Сидельников В.М. Теория кодирования. Справочник по принципам и методам кодирования
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U
d−1
= T
τ
S
τ
0
e
l,β
L
U
d−1
= e
l+τ,β+τ
0
L
e
l
0
,β
0
L
, (l
0
, β
0
) ∈ Γ
β + β
0
+τ
0
≡ 0 mod L l
0
+l +τ ≡ 0 mod L
⊥
e
l,β
L
U
d−1
= e
l+τ,β+τ
0
L
= ±e
l
0
,β
0
L
e
l,β
L
, e
l
0
,β
0
L
¤
L F
n
2
≥ d F (L
⊥
)
c
Q
L
(Γ) Γ = F ×0 d
f + f
0
+ τ 6≡ 0 mod L
⊥
f f
0
F τ
wt(τ) < d F (L
⊥
)
c
Q
L
(Γ) d
f + τ 6≡ 0 mod L
⊥
f ∈ F τ wt(τ) < d
Q
L
(Γ) ≥ d
d(F \ {0}, L
⊥
) F \ {0} L
⊥
d
L
⊥
⊂ L
˜
L L
˜
L ∩L
⊥
= {0}
˜
L L
⊥
L
L = L
⊥
⊕
˜
L
L d d(
˜
L\{0}, L
⊥
) ≥ d
L
⊥
⊂ L
F
˜
L Q
L
(Γ) Γ =
˜
L ×0
d = d(L) |
˜
L| =
|L|
2
2
n
Q
L
(Γ) dimL − dimL
⊥
= n − 2l l = n − k
L
Q
L
(Γ)
U
d−1
= T
(τ)
S
(τ
0
)
wt(τ) < d
wt(τ
0
) < d U
d−1
d
wt(τ ∨ τ
0
) < d
Γ Γ =
˜
L × 0
n = 2
m
C
H
n = 2
m
4 k = 2
m
−1−m
C
ch
n 2 k
0
= 2
m
− 1
M = {x
1
, . . . , x
n
} ⊂ F
n
2
1 y
M C
ch
C
H
C
ch
=
[
x∈M
(C
H
+ x + y).
M 2−
˙
+ modC
H
x
˙
+x
0
=
z x + y + x
0
+ y + z + y ≡ 0 mod C
H
F
n
2
, n = 2
m
, F
2
m
=
{α
1
, . . . , α
n
} a = (a
α
1
, . . . , a
α
n
), a
α
j
∈ F
2
, C
H
a
α
1
α
1
+ ··· + a
α
n
α
n
= 0 a
α
1
+ ··· + a
α
n
= 0 a
α
M
α a
0
= y
a
α
˙
+a
β
= a
γ
a
α
+ y + a
β
+ y + a
γ
+ y ∈ C
H
a
α
+ y + a
β
+ y +
a
γ
+ y ≡ 0 mod C
H
y α, β, γ
α + β + γ = 0 M
2−
˙
+
θ F
2
m
M
˙
× M x
˙
×y = θ(θ
−1
(x) ·θ
−1
(y)) ·
F
m
2
M
˙
×
˙
+
˙
+ modC
H
Q
L
(Γ) L
L = C
H
(C
H
)
⊥
⊂ C
H
, L = (C
H
)
⊥
⊕
˜
L
dim
˜
L = 2
m
− 2m − 2
z M θ
−1
(z) 6= 0, 1 ∈ F
2
m
Γ = Γ(z) =
[
x∈M
{(
˜
L + z
˙
×x) × (x + y)}.
