Сидельников В.М. Теория кодирования. Справочник по принципам и методам кодирования
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(T r(s(ax) + bx + ε), T r(b
0
x + ε
0
) = (−1)
ε+ε
0
T (a, b + b
0
)
(T r(s(ax) + (a + b)x + γ + ε)), T r(+b
0
)x + γ
0
+ ε
0
)) =
= (−1)
ε+ε
0
+γ+γ
0
T (a, b + b
0
+ a)
0 2
t
= 2
m+1
2
a = a
0
, b + b
0
6= 0
(Υ(a, b, ε, γ), Υ(a, b
0
, ε
0
, γ
0
))
a = a
0
, b = b
0
, (ε, γ) 6= (ε
0
, γ
0
) (Υ(a, b, ε, γ), Υ(a, b, ε
0
, γ
0
)) = 0
γ + γ
0
6= 0 (Υ(a, b, ε, γ), Υ(a, b, ε
0
, γ
0
)) = −2
m
γ + γ
0
= 0 ε + ε
0
6= 0
(a, b, ε, γ) (a
0
, b
0
, ε
0
, γ
0
)
4n
2
³
¡
n
2
¢
2
−
n
2
´
= n
4
− 2n
3
|(Υ(a, b, ε, γ), Υ(a, b, ε
0
, γ
0
))| = 2
m+1
2
K
m
Υ
Υ, Υ
0
∈ K
m
(Υ(a, b, ε, γ), Υ(a, b, ε
0
, γ
0
)) = −2
m
2
(Υ(a, b, ε, γ), Υ(a, b, ε
0
, γ
0
)) = 2
m+1
2
(a, b, ε, γ) (a
0
, b
0
, ε
0
, γ
0
)
8n
³
¡
n
2
¢
2
−
n
2
´
= 2n
3
− 4n
2
(a, b, ε, γ) (a
0
, b
0
, ε
0
, γ
0
)
(Υ(a, b, ε, γ), Υ(a, b, ε
0
, γ
0
)) = 0 8
¡
n
2
¢
2
= 2n
2
(Υ(a, b, ε, γ), Υ(a, b, ε
0
, γ
0
)) = ±n 2n
2
¤
Z
4
4
Z
4
Υ ∈ K(m)
RM
2,m+1
m + 1 = 2t Υ
f(x), x ∈ F
2
m+1
2
f(x)
2
T r(s(ax) + bx) x = x
1
ω
1
, . . . , x
m
ω
m
f(x
1
, . . . , x
m
)
K(m) Υ
g
Υ
(x
1
, . . . , x
m
, x
m+1
) = f(x
1
, . . . , x
m
) + x
m+1
l(x
1
, . . . , x
m
) + ε, ε ∈ F
2
x
m+1
= γ l(x
1
, . . . , x
m
) F
2
T r(ax), x = x
1
ω
1
+ ··· + x
m
ω
m
g
Υ
(x
1
, . . . , x
m
, x
m+1
) s(ax) 6=
0 m + 1 x = yA + α x =
(x
1
, . . . , x
m
, x
m+1
) A
g((x
1
, . . . , x
m
, x
m+1
)A + α) =
P
t
i=1
y
2i−1
y
2i
+ ε
g
Υ
(x
1
, . . . , x
m
, x
m+1
) +
g
Υ
0
(x
1
, . . . , x
m
, x
m+1
) m+ 1
m n = 2
m
− 1 X ⊂ F
2
m
Y ⊂ F
2
m
F
2
m
ϕ
X
(x) =
(
1, x ∈ X ,
0, x 6∈ X
X
F
2
m
= (α
1
, α
2
, . . . , α
2
m
)
F
2
m
(X, Y ) = (ϕ
X
, ϕ
Y
)
2
m+1
ϕ
X
= (ϕ
X
(α
1
), ϕ
X
(α
2
), . . . , ϕ
X
(α
2
m
)) ϕ
Y
= (ϕ
Y
(α
1
), ϕ
Y
(α
2
), . . . , ϕ
Y
(α
2
m
)).
(X, Y )
X Y
P
m
P
m
(X, Y ) ⊂ F
m+1
2
X Y
P
α∈X
ϕ
X
(α) =
=
P
α∈Y
ϕ
X
(α) = 0
P
x∈X
x =
P
y∈Y
y.
P
x∈X
x
3
+
¡
P
x∈X
x
¢
3
=
P
y∈Y
y
3
.
