Сидельников В.М. Теория кодирования. Справочник по принципам и методам кодирования
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λ
H,a
F
n
p
H = Φ
p−1
2
p = 4t − 1
λ
H,a
λ
H,a
S = S(X, R
0
, . . . , R
m
) Y ⊆ X X
α
i
= α
i
(Y ) = |Y × Y ∩ R
i
|, i = 0, . . . , m.
α
S
(Y ) = (α
0
, . . . , α
m
)
S Y S = H
n
q
α
S
(Y )
Y Y
Y = K n F
q
K
⊥
ν(K) = α
i
(K)
ν(K
⊥
) K K
⊥
f
F
n
q
F
n
q
f
F
n
q
f
F
n
q
F
q
F
n
q
F
q
F
n
q
F
p
θ
a
: x = (x
1
, . . . , x
n
) →
ha, xi, a = (a
1
, . . . , a
n
), x ∈ F
n
q
ha, xi F
q
q = p
f
F
n
q
F
n
p
K
F
n
q
K
⊥
⊆
f
F
n
q
θ
a
(x) ∈
f
F
n
q
K
K
f
G
n
G
n
f
G
n
G
n
G
f
G
n
θ(x) ∈
f
G
n
θ(x) · θ
0
(x) ∈
f
G
n
θ(x), θ
0
(x) ∈
f
G
n
f
G
n
G
G
n
G
f
G
n
G
G
n
→ G
G G
n
K G
n
G
n
e
K K
f
G
n
θ(x) ∈
f
G
n
e ∈ G K
S
H
(G
n
) H
G
n
H
e
H
f
G
n
S
e
H
(
f
G
n
)
S
H
(G
n
)
H S
H
(G
n
)
α
S
(K) S
H
(G
n
) K
α
e
S
(
e
K)
e
K
α
S
(K)
α
e
S
(
e
K)
q−
H
L
2− E
⊗n
2
2m+1
L F
n
2
≥ d
Z/4Z
Q
L
(Γ) Γ ⊂ {(L
⊥
)
c
× L
c
} L
F
n
2
L
c
L F
n
2
= L ⊕ L
c
{e
α,β
L
; α, β ∈ Γ} e
α,β
L
H
L
E
⊗n
H
L
L
⊥
⊂ L
e
α,0
L
Q
L
(Γ)
≥
n = 2
m
n q− K = 2
n−m−2
4
3
n = 8
K = 2
3
= 2
8−3−2
C K
K− 2
n
− V = C
2
n
n q− n
V n−
F
n
2
{e
α
|α ∈ F
n
2
}
V e
α
= (0, . . . 0, 1, 0, . . . , 0) 1
α
½
I
2
= ±
µ
1 0
0 1
¶
, σ
x
= ±
µ
0 1
1 0
¶
, σ
y
= ±
µ
0 −i
i 0
¶
, σ
z
= ±
µ
1 0
0 −1
¶¾
.
⊗
2 × 2− 4 × 4−
µ
a b
c d
¶
⊗
µ
a
0
b
0
c
0
d
0
¶
=
a
µ
a
0
b
0
c
0
d
0
¶
b
µ
a
0
b
0
c
0
d
0
¶
c
µ
a
0
b
0
c
0
d
0
¶
d
µ
a
0
b
0
c
0
d
0
¶
.
n ×m− n
0
×m
0
−
nn
0
× mm
0
− n
n
0
nn
0
U = σ
1
⊗ σ
2
⊗ ··· ⊗ σ
n
,
V (2 × 2)− σ
i
U d −1
U
d−1
d−1
q− {±σ
x
, ±σ
y
, ±σ
z
, }
hv|U|wi, w, v ∈ V, vUw
T
C d
hv|U
d−1
|wi = 0
v w C U
d−1
d − 1 q
C d
d − 1
q−
C d
hv|U
d−1
|vi = hw|U
d−1
|wi
v w C U
d−1
d − 1 q−
v w
v −w v +w
0 = h(v − w)|U
d−1
|(v + w)i = hv|U
d−1
|vi −hw|U
d−1
|wi),
v w
v + w, v − w, v − iw
0 = h(v + w)|U
d−1
|(v + w)i−h(v −w)|U
d−1
|(v −w)i = 2(hv|U
d−1
|wi+ hw|U
d−1
|vi)
0 = h(v + w)|U
d−1
|(v + w)i − h(v − iw)|U
d−1
|(v − iw)i =
(1 + i)hv|U
d−1
|wi + (1 − i)hw|U
d−1
|vi.
