Сидельников В.М. Теория кодирования. Справочник по принципам и методам кодирования
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b
r
=
1
|∆
r+1
||∆
r
|
S
B
(x · O
2
r
(x))
S
B
(x · O
2
r
(x)) = S
B
(o
r,r
x
r+1
O
r
(x) + o
r,r−1
x
r
O
r
(x)) = o
r,r
r
X
j=0
o
r,j
m
j+r+1
+ o
r,r−1
r
X
j=0
o
r,j
m
j+r
.
b
r
O(r) F
q
O
r
(x)
β
r
O
r+1
(x)
O
r
(x), O
r−
(x) O(r) F
q
O
r+1
(x) O
r+2
(x)
O
u
(x)
O
u
(x)
F
q
O
u
(x) O(u
2
)
|∆
j
| 6= 0, j =
0, . . . , r + 1
O
u
(x) |∆
r
| = 0
r ≤ u − 1
O
u
(x) O
r
(x), r < u,
O
u
(x)
O
r
(x), r = 0, . . . , u −1,
S
B
(O
r
(x)O
r
0
(x)) = 0 r 6= r
0
S
B
(O
r
(x)f(x)) = 0 deg f(x) < r
¤
O
r
(x)
r S
B
(O
2
r
(x)) = 0 deg O
r
(x) < r
O
r
(x)
O
0
u−1
(x) = O
u−1
(x)
O
0
r
(x) =
(
x
r
+ O
r
(x), deg O
r
(x) < r
O
r
(x), deg O
r
(x) = r
, r = 0, . . . , u − 2.
deg O
0
r
(x) = r S
B
(O
r
(x))O
0
s
(x)) = 0 r > s
O
r
(x) 6= 0
xO
r
(x) =
r
0
+1
X
k=0
c
r,k
O
0
k
(x) = c
r,r+1
O
0
r+1
(x) + c
r,r
O
0
r
(x) + c
r,r−1
O
0
r−1
(x), r = 0, . . . , u −2,
r
0
O
r
(x) c
r,k
O
0
k
(x)
S
B
xO
r
(x) =
r
0
+1
X
k=0
c
r,k
O
0
k
(x) = c
r,r+1
O
0
r+1
(x) + c
r,r
O
0
r
(x) + c
r,r−1
O
0
r−1
(x), r = 0, . . . , u −2,
m
j
m
0
j
|∆
0
r
| 6= 0, j =
0, . . . , u O
u
(x) = O
u
(x, m
0
, m
1
, . . . , m
2u−1
)
O
0
u
(x) = O
u
(x, m
0
0
, m
0
1
, . . . , m
0
2u−1
)
O
r
(x) O
0
r
(x), r < u,
β
0
j
= β
j
+γ m
j
m
0
j
=
j
X
i=0
m
i
µ
j
i
¶
γ
j−i
.
O
0
u
(x) = O
u
(x, m
0
0
, m
0
1
, . . . , m
0
2u−1
)
O
0
u
(x − γ, m
0
0
, m
0
1
, . . . , m
0
2u−1
) = C
u
O
u
(x, m
0
, m
1
, . . . , m
2r−1
), C
u
6= 0.
|∆
0
r
(γ)|
∆
r
(γ) =
m
0
0
m
0
1
··· m
0
r−1
m
0
1
m
0
2
··· m
0
r
···
m
0
r−1
m
0
r
··· m
0
2r−2
,
γ
Q
u−1
r=1
∆
r
(γ)
γ
F
q
|∆
r
| 6= 0, r = 1, . . . , u
a
0
r
O
0
r+1
(x) =
k
j
k
j
=
−
1
T
u
(β
j
, β
j
, B)
.
k
j
k
j
B
m
i
, i = 1, . . . , u
k
j
B
O(u
4
)
O(u
3
) F
q
T
u
(x, y, B)
a
u−1
(P
u
(x)P
u−1
(y) −P
u
(y)P
u−1
(x)) = (y − x)
u−1
X
r=0
P
r
(x)P
r
(y),
a
u−1
x
y
yP
r
(y) = a
r
P
r+1
(y) + b
r
P
r
(y) + a
r−1
P
r−1
(y).