P
m
(X, Y ), (X
0
, Y
0
) ∈
P
m
(X + X
0
, X + Y
0
) P
m
P
m
6
P
m
2
2
m
−2m
(X, Y ), (X
0
, Y
0
) ∈ P
m
wt(ϕ
X
+ ϕ
X
0
) + wt(ϕ
Y
+ ϕ
Y
0
) > 4
d((X, Y ), (X
0
, Y
0
)) ≥ 6
(ϕ
X
|ϕ
Y
) (ϕ
X
0
|ϕ
Y
0
)
(X, Y ), (X
0
, Y
0
)
P
m
wt(ϕ
X
+ ϕ
X
0
) + wt(ϕ
Y
+ ϕ
Y
0
) ≤ 4
ϕ
X
+ ϕ
X
0
= ϕ
X4X
0
X4X
0
= (X ∪ X
0
) \ (X
0
∩ X)
X X
0
(X, Y ), (X
0
, Y
0
) ∈ P
m
wt(ϕ
X
+ϕ
X
0
) = 2, wt(ϕ
Y
+ϕ
Y
0
) = 0
(X, Y ), (X
0
, Y
0
) ∈ P
m
wt(ϕ
X
+ ϕ
X
0
) +
wt(ϕ
Y
+ ϕ
Y
0
) = 4
wt(ϕ
Y
+ ϕ
Y
0
) = 0 Y = Y
0
X
x∈X
ϕ
X
(x) +
X
x
0
∈X
0
ϕ
X
0
(x
0
) = 0,
X
x∈X
x +
X
x
0
∈X
0
x
0
= 0,
X
x∈X
x
3
+
X
x
0
∈X
0
x
03
= 0.
X
z∈Z
ϕ
Z
(z) = 0,
X
z∈Z
z = 0,
X
z∈Z
z
2
= 0,
X
z∈Z
z
3
= 0,
X
z∈Z
z
4
= 0,
Z = X 4 X
0
|Z| > 4
wt(ϕ
X
+ ϕ
X
0
) + wt(ϕ
Y
+ ϕ
Y
0
) > 4
wt(ϕ
Y
+ ϕ
Y
0
) = 2 wt(ϕ
X
+ ϕ
X
0
) > 2
wt(ϕ
X
+ ϕ
X
0
) = 2
X
z∈Z
z =
X
z
0
∈Z
z
0
,
X
z∈Z
z
3
+ (
X
x∈X
x)
3
+ (
X
x∈X
x +
X
z∈Z
z)
3
=
X
z
0
∈Z
0
z
03
,
Z = X4X
0
Z
0
= Y 4Y
0
|Z| = |Z
0
| = 2
P
x∈X
x = 0
X
z∈Z
z =
X
z
0
∈Z
z
0
,
X
z∈Z
z
3
+ (
X
z∈Z
z)
3
=
X
z
0
∈Z
0
z
03
, |Z| = |Z
0
| = 2,
P
z∈Z
z
3
+ (
P
z∈Z
z)
3
= z
1
z
2
2
+
z
2
1
z
2
P
z
0
∈Z
0
z
03
= (z
1
+ z
2
)
3
+ z
0
1
z
0
2
(z
1
+ z
2
) Z = {z
1
, z
2
}, Z
0
= {z
0
1
, z
0
2
}
z
1
z
2
+ z
0
1
z
0
2
+ (z
1
+ z
2
)
2
= 0 z
0
1
= z
1
+ h, z
0
2
= z
2
+ h
h ∈ F
2
m
h
2
+ h(z
1
+ z
2
) + (z
1
+ z
2
)
2
= 0
1 +
h
z
1
+ z
2
+
µ
h
z
1
+ z
2
¶
2
= 0.