Q
L
(Γ(z)) n = 2
m
3
2
n
4n
τ, τ
0
∈ F
n
2
, wt(τ ∨τ
0
) ≤ 2
U
2
(l + z
˙
×x
i
, x
i
+y), (l
0
+z
˙
×x
j
, x
j
+y), x
i
, x
j
∈ M,
Γ
x
i
+ x
j
+ τ
0
6∈ L l + l
0
+
z
˙
×x
i
+ z
˙
×x
j
+ τ 6∈ L L =
˜
L + L
⊥
τ
0
= 0, 0 < wt(τ) ≤ 2 (x
i
+y) + (x
j
+y) ∈ L
x
i
= x
j
l + l
0
+ τ 6∈ L τ
4 l + l
0
6= 0 L 4
wt(τ
0
) = 1 (x
j
+ y) + (x
i
+ y) + τ
0
6∈ L L
(x
j
+ y) + (x
i
+ y) + τ
0
wt(τ
0
) = 2, wt(τ ) = 2 x
i
+ x
j
+ τ
0
∈ L (l + z
˙
×x
i
, x
i
+ y), (l
0
+
z
˙
×x
j
, x
j
+ y) Γ
x
j
6= x
i
wt(τ ∨ τ
0
) ≤ 2 τ = τ
0
l + l
0
+ z
˙
×x
i
+ x
i
+ z
˙
×x
j
+ x
j
≡ (z
˙
+z
0
)
˙
×x
i
+ (z
˙
+z
0
)
˙
×x
j
mod L
L z
0
∈ M θ
−1
(z
0
) = 1 ∈ F
m
2
((z
˙
+z
0
)
˙
×x
i
(z
˙
+z
0
)
˙
×x
j
1 (z
˙
+z
0
)
˙
×x
i
+ (z
˙
+z
0
)
˙
×x
j
6≡ 0 mod L L
4 wt((z
˙
+z
0
)
˙
×x
i
+ (z
˙
+z
0
)
˙
×x
j
) = 2
wt(τ
0
) = 2, wt(τ) = 1 l+l
0
+z
˙
×x
i
+
z
˙
×x
j
+ τ ∈ L l + l
0
+ z
˙
×x
i
+ z
˙
×x
j
+ τ
L
wt(τ
0
) = 2, wt(τ) = 0
l + l
0
+ z
˙
×x
i
+ z
˙
×x
j
6≡ 0 mod L x
j
6= x
i
wt(z
˙
×x
i
+ z
˙
×x
j
) = 2
Q
L
(Γ) ¥
Q
L
(Γ(z)), θ
−1
(z) 6= 0, 1 ∈ F
2
m
, 2
m
− 2
Q
L
(Γ)
K n
q− d = 2t + 1
K
t
X
j=0
3
j
µ
n
j
¶
≤ 2
n
.
4
3
n =
4
m
−1
3
k = 2m d = 3
5
n = 2
m
5
K = 2
m
− 3m − 3
K = 2
m
−4m−2
K ≤ 2
m
− 2m + const 5
C
6
n = 2
m
6 k = 2
m
− 1 − 2m
B C
6
B =
1 1 1 1 1
α
1
α
2
. . . α
N
0
α
3
1
α
3
2
. . . α
3
N
0
, N = 2
m
− 1, α
j
∈ F
2
m
\ {0}.
F
n
2
, n = 2
m
,
F
2
m
= {α
1
, . . . , α
n
} x
α
F
n
2
, n = 2
m
, α 1
0
M
0
= {x
α
+ x
α+1
|α ∈ F
2
m
} M
0
2
m−1
z
α
= x
α
+ x
α+1
F
n
2
2 z
α
= z
α+1
z
α
M
0
α
{y
m−1
, . . . , y
0
} F
m
2
F
2
y
0
0
Q
L
(Γ) 5
L L = C
6
(C
6
)
⊥
⊂ C
6
, L = (C
6
)
⊥
⊕
e
L
dim
e
L = 2
m
− 4m − 2
Γ =
[
α∈M
0
{(
e
L + x
α
) × (z
α
)}.