P
r
(y) P
r
(x)
(y − x)P
r
(x)P
r
(y) = a
r
(P
r+1
(x)P
r
(y) −P
r+1
(y)P
r
(x)) + a
r−1
(P
r−1
(x)P
r
(y) −P
r−1
(y)P
r
(x))
r = 0 r = u − 1
¤
h
u
(x, y) =
u
X
r=0
P
r
(x)P
r
(y)
y
S
B
(h
u
(x, y)f(x)) = f(y), deg f(x) ≤ u.
T
u
(x, y, B)
S
B
(T
u
(x, y, B)x
r
) = −y
r
, r = 0, . . . , u.
−T
u
(x, y, B) =
u
X
r=0
P
r
(x)P
r
(y), x, y ∈ F
q
,
−T
u
(β
j
, β
j
, B) =
u
X
r=0
P
2
r
(β
j
) =
1
k
j
.
h
u
(x, y)
T
u
(x, y, B) −T
u
(x, y, B) h
u
(x, y) F
q
× F
q
¤
a
r
, b
r
h
u
(β
j
, β
j
)
1
k
j
O(u)
F
q
P
r
(β
j
), r = 1, . . . , u
k
j
O(u
2
)
u
a = (a
0
, a
1
, . . . , a
N
), a
j
∈ F
q
,
g(x) = g
u
+ g
u−1
x + ··· + g
1
x
u−1
+ x
u
∈ F
q
[x], g
u
6= 0, u < N
a
j+u
= −(a
j+u−1
g
1
+ ··· + a
j−1
g
u−1
+ a
j
g
u
), j = 1, . . . , N − u.
u a
u q
g(x)
bg(x) = x
u
g(x
−1
) n bg(x)
x
n
− 1 n g
u
6= 0
K ⊆ F
n
q
n
f(x) =
x
n
−1
bg(x)
a
K f(x)
N ≤ n
a ∈ K
a
j
=
u
X
i=1
α
i
θ
j
i
, α
i
∈ F
q
n
\ {0}, j = 0, . . . , n −1,
θ
i
bg(x)
a
j
α
i
θ
i
Θ
r−1
(a) =
a
0
a
1
··· a
r−1
a
1
a
2
··· a
r
···
a
r−1
a
r
··· a
2r−2
,
∆
r−1
|Θ
r−1
(a)| Θ
r−1
(a) r = u r > u
|Θ
r−1
(a)| = 0 0 < r < u |Θ
r−1
(a)|
u a
(|Θ
u−1
(a)|, |Θ
u
(a)|, . . . , |Θ
r
(a)|)
0
u
a a u
|Θ
u−1
(a)|
g
j
u a O
u
(x, a
0
, a
1
, . . . , a
2u−1
)
1
∆
u−1
O
u
(x, a
0
, a
1
, . . . , a
2u−1
) = g
r
+ g
r−1
x + ···+ g
1
x
r−1
+ x
r
g
j
g
j
g
j
Θ
u−1
(a)
g
j
e
a = a
0
− e
x
RS
q
(n, d), n = q,
a
f
f(x) n −d
A = F
q
a
β
j
= f (β
j
)
f(x) β
j
, ∈ F
q
, j = 1, . . . , n − d + 1
f(x)
a
f
∈ RS
q
(n, d) f(x)
a
β
j
= f (β
j
), j = 1, . . . , n − d + 1
f(x)
n − d + 1
f(x)
f(x)
r(x) =
Q
n−d+1
j=1
(x − β
j
) r
j
(x) =
r(x)
x−β
j
, j = 1, . . . , n − d + 1
γ
j
= r
j
(β
j
) 6= 0, r
j
(β
i
) = 0 i 6= j
f(x) n −
d a
β
j
β
j
, j = 1, . . . , n − d + 1
f(x) =
a
β
1
γ
1
r
1
(x) +
a
β
2
γ
1
r
2
(x) + ··· +
a
β
n−d+1
γ
n−d+1
r
n−d+1
(x).