h, z
1
, z
2
, z
1
+ z
2
6= 0
T r(1 +
h
z
1
+z
2
+
³
h
z
1
+z
2
´
2
) = T r(1) = 1
T r(1 +
h
z
1
+z
2
+
³
h
z
1
+z
2
´
2
) = T r(0) = 0
P
m
(ϕ
X
, ϕ
Y
), (ϕ
X
0
, ϕ
Y
0
) wt(ϕ
X
+ ϕ
X
0
) = wt(ϕ
Y
+ ϕ
Y
0
) = 2
P
x∈X
x = 0
S =
P
x∈X
x 6= 0
ˆ
X = X ∪ {S}
ˆ
Y = Y ∪ {S},
ˆ
X
0
= X
0
∪ {S}
ˆ
Y
0
= Y
0
∪ {S}
Z =
ˆ
X4
ˆ
X
0
Z
0
=
ˆ
Y 4
ˆ
Y
0
wt(ϕ
ˆ
Y
+ ϕ
ˆ
Y
0
) = 2 wt(ϕ
Y
+ ϕ
Y
0
) = 2
wt(ϕ
ˆ
X
+ ϕ
ˆ
X
0
) = 2 wt(ϕ
X
+ ϕ
X
0
) = 2
ˆ
X,
ˆ
Y ,
ˆ
X
0
,
ˆ
Y
0
X, Y, X
0
Y
0
ˆ
X
P
x∈
ˆ
X
x = 0
P
x∈X
x 6= 0
Y = Y
0
(ϕ
X
, ϕ
Y
) P
m
x = (x
α
1
, . . . , x
α
2
m
) ∈ F
2
m
2
α
j
F
2
m
= {α
1
, . . . , α
2
m
} X(x) = { α
j
|x
α
j
= 1}
x 1
x ↔ X(x) x =
x(X) X
P
α∈F
2
m
ϕ
X
(α),
P
α∈X
α,
P
α∈X
α
3
X
α∈F
2
m
x
α
,
X
α∈F
2
m
x
α
α,
X
α∈F
2
m
x
α
α
3
,
X Y
2
2
m
−2m−1
X
α∈F
2
m
x
α
= 0,
X
α∈F
2
m
x
α
α = 0,
X
α∈F
2
m
x
α
α
3
= 0
2
m
− 2m − 1 Y X
X
2
2
m
−1
¤
P
0
m
n = 2
m+1
− 1 m
P
m
5 2
n−2m−2
P
0
m
P
m
K
K
⊥
P
m
K K
⊥
Z
4
= Z/4Z 4 Z
2
K K
⊥
Z
4
2
m+1
− 2
2
m = 2t − 1
K
m
n = 2
m
Υ = T r(x
3
+ bx + ε), a, b ∈ F
2
m
, ε ∈ F
2
,
K
m
2m+1 d = 2
m−1
−2
1
2
(m+1)
=
1
2
(n −
√
2n)
K
m
d =
1
2
(n −
√
2n)
K
m
0,
1
2
(n ±
√
2n)
n
2
Υ ∈ K
m
1
2
(n −
√
2n)
n
2
1
2
(n +
√
2n) n
Υ 1
1
2
n(n − 1) n
2
+n−2
1
2
n(n − 1) 1
K
m
(Υ, Υ
0
) = n − 2d(Υ, Υ
0
)
¤
T r(x
3
+ bx + ε)
F
2
m
F
2
m
= {α
1
, α
1
, . . . , α
2
m
}, α
2
m
= 0
F
2
m
θ
α
j
= θ
j−1
, j = 1, . . . , 2
m
− 1,
T r(f(x))
0
, f(x) = x
3
+ bx + ε
0 ∈ F
2
m
T r(x
3
+ bx + ε)
0
= (T r(f(θ
0
)), T r(f(θ
1
), . . . , T r(f(θ
2
m
−2
))).
K
0
m
n = 2
m
− 1, m = 2t − 1 > 1
2m + 1 d =
1
2
(n −
√
2n) − 1
x = (x
0
, x
1
, . . . , x
n−1
), y = (y
0
, y
1
, . . . , y
n−1
)
C x, y ∈ C
n
T
x,y
(j) = <{x
0
y
j
+ x
1
y
j+1
+ ··· + x
n−1
y
n−1+j
},
y
j+k
mod n <{x}
x y
k
y
k
x y
x = y T
x
(j) = T
x,x
(j)
x T
x
(0) = |x
0
|
2
+ ··· + |x
n−1
|
2
= |x|
2
T
x,y
(j)
e
T
x,y
(j) = |x
0
y
j
+ x
1
y
j+1
+ ··· + x
n−1
y
n−1+j
|.
