(2,n)
s
(j)K
(2,n)
s−1
(c
s
) − K
(2,n)
s−1
(x)K
(2,n)
s
(c
s
)
´
s−1
X
r=0
K
(2,n)
r
(j)K
(2,n)
r
(c
s
)
¡
n
r
¢
=
1
2
n
n
X
j=0
2
s
³
−K
(2,n)
s−1
(j)K
(2,n)
s
(c
s
)
´
K
(2,n)
s−1
(j)K
(2,n)
s−1
(c
s
) = −
2
s
µ
n
s − 1
¶
K
(2,n)
s
(c
s
)K
(2,n)
s−1
(c
s
).
f(0) =
³
¡
n
s
¢
K
(2,n)
s−1
(c
s
) −
¡
n
s−1
¢
K
(2,n)
s
(c
s
)
´
2
c
s
,
K
(2,n)
s
(0) =
¡
n
s
¢
f(0)
α
0
=
s
¡¡
n
s
¢
+
¡
n
s−1
¢¢
2
2c
s
¡
n
s−1
¢
≤
(n − s + 1)
¡
n
s
¢
2x
s
1
.
s x
s
1
≤ d
λ, 0 < λ <
1
2
,
δ ≥
p
λ(1 − λ)
λ =
1
2
−
p
δ(1 −δ)
¤