Сидельников В.М. Теория кодирования. Справочник по принципам и методам кодирования
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q− X = {a
0
, . . . , a
q−1
}
X
n
n− n− X
|X
n
| = q
n
X
n
a = (a
0
, . . . , a
n
), b = (b
0
, . . . , b
n
)
X
n
d(·, ·)
d(a, b) = j a
j
6= b
j
.
d(·, ·)
d(a, b) ≤ d(a, c) + d(c, b) c ∈ X
n
X
n
d
K
X
n
d = d(K) K
K
d(K) = min
a,b∈K
a6=b
d(a, b)
X
X
X
n
X F
q
, q = p
l
p
X
n
n−
F
q
F
n
q
q = 2 X
n
X
X 4 X
X X
n
F
n
q
K
k = dim K K ω = {ω
1
, . . . , ω
k
}
K A = A(K)
{ω
1
, . . . , ω
k
} K
x K
x = zA,
z k− F
k
q
A K
A
wt(a) = a
a
wt(a)
K ⊂ F
n
q
K ⊂ F
n
q
K
d(K) = min
a∈K,a6=0
wt(a).
d(a, b) = wt(c)
c = a − b ∈ K ¤
a, b
K
K ⊆ F
n
q
F
q
, q = p
l
.
hx, yi F
q
x = (x
1
, . . . , x
n
) y = (y
1
, . . . , y
n
)
F
n
q
hx, yi = x
1
y
1
+ ··· + x
n
y
n
. F
q
x, y hx, yi = 0
K
⊥
K
K
K
⊥ ⊥
= K
K
⊥
F
q
n − k k = dim K
x, y a
F
q
a
K
⊥
x
A K
x ∈ K
⊥
xA
T
= 0,
A
T
A
n x k
n −k
0
k
0
A
k × n− A k
K ¤
K (n −
k) ×n− B K
⊥
K
x K
xB
T
= 0.
A · B
T
k × k−
n −k
x
B
x
K
K B
d − 1 B
F
q
d(K) K B
d
d B d(K) = d
d d − 1
B
x K d
x
d xB
T
= 0 K
x
d(K) ≥ d
¤
d −1
B
H
m ×2
m
−1
a
T
, a ∈ F
m
2
, m F
2
2
m
− 1
B
H
2
n−m
= 2
n−log
2
(n+1)
2
m
− 1 3
B
H
F
2
2
B
H
B
H
¤
2
m
4 2
m
2
m
− m − 1
J
w,n
w n
a b J
w,n
d(a, b) = 2w − 2u,
u j a
j
= b
j
= 1 j(a, b) =
1
2
d(a, b)
a, b ∈ J
w,n
S
n−1
n −1− S
n−1
R
n
n− R
n
(x, y) = x
1
y
1
+ ··· + x
n
y
n
.
|x| x = (x
1
, . . . , x
n
) ∈ R
n
|x| =
p
x
2
0
+ ··· + x
2
n
=
p
(x, x) λ(a, b) R
n
λ(x, y) = |x −y, x − y| =
p
(x
1
− y
1
)
2
+ ··· + (x
n
− y
n
)
2
.
x, y ∈ S
n−1
λ(x, y) = 2 −2(x, y).
R
n
r
(n − 1)− r
S
n−1
(r) r = 1 S
n−1
(n−1)− U
n−1
(r)
r C
n
(x, y)
C
n
(x, y)
C
= (x, y) = x
1
y
1
+ ··· + x
n
y
n
,
C a + ia
0
= a − ia
0
, a, a
0
∈ R i =
√
−1
|x| x = (x
1
, . . . , x
n
) ∈ C
n
|x| =
p
|x
0
|
2
+ ··· + |x
n
|
2
=
p
(x, x)
C
, x
j
= x
0
j
+ ix
00
j
|x
j
| =
q
x
0
j
2
+ x
00
j
2
λ
C
λ
U
n−1
(r) r
C
n
λ
C
r
U
n−1
(r) = {x ; |x| = λ
C
(0, x) = r}.
x, y ∈ U
n−1
(r) λ
C
(x, y) x, y
(x, y)
C
λ
C
(x, y) =
p
2r
2
− 2<(x, y)
C
,
<z z
λ
2
C
(x, y) = (x −y, x − y) = |x|
2
+ |y|
2
− (x, y)
C
− (y, x)
C
= 2r
2
− 2<(x, y)
C
,
C
n
R
2n
x
0
+ ix
00
→ (x
0
, x
00
) R
2n
R
2n
C
n
(x
0
, x
00
) → x
0
+ ix
00
R
2n
C
n
U
n−1
S
2n−1
A
A
T
A = E, , U
∗
U = E ,
A
T
A U
∗
= U
T
U E
A U 1
S
n−1
U
n−1
R
n
C
n
A U
λ
A U
λ(x, y) = λ(xA, yA) (λ
C
(x, y) = λ
C
(xU, yU)).