T
x,y
(j)
e
T
x,y
(j)
x = (1, 1, exp −
2π i
3
, exp
2π i
3
, exp
2π i
3
, exp −
2π i
3
, 1)
7 3− 1
T
x
(j)
T
x
(j) j ≡ 0 mod 7
j 6≡ 0 mod 7
x
|y| = |x| = n x, y
U
n−1
√
n
T
x,y
(j)
C
n
x y
(j)
y j
|y| = |x| = n T
x,y
(j)
λ
C
(x, y) x y
(j)
T
x,y
(j) = n(1 −
1
2
λ
2
C
(x, y
(j)
)
x y
m− 1 x
j
= exp
2π i a
j
m
, i =
√
−1 a
i
, 0 ≤ a
i
< m,
m = p p
x y exp
2π i a
j
p
x
p− a = (a
0
, a
1
, . . . , a
n−1
) ∈ F
n
p
p = 2 x
j
= (−1)
a
j
x a
a = (a
0
, . . . , a
n−1
) ∈ F
n
p
F
n
p
f : a →
b
a =
1
√
n
(exp
2π i a
0
p
, . . . , exp
2π i a
n−1
p
) F
n
p
U
n−1
F
n
p
d U
n−1
F
n
p
f λ
C
(x, y)
p ≫ 2
a
0
= (0, . . . , 0) a
1
= (1, . . . , 1) d
λ
C
b
a
0
b
a
1
p = 2 d λ
C
(x, y)
ρ(x) ρ(λ
C
(
b
a,
b
b)) = d(a, b)
ρ(x)
W = {a
1
, . . . , a
M
} ⊂ F
n
p
F
n
p
a
(j)
s
6= a
s
, j = 1, . . . , n − 1,
c
W = {
b
a
1
, . . . ,
b
a
M
} ⊂ U
n−1
W f
τ(
c
W ) = max
x, y ∈
c
W , 0 ≤ t < n
t > 0, x = y
T
x,y
(j).
τ(
c
W )
c
W
W
(c)
W
τ(
c
W )
η(
c
W
(c)
) = max
x,y∈W
(c)
,x6=y
<(
b
x,
b
y).
η(
c
W
(c)
) = 1 −
1
2
λ
2
C
(
c
W
(c)
)
λ
C
(
c
W
(c)
) λ
C
c
W
(c)
c
W λ
C
c
W
(c)
λ
C
(
c
W
(c)
) τ(
c
W )
c
W τ(
c
W )
λ
C
ξ
F
p
p a = (ξ
0
, ξ
1
, . . . , ξ
p−2
) ∈ F
p−1
p
b
a =
1
√
p−1
(exp
2π i a
0
p
, exp
2π i a
1
p
, . . . , exp
2π i a
p−2
p
) a
j
= ξ
j
T
x
(j) =
(
1, j ≡ 0 mod n ,
−
1
p−1
, j 6≡ 0 mod n
.
x ∈ {1, −1}
n
j
n
j
, j = 1, 2 . . . T
x
(j) = −1 j 6≡
0 mod n τ({x}) = max
1<k<n
j
(x, x
(k)
) = −1
F
q
q = p
l
F
∗
q
= F
q
r {0}
F
∗
q
ξ
F
∗
q
= {ξ
j
|j = 0, . . . , p −2}
m|p −1 ψ(x) F
∗
q
m− 1 m−
F
∗
q
ψ(x) m
Φ = {exp
2π i a
p−1
|a = 0, . . . , p −2}
ψ(x ·y) = ψ(x) ·ψ(y) x, y ∈ F
∗
q
.
ψ(x)
(F
∗
q
, C
∗
)
x = 0 ψ(x) 0
F
q
ψ(0) = 0
ψ(x)
ψ(x), x ∈ F
∗
q
,
ψ(x) = exp
2π i t ind x
m
, (t, m) = 1,
ind
ξ
x = ind x y, 0 ≤ y ≤ p − 2 x = ξ
y
ind
ξ
x ξ
x
q m = 2 ψ(x)
±1
³
x
p
´
µ
x
p
¶
= x
p−1
2
.
1 −1 F
q
1 −1
F
q
³
x
p
´
³
0
p
´
= 1
³
x
p
´
[
x ∈ {1, −1}
p
x =
µµ
0
p
¶
[
,
µ
1
p
¶
[
,
µ
2
p
¶
[
, . . . ,
µ
p − 1
p
¶
[
¶
.
x
(j)
x j
x
(j)
=
µµ
j
p
¶
[
,
µ
j + 1
p
¶
[
,
µ
j + 2
p
¶
[
, . . . ,
µ
j + p − 1
p
¶
[
¶
.
S =
X
x∈F
q
µ
x(x + a)
p
¶
= −1, a 6= 0.
³
y
p
´
= 0 y = 0
³
x
p
´
=
³
x
−1
p
´
, x ∈ F
∗
p
S =
X
x∈F
q
µ
x
−1
(x + a)
p
¶
=
X
x∈F
q
µ
(1 +
a
x
)
p
¶
=
X
y∈F
q
, y6=0
µ
1 + y
p
¶
= −1.
¤
p = 4t − 1 x
T
x
(j) =
(
p, j ≡ 0 mod p ,
−1, j 6≡ 0 mod p
.