λ
(0)
=
µ
1 0
0 1
¶
= I
2
, λ
(1)
=
µ
i 0
0 −i
¶
= iσ
z
,
λ
(2)
=
µ
0 1
−1 0
¶
= iσ
y
, λ
(3)
=
µ
0 i
i 0
¶
= iσ
x
,
Q (2 × 2)−
Q = {±λ
(0)
, ±λ
(1)
, ±λ
(2)
, ±λ
(3)
}
8
1, i, j, k
Q
ψ
ψ(±λ
(0)
) = ψ(±λ
(2)
) = 1, ψ(±λ
(1)
) = ψ(±λ
(3)
) = −1.
τ
(0)
=
µ
1 0
0 1
¶
= I
2
, τ
(1)
=
µ
1 0
0 −1
¶
,
τ
(2)
=
µ
0 1
−1 0
¶
, τ
(3)
=
µ
0 1
1 0
¶
,
λ
(j)
λ
(j)
= J
ω(λ
(j)
)
τ
(j)
,
J =
µ
i 0
0 i
¶
ω(λ
(j)
) =
1 − ψ(λ
(j)
)
2
=
0 λ
(j)
1
E = {±τ
(0)
, ±τ
(1)
, ±τ
(2)
, ±τ
(3)
}
8 2−
E
E
U
d−1
D · U
d−1
, D = diag(d, . . . , d) ∈ U
2
n
, d ∈ C, |d| = 1
hv|U
d−1
|wi = 0 hv|D · U
d−1
|wi = dhv|U
d−1
|wi = 0
U
S
S = τ
1
⊗ τ
2
⊗ ··· ⊗ τ
n
,
τ
i
∈ E
τ
λ
Q
E
⊗n
E
⊗n
= E ⊗ ··· ⊗ E n
(2
n
× 2
n
) S
2
S S
0
(τ
1
⊗ τ
2
⊗ ··· ⊗ τ
n
)·(τ
0
1
⊗ τ
0
2
⊗ ··· ⊗ τ
0
n
) = τ
1
τ
0
1
⊗ τ
2
τ
0
2
⊗ ··· ⊗ τ
n
τ
0
n
,
E
⊗n
E
⊗n
2
2n+1
G = E × ··· × E/H H = {²
1
I
2
× ··· × ²
n
I
2
| ²
1
···²
1
= 1}, ²
1
∈
{1, −1}, 2
n−1
E
⊗n
τ
1
⊗ τ
2
⊗ ··· ⊗ τ
n
= ²
1
τ
1
⊗ ²
2
τ
2
⊗ ··· ⊗ ²
n
τ
n
, ²
1
···²
1
= 1.
τ
1
⊗ τ
2
⊗ ··· ⊗ τ
n
= τ
0
1
⊗ τ
0
2
⊗ ··· ⊗ τ
0
n
,
τ
j
= ±τ
0
j
n
¤
wt(S) S ∈ E
⊗n
±τ
(0)
τ
j
wt(S)
S
wt(S)
wt(τ
1
⊗ τ
2
⊗ ··· ⊗ τ
n
) = wt(τ
0
1
⊗ τ
0
2
⊗ ··· ⊗ τ
0
n
),
τ
1
⊗ τ
2
⊗ ··· ⊗ τ
n
= τ
0
1
⊗ τ
0
2
⊗ ··· ⊗ τ
0
n
F
n
2
= {x
1
, . . . , x
2
n
}
F
n
2
T
(α)
S
(β)
α, β ∈ F
n
2
T
(α)
T
(α)
= diag
¡
(−1)
(α,x
1
)
, . . . , (−1)
(α,x
2
n
)
¢
,
S
(β)
= (s
γ,δ
) (2
n
× 2
n
)
x → x + β F
n
2
β s
γ,δ
= 1
γ = δ + β s
γ,δ
= 0
T
(α)
S
(β)
F
n
2
E
⊗n
T
(α)
=
¡
τ
(1)
¢
α
1
⊗
¡
τ
(1)
¢
α
2
⊗ ··· ⊗
¡
τ
(1)
¢
α
n
S
(β)
=
¡
τ
(3)
¢
β
1
⊗
¡
τ
(3)
¢
β
2
⊗ ··· ⊗
¡
τ
(3)
¢
β
n
T
(α)
S
(β)
= (−1)
(α,β)
S
(β)
T
(α)
,
(α, β) α β F
n
2
E
⊗n
T
(α)
S
(β)
E
⊗n
= hT
(α)
, S
(β)
|α, β ∈
F
n
2
i
S = ±T
(α)
· S
(β)
wt(S) = wt(α ∨ β) wt(γ)
γ α ∨ β = (α
1
∨ β
1
, . . . , α
n
∨ β
n
)
S = T
(α)
S
(β)
2
n
− V
e
γ
= (0, . . . , 0, 1, 0, . . . , 0), γ ∈ F
n
2
, γ
V
e
γ
S = e
γ
T
(α)
S
(β)
= (−1)
(γ,α)
e
γ+β
.
