Сидельников В.М. Теория кодирования. Справочник по принципам и методам кодирования
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λ
2
(π(a), π(b)) = |a − b, a − b|
2
= 2 − 2<(a, b) =
½
2q
q−1
, a 6= b
0, a = b
<x x
ρ ρ(x) =
q−1
2q
x
U
q−2
q − 1− n
0
= q − 1
U
q−2
Y
1
2(q −1)−
S
2q−3
λ
1
(Y
1
) =
q
2q
q−1
Y
1
X, q > 2,
q
q − 1
2(q − 1)−
X
X
b
X
b
X
K(G, a) G
g
a
= diag
µ
exp
µ
2π i a
q
¶
, exp
µ
2π i 2a
q
¶
, . . . , exp
µ
2π i (q − 1)a
q
¶¶
, a ∈ Z
q
.
a a =
1
√
q−1
(1, . . . , 1) ∈ U
q−2
K(G, a)
q
X Y
1
X
U
q−2
C
q−1
ϕ
ϕ G G
0
2(q − 1) × 2(q − 1)− G
0
a
0
=
1
√
q−1
(ϕ(1), . . . , ϕ(1)) =
1
√
q−1
(1, 0, 1, 0, . . . , 1, 0) ∈ S
2(q−1)−1
K(G
0
, a
0
)
S
2(q−1)−1
G
0
2 × 2− ϕ(exp
³
2π i ab
q
´
), a, b ∈ Z
q
G G
0
ϕ
K(G, a) ⊂ U
q−2
K(G
0
, a
0
) ⊂ S
2(q−1)−1
K(G
0
, a
0
)
X
S
2(q−1)−1
p = 2 Y
1
F
2
Y
X, |X| = q,
S
q−2
q − 1−
π(x) x ∈ X
q − 1−
a
j
= e
j
−
1
q
P
q
s=1
e
s
, j = 1, . . . , q
e
j
= (0, . . . , 0, 1, 0, . . . , 0) ∈ R
q
j−
q − 1− x
1
+ ··· + x
q
= 0
q − 1−
a
j
{a
1
, . . . , a
q
}
R
q−1
X S
q−1
q − 1−
Y
Y
G
X S
q−2
X
π(X)
X
G q +
π
X S
q−2
ω = {ω
1
, . . . , ω
q
} q−
R
q
ω
1
=
1
√
q
(1, . . . , 1) ω
j
, j > 1,
R
q
= L
1
⊕ L
q−1
q− R
q
L
1
e
1
+···e
q
=
√
qω
1
q−1− L
q−1
ω
2
, . . . , ω
q
ω a
j
α
0
j
= (0, α
(j)
2
, . . . , α
(j)
q
) = a
j
A
A e = {e
1
, . . . , e
q
}
ω
α
j
= (α
(j)
2
, . . . , α
(j)
q
) (a
j
, a
j
) = (α
j
, α
j
) =
q−1
q
q −
1− α
j
q
q−1
q
b
j
=
r
q
q − 1
α
j
S
q−2
(b
i
, b
j
) =
q
q−1
(α
i
, α
j
) =
q
q−1
(e
i
, e
j
) = −
1
q−1
λ(b
i
, b
j
) =
p
(b
i
− b
j
, b
i
− b
j
) =
p
2 − 2(b
i
, b
j
)
b
i
, b
j
, i 6= j,
q
2q
q−1
B = {b
1
, . . . , b
q
}
S
q−2
X B
G
R
q
g ∈ G L
1
L
q−1
1 q −1 a ∈ L
j
, j = 1, q −1
ag ∈ L
j
g ∈ G
G
G G
ω = {ω
1
, . . . , ω
q
}
g ∈ G
g(g) =
µ
g
1
(g) 0
0 g
q−1
(g)
¶
,
g
1
1 ×1− g
q−1
q −1 ×q −1−
L
1
L
q−1
G
q−1
q −1 × q − 1− g
q−1
q −1 × q − 1−
g
q−1
G
q−1
G
L
q−1
G q + G = {g(g)|g ∈ G}
g(g) = g ∈ G
q × q−
G σ
g
E = {e
h
|h ∈ G} R
q
e
h
g(g) = e
h+g
.
G E E
L
1
e
1
=
P
h∈G
e
h
= (1, 1, . . . , 1)
∈ R
q
G
R
q
e
1
g(g) = e
1
L
1
L
q−1
q − 1
G R
q
R
q
= L
1
⊕ L
q−1
G
q−1
g
q−1
= g
q−1
(g)
G q −1−
G A = {a
1
, . . . , a
q
} a
j
=
e
j
−
1
q
P
q
s=1
e
s
, j = 1, . . . , q
E A G
G
q−1
B = {b
1
, . . . , b
q
} ⊂ L
q−1
A
G b
h
g
q−1
(g) = b
h+g
B G
q−1
B = {b
1
h|h ∈ G
q−1
}
π(h) = b
h
= b
1
bg(h), h ∈ G = X b
h
π
X = G S
q−2
¤
q = p = 4t + 1
Y
3
X = (F
p
, +), p > 2, p
p−1
2
−
F
p
π(F
p
) ⊂ U
p−3
2
F
p
, p > 2, G
C
p−1
2
π(F
p
)
G
F
p
p −1−
G
2 × 2−
Y
3
π(a)
π(a) =
µ
p − 1
2
¶
−
1
2
Ã
exp
µ
2π i z
1
a
p
¶
, exp
µ
2π i z
2
a
p
¶
, . . . , exp
Ã
2π i z
p−1
2
a
p
!!