Γ 2
m−1
|
e
L| = 2
2
m
−3m−3
Q
L
(Γ) n = 2
m
5
K = 2
m
− 3m − 3
τ, τ
0
∈ F
n
2
, wt(τ ∨τ
0
) ≤ 4
U
2
(l + x
α
, z
α
), (l
0
+ x
β
, z
β
), α, β ∈ M
0
,
Γ
x
α
+ x
β
+ τ
0
6∈ L l + l
0
+ z
β
+ z
α
+ τ 6∈ L
L =
e
L + L
⊥
x
α
1
+ ···+ x
α
s
∈ L
α
1
+ ··· + α
s
= 0, α
1
3
+ ··· + α
s
3
= 0, s
wt(τ
0
) = 0, 0 < wt(τ) ≤ 4 z
α
+ z
β
∈ L
x
α
= x
β
β = α l + l
0
+ τ 6∈ L
τ 5 L
6
wt(τ
0
) = 1, 3 z
α
+z
β
+τ
0
6∈ L L
x
α
+ x
β
+ τ
0
wt(τ
0
) = 2, τ
0
= x
γ
+x
δ
, 0 ≤ wt(τ) ≤ 4 z
α
+z
β
+τ
0
∈ L
α + α + 1 + β + β + 1 + γ + δ = γ + δ = 0
γ + δ 6= 0
wt(τ
0
) = 4, τ
0
= x
γ
1
+ x
γ
2
+ x
γ
3
+ x
γ
4
, wt(τ) = 4 wt(τ
0
∨τ) ≤ 4
τ
0
= τ
z
α
+ z
β
+ τ
0
∈ L α + α + 1 + β + β + 1 + γ
1
+ γ
2
+ γ
3
+ γ
4
= γ
1
+ γ
2
+ γ
3
+ γ
4
= 0
x
α
+ x
β
+ τ ∈ L α + β + γ
1
+ γ
2
+ γ
3
+ γ
4
= 0
γ
1
+ γ
2
+ γ
3
+ γ
4
= 0 α + β 6= 0
K Q
L
(Γ) square
Q
L
(Γ)
t = 2
m + const
RS
q
(n, d), n = q
a K a
0
K σ
σ(a) = a
0
∈ K a ∈ K
K σ
0
σ · σ
0
K Σ
K
K
F
n
q
· S
n
n!
σ Γ
σ
= Γ =
kγ
i,j
k
γ
i,j
1 i
σ j γ
i,j
= 0
Γ
1 Γ
σ a σ(a) = aΓ
G = G
K
Γ
σ
σ ∈ Σ
K
Γ ∈ G
K
B K B · Γ
K
B ·Γ = h ·B h n − k ×n −k
B
B
0
B
0
= h · B
Γ → h
G
K
K n×n h
n −k ×n −k J(K) Γ
K h G
K
B ·Γ = h·B G
K
/J(K)
J(K)
h
G
K
RS
q
(n, d) BCH
q
(n, d)
B
K
B = {B · Γ|Γ ∈ S
n
} B
N
q
(n, d) K =
RS
q
(n, d) B
K
N
q
(n, d) =
n!
|G
K
|
,
K = RS
q
(n, d)
G
K
K = RS
q
(n, d) G
RS
q
(n,d)
x → ax, a ∈ F
q
\{0} = F
∗
q
,
F
q
F
q
n = q −1
A
RS
q
(n, d)
h d − 1 × d − 1
B hB
RS
q
(n, d)
B hB h 6= E
RS
q
(n, d) N
q,d−1
N
q,s
h
s × s
N
q,s
N
q,s
= (q
s
− 1)(q
s
− q) ···(q
s
− q
s−1
).
h F
q
s×s q
s
−1 s
q
s
− q
q
s
−q
2
2 F
s
q
h q
s
− q
s−1
s −1− s −1 h
RS
q
(n, d)
RS
q
(n, d), n = q +
1
K
Γ Λ = Γ·D D
Λ
F
q
d(a, b) = d(aΛ, bΛ)
RS
q
(n, d)
RS
q
(n, d)
Λ aΛ = a
0
∈ K a ∈ K
K Λ
0
Λ·Λ
0
K Ξ
K
K Ξ
K
n × n H
K
Ξ
K
F
q
n − k × n − k Λ Ξ
K
h = h
Λ
h
Λ
· B = B · Λ.
Λ·Λ
0
Ξ
K
g(Λ·Λ
0
) = h
Λ
0
·h
Λ
H
K
H
K
Ξ
K
g : Λ → h
Λ
Ξ
K
n−k×n−k
F
q
K = RS
q
(n, d) g
|Ξ
K
| = |H
K
|
g
B B 6= B · Λ
Λ
Λ a = aΛ a ∈ RS
q
(n, d)
Ξ
K
K = RS
q
(n, d)
N
q,d−1
N
q,s
h
s × s F
q
|Ξ
K
| = |H
K
| |Ξ
K
| ≤ N
q,n−k
, k =
n −d + 1 |H
K
| d −1 ×d −1
F
q
Ξ
K
A
K
, K = RS
q
(n, d),
B = {BΛ|Λ ∈ U
q,n
} B
U
q,n
F
q
A
K
|U
q,n
| = n!(q − 1)
n
A
K
A
q
(n, d)
K = RS
q
(n, d) A
K
A
q
(n, d) =
n!(q − 1)
n
|Ξ
K
|
.