H
L
E
⊗n
H E
⊗n
H
H
L
= {T
α
· S
β
|α ∈ L
⊥
, β ∈ L}, |H
L
| = 2
n
L k
F
n
2
H
L
P ∈ E
⊗n
P 6∈ H
L
hH
L
, P i
H
L
E
⊗n
L
c
, dim L
c
= n −k, L F
n
2
L
c
F
n
2
= L ⊕ L
c
(L
c
)
c
= L
L
d
= (L
⊥
)
c
, dim L
d
= k L
d
∩
L
⊥
= 0 L
d
(β, x), x ∈
L L β ∈ L
d
\ {0}
(L
d
)
d
= L
L
d
L = L
k
D L
k
= {(α
1
, . . . , α
k
, 0,
. . . , 0)|α
j
∈ F
2
} F
n
2
D
L
k
L L
d
= L
k
(D
T
)
−1
(β, x) (β, x) = (β
0
(D
T
)
−1
, x
0
D) = β
0
1
x
0
1
+ ··· + β
0
k
x
0
k
L β ∈ L
L F
n
2
e
l,τ
L
, l, τ ∈ F
n
2
e
l,τ
L
=
X
γ∈L
(−1)
(l,γ)
e
γ+τ
,
L
l,τ
⊂ R
2
n
2
n
−
e
l,τ
L
= (−1)
(β,l)
e
l+α,τ +β
L
,
β ∈ L α ∈ L
⊥
e
l,τ
L
, e
l
0
,τ
0
L
τ, τ
0
∈ L
c
, l, l
0
∈ (L
⊥
)
c
(l , τ) 6= (l
0
, τ
0
) L
l,τ
2
n
L
l,τ
, l, τ ∈ F
n
2
,
H
L
S = T
α
S
β
∈ H
L
α ∈ L
⊥
, β ∈ L
e
l,τ
L
S =
X
γ∈L
(−1)
(l+α,γ)
e
γ+β+τ
=
X
γ∈L
(−1)
(l+α,γ+β)
e
γ+τ
= (−1)
(l,β)
e
l,τ
L
,
L
l,τ
¤
L
l,τ
2
n
H
L
ϕ
l,τ
(S), S ∈ H
L
e
l,τ
S =
ϕ
l,τ
(S)e
l,τ
H
L
Q
L
(Γ) 2
n
−
V {e
l,β
L
|(l , β) ∈ Γ} Γ
(L
⊥
)
c
× L
c
e
l,β
L
L
F
n
2
U
d−1
= T
(τ)
S
(τ
0
)
d − 1 q− wt(τ ∨ τ
0
) < d
e
l,β
L
U
d−1
=
X
γ∈L
(−1)
(l+τ,γ)
e
γ+β+τ
0
= e
l+τ,β+τ
0
L
.
e
l,β
L
e
l
0
,β
0
L
, l
0
, l ∈ (L
⊥
)
c
, β
0
, β ∈ L
c
(l , β) 6= (l
0
, β
0
)
H
L
β
0
6≡ β mod L β
0
6= β + α α ∈ L β
0
= β + α
α ∈ L β
0
≡ β mod L
e
l
0
,β
0
L
e
l,β
L
l
0
6≡ l mo d L
⊥
, β 6≡ β
0
mod L.
l
0
≡ l mod L
⊥
β ≡ β
0
mod L e
l
0
,β
0
L
e
l,β
L
e
l
0
,β
0
L
e
l,β
L
H
L
L
l
0
,β
0
L
l
0
,β
0
e
l,β
L
e
l
0
,β
0
L
e
l,β
L
¤
Q
L
(Γ), Γ ⊆ (L
⊥
)
c
× L
c
d τ, τ
0
wt(τ ∨ τ
0
) < d (l, β) (l
0
, β
0
) Γ
β + β
0
+ τ
0
6≡ 0 mod L
l
0
+ l + τ 6≡ 0 mod L
⊥