,
{z
1
, . . . , z
p−1
2
} F
p
a 6= b
(π(a), π(b))
−1
p−1
λ(π(a), π(b)) =
r
2p
p − 1
.
K(G, a)
p−1
2
×
p−1
2
−
g
a
= diag (exp
µ
2π i z
1
a
p
¶
, exp
µ
2π i z
2
a
p
¶
, . . . , exp
Ã
2π i z
p−1
2
a
p
!
), a ∈ F
p
,
{z
1
, . . . , z
p−1
2
} F
p
p−1
2
− a =
¡
p−1
2
¢
−
1
2
(1, . . . , 1), π(F
p
)
F
p
X
n
n ≥ 1
a = (a
1
, . . . , a
n
) ∈ X
n
π
Y
i
, i = 1, 2, 3, X, |X| = q,
S
n
0
U
n
0
π(a) =
1
√
n
(π(a
1
), . . . , π(a
n
)) ∈ S
n
0
n−1
, (U
n
0
n−1
)
X
n
S
n
0
n−1
.
λ
2
(π(a), π(b)) = 2 −
2
n
<{(π(a
1
), π(b
1
))
C
+ ··· + (π(a
n
), π(b
n
))
C
} =
1
n
(λ
2
(π(a
1
), π(b
1
)) + ··· + λ
2
(π(a
n
), π(b
n
)) =
1
n
d(a, b),
(·, ·)
C
C
n
0
<x
x
n > 1
K ⊆ F
n
q
S
n
0
n−1
F
n
q
K
K ∈ X
n
n · n
0
n·n
0
n · n
0
K ∈ X
n
S
m−2
R
m−1
m = |K|
a
j
, j = 1, . . . , m
R
m−1
π(x)
|K| a
j
, j = 1, 2, . . . , m
d(x, y) = cλ(π(x), π(y)) K
|K| ≤ n + 1 q−
|K| n d < n n > 1 q > 2
|K| (q −1)n
K G K
S
m−2
⊂ R
m−1
x ∈ K
G
Y K ⊆ F
n
q
π
K(F
q
, a) a Y
n
K
π
K ⊆ F
n
q
π S
n
0
n−1
X
n
, n > 1, S
N−1
N
n
0
n n
0
X
J
w,n
a b
J
w,n
n
0
= n
J
w,n
, 0 < 2w ≤ n,
a, b wa
2
+ (n − w)b
2
= 1
x ∈ J
w,n
π : x → π(x)
x −b a J
w,n
π(J
w,n
) S
n−1
x, y ∈ J
w,n
π
(π(x), π(y)) = (w−
d(x, y)
2
)a
2
+(n−w −
d(x, y)
2
)b
2
−d(x, y)ab = 1−
d(x, y)
2
(a+b)
2
.
λ(π(x), π(y)) = d(x, y)(a + b)
2
= 2j(x, y)(a + b)
2
,
j(x, y)
(a + b)
2
wa
2
+ (n −w)b
2
= 1
n
w(n−w)
a =
q
n−w
wn
b =
q
w
n(n−w)
π a, b
(a + b)
2
K ⊆ J
w,n
j
π(K)
2jn
w(n−w)
¤
M
q
(n, d) q− K ⊆
X
n
, |X| = q, n d
V
t,n
(x) t x X
n
x t
|V
t,n
(x)| =
t
X
s=0
(q − 1)
s
µ
n
s
¶
.
V
t,n
(x)
d = 2t + 1
M
q
(n, d) ≤
q
n
|V
t,n
(x)|
=
q
n
P
t
s=0
(q − 1)
s
¡
n
s
¢
.
d(x, x
0
) ≥ 2t + 1, x, x
0
∈ X
n
V
t,n
(x) V
t,n
(x
0
)
d
K = {x
1
, . . . , x
M
} d = 2t + 1
V
t,n
(x
i
) V
t,n
(x
j
), i 6= j,
M
[
s=1
V
t,n
(x
s
) ⊆ X
n
.
M
X
s=1
|V
t,n
(x
s
)| = M|V
t,n
(x
s
)| ≤ |X
n
| = q
n
,
¤
K
t
X
n
d = 3, q = 2 U
n
=
2
n
n+1
n = 2
m
− 1 U
n
= 2
n−log
2
(n+1)
n
3
(0, . . . , 0)
(1, . . . , 1)
7 2
12
11 3
6
V
3,23
(x) V
2,11
(x) 3 2
2
11
= 1+
¡
23
1
¢
+
¡
23
2
¢
+
¡
23
3
¢
3
5
= 1+ 2
¡
11
1
¢
+2
2
¡
11
2
¢
3