Ξ
K
A
q
(n, d)
A
q
(n, d) K = RS
q
(n, d)
A
K
A
q
(n, d) ≥
n!(q − 1)
n
N
q,k
=
n!(q − 1)
n
(q
d−1
− 1)(q
d−1
− q) ···(q
d−1
− q
d−2
)
,
k = n − d + 1 K = RS
q
(n, d) N
q,k
k × k
Ξ
K
Φ
q
b
F
q
= F
q
∪ {∞} Φ
q
ϕ(x) =
ax+b
cx+e
µ
a b
c e
¶
·
ϕ(x) ·ϕ
0
(x) = ϕ(ϕ
0
(x)) φ(x)
b
F
q
Φ
q
P GL(2, q) (q + 1)q(q − 1)
Φ
q
(a
1
, a
2
, a
3
) (b
1
, b
2
, b
3
), a
i
, b
i
∈ F
0
q
,
Φ
q
φ
φ(a
i
) = b
i
, i = 1, 2, 3
Ξ
K
K = RS
q
(n, d), n = q + 1,
B
F
0
q
B
F
0
q
B(α
j
) = (α
0
j
, α
1
j
, . . . , α
d−2
j
)
T
α
j
φ(x) =
ax+b
cx+e
Γ
φ
x → φ(x) F
0
q
D
φ
=
diag((cα
1
+ e)
d−2
, (cα
2
+ e)
d−2
, . . . , (cα
n
+ e)
d−2
)
φ(x) F
0
q
B(α
j
) · Γ
φ
· D
φ
= ((cα
j
+ e)
d−2
, (aα
j
+ b)(cα
2
+
e)
d−3
, . . . , (aα
j
+b)
d−3
(cα
2
+e), (aα
j
+b)
d−2
)
T
(ax+b)
d−2−i
(cx+e)
i
1, x, , . . . , x
d−2
B(α
j
)·
Γ
φ
· D
φ
B(α
j
)Γ
φ
· D
φ
= h(1, x, x
2
, . . . , x
d−2
)
T
B · Γ
φ
· D
φ
B · Γ
φ
· D
φ
= h · B
h = {h
i,j
} (ax + b)
d−2−i
(cx + e)
i
=
P
d−2
i=0
h
i,j
x
j
φ Γ
φ
· D
φ
Ξ
K
Λ
φ
= Γ
φ
·D
φ
Φ
q
Γ
−1
φ
·D
φ
0
·Γ
φ
= diag((c
0
φ(α
1
) + e
0
)
d−2
, . . . , (c
0
φ(α
n
) + e
0
)
d−2
) = D
φ
0
,φ
φ
0
(x) =
a
0
x+b
0
c
0
x+e
0
D
φ
0
·Γ
φ
= Γ
φ
·D
φ
0
,φ
Γ
φ
0
·D
φ
0
·Γ
φ
·D
φ
= Γ
φ
0
⊗φ
·
D
φ
0
,φ
·D
φ
D
φ
0
,φ
·D
φ
= D
φ
0
⊗φ
Γ
φ
· D
φ
Φ
q
¤
Ξ
K
(β
1
, β
2
, β
3
) (γ
1
, γ
2
, γ
3
) {β
1
, β
2
, β
3
}, {γ
1
, γ
2
, γ
3
} ∈
A = {α
1
, α
2
, . . . , α
n
} = F
0
q
Λ
φ
∈ Ξ
K
x
β
1
, x
β
2
, x
β
3
x = (x
α
1
, x
α
2
, . . . , x
α
n
)
x
γ
1
, x
γ
2
, x
γ
3
xΛ
φ
D
φ
= diag(d
α
1
, d
α
2
, . . . , d
α
n
)
Λ
φ
x {β
1
= 1, β
2
= 0, β
3
= ∞} γ
1
= α
1
, γ
2
= α
2
, γ
3
= α
3
xΛ
φ
= (d
α
1
x
1
, d
α
2
x
0
, d
α
3
x
∞
, d
α
4
x
φ(α
4
)
, . . . , d
α
n
x
φ(α
n
)
)
φ(x)
a
0
a
a
0
= a + e, a ∈ K, wt(e) ≤ t,
a e
a e a
0
K {e|wt(e) ≤ t}