O(n) U(n)
U(n)
ϕ C
µ
a b
−b a
¶
ϕ : a + ib →
µ
a b
−b a
¶
.
ϕ C
µ
a b
−b a
¶
ϕ(ab) = ϕ(a)ϕ(b), ϕ(a + b) = ϕ(a) + ϕ(b), a, b ∈ C.
U = (a
i,j
)
i,j=1,...,n
a
i,j
2 ×2− ϕ 2n ×2n−
b
U
b
U
c
U
0
=
d
UU
0
ϕ : U →
b
U
U(n) O(2n) ϕ(U(n)) U(n)
O(2n)
ϕ(U(n)) 6= O(2n) n > 1 O(2n)
ϕ(U(n))
K S
n−1
S
n−1
x, y ∈ S
n−1
λ(K) K
λ(K) = min
x,y∈K,x6=y
λ(x, y).
G O(n) a
S
n−1
K = {ag|g ∈ G} ⊂ S
n−1
G
K = K(G, a) a
K = {ag|g ∈ G} ⊂ S
n−1
O(n) K(G, a)
G a
K(G, a)
K(G, a) = K(G, b) b ∈ K(G, a)
St
a
a G St
a
= {g|(g ∈
G)&(ag = a)} St
a
G St
b
= g
−1
St
a
g b = ag
St
b
, b ∈ K, G
G
|K| =
|G|
|St
a
|
.
wt(x) $(x)
K(G, a)
$
a
(x) = $(x) = λ(a, x), x ∈ K(G, a).
x = ag, y = ag
0
, g, g
0
∈
G $
λ(x, y) = λ(ag, y) = λ(a, yg
−1
) = $(yg
−1
) = $(ag
0
g
−1
).
K = K(G, a)
$(x), x ∈ K a
λ(K) = min
x∈K,x6=a
$(x).
¤
a, b
K
K ⊂ F
n
p
S
n
0
−1
n
0
S
n
0
−1
F
n
p
K
K S
n
0
−1
K
G
K ≤ G = X
n
X
n
G
· e K
K ≤ X
n
X
n
n−
F
p
· G
K F
p
π X
n
S
n
0
−1
π(K) ⊂ S
n
0
−1
π(K)
π(e) ∈ S
n
0
−1
G
S
n
0
−1
St
e
e G
G π(x) = π(e)g π(y) = π(e)g
0
π(x ·y) = π(e)g
00
x, y ∈ G G
g · g
0
∈ St
e
g
00
,
π(K)
K G = X
n
S
n
0
−1
St
e
e G
g · g
0
= g
00
τ G G
τ : x ↔ g, π(x) = π( e)g .
St
e
E G
K G/St
e
π(K)
G/St
e
K π(K)
K ⊆ X
n
S
n
0
−1
b
K ⊂ S
n
0
−1
, |
b
K| = |K|, n
0
6= n n
0
n
K ⊆ X
n
S
n
0
−1
f : K →
b
K
ρ
λ(f(x), f(y)) = ρ(d(x, y)),
d(·, ·) λ(·, ·) X
n
S
n
0
−1
K ⊆ X
n
U
n
0
−1
ρ ρ(x) = cx, c > 0
K ⊆ X
n
S
n
0
−1
K ⊆ G = X
n
G
b
K = π(K), |
b
K| = |K|, S
n
0
−1
G S
n
0
−1
G S
n
0
−1
G S
n
0
−1
b
K
K
K
X
n
K X
n
X
n
X
n
Y d(x, y) = d(x
0
, y
0
)
λ(π(x), π(y)) = λ(π(x
0
), π(y
0
))
X
n
c
X
n
n
0
= n
0
(n)
n
0
n
0
X
n
S
n
0
−1
λ(f(x), f(y))
π(x), π(y) x, y X
n
λ
n
(n
0
) = max λ(π(x), π(y))
d(x, y) = 1 Y
X
n
S
n
0
−1
λ
n
(Y) = λ(π(x), π(y)), d(x, y) = 1,
Y Y
F
n
q
S
n
0
−1
S
n
0
−1
S
n
0
−1
π
Y
1
X = {0, . . . , q − 1}
U
n
0
−1
, n
0
= q − 1 X Z
q
mod q
a ∈ X q − 1−
π(a) =
1
√
q − 1
µ
exp
µ
2π i a
q
¶
, exp
µ
2π i 2a
q
¶
, . . . , exp
µ
2π i (q − 1)a
q
¶¶
U
q−2
⊂ C
p−1
(π(a), π(b)) =
1
q − 1
p−1
X
k=1
exp
µ
2π i (a − b)k
q
¶
=
½
−1
q−1
, a 6= b;
1, a = b,