u = wt(e) a
0
= a + e, a ∈ RS
q
(n, d),
O
u
(x, b
0
, . . . , b
2u−1
) F
B
(x)
β ∈ A ⊆ F
q
O
u
(β, b
0
, . . . , b
2u−1
) 6= 0 β 6∈ B
a
0
β
F
q
= A n−d+1
β
j
∈ A a
β
j
a
0
f(x) a
e
t
d−1
2
a
0
= a + e K(B
(d)
A
) = RS
q
(n, d), A = {α
1
, . . . , α
n
} ⊆
F
q
,
t wt(e) ≤ t d ≥ 2t + 1
t
a = (a
1
, . . . , a
n
) = (f(α
1
), . . . , f(α
n
)),
a ∈ RS
q
(n, d)
f(x) s = n −d A
c
j
j− a
0
c
j
= a
j
+ e
j
P = {(a
1
, c
1
), . . . , (a
n
, c
n
)}
F
q
× F
q
D(x, y) ∈ F
q
[x, y]
P D(x, y) D(a
j
, c
j
) = 0 j = 1, . . . , n
e = 0 P
y −f(x) f(x) a
F (x) a
0
P F (x)(y−f(x)), deg F (x) ≤ t, deg f(x) ≤ n−d
d < 2t+1 (a, e)
D(x, y)
D(x, y) = R(x, y)(y −f(x)), deg f(x) ≤ n − d,
(y − f(x))
P
D(x, y) P
P
Λ(x, y)
max i + j
x
i
y
j
Λ(x, y)
x y
D(x, y) x
n − d y
Λ(x, y) =
P
∞
i=0
P
∞
j=0
= ω
i,j
x
i
y
j
F
q
s, u (s, u )− N
s,u
Λ(x, y) max is +
ju (i, j) ω
i,j
6= 0
u = s = 1 N
1,1
Λ(x, y)
s = n − d, u = 1
(s, 1)−
h N
s,1
(h) N
s,1
(h)
x
i
y
j
is + j ≤ h N
s,1
(h)
h N
s,1
(h)
h
2
2s
D
P
(x, y) (s, 1)−
h P
ω
i,j
Λ(x, y) =
P
∞
i=0
P
∞
j=0
= ω
i,j
x
i
y
j
(s, 1)−
h P
X
ui+j≤h
ω
i,j
α
i
β
j
= 0 (α, β) P .
N
s,1
(h) ω
i,j
n = |P|
h
Q
P
(x, y)
N
s,1
(h) ≥ n
Q
P
(x, y) y−f(x), deg f(x) <
s
f(x) a
f
a
f
L(Q
P
(x, y))
a
0
RS
q
(n, d)
L(Q
P
(x, y)) a
0
n − δ >
h
n
³
1 −
√
2R
´i
= n −
p
2n(n − d),
R =
n−d
n
RS
q
(n, d) δ
N
s,1
(δ) >
n
2
Q
P
(x, y) Q
P,m
(x, y)
P m
N
s,1
>
nm(m − 1)
2
,
Q
P,m
(x, y)
P m
Q
P
(x, y) Q
P,m
(x, y)
a
0
RS
q
(n, d)
L(Q
P,m
(x, y)) a
0
n −
·
δ
m
¸
>
"
n
Ã
1 −
r
R
m + 1
m
!#
= n −
r
m + 1
m
n(n − d),
R
=
n−d+1
n
RS
q
(
n, d
) δ
N
s,1
(δ) >
n
2
m → ∞
t < n − n
√
R
t <
n(1−R)
2
=
d
2
d
2
O
u
(x, m
0
, m
1
, . . . , m
2u−1
) β
j
m
i
, i = 0, . . . , 2u − 1
m
i
=
u
X
j=1
k
j
β
i
j
, k
j
∈ F
q
r {0}, u ≥ 1, i = 0, . . . , 2u − 1,
k
j
F
q
F
q
O
u
(x, m
0
, m
1
, . . . , m
2u−1
) 2u
m
i
2u u β
j
N
u,r
= N(m
0
, m
1
, . . . , m
2u−r
) u−
B = {β
1
, . . . , β
u
} ⊆ F
q
k
j
∈ F
q
r {0}
i = 0, . . . , 2u − r B
u− N
u,r
u−
O
u,r
(x
1
, . . . , x
r
) =
O
u
(x
1
, . . . , x
r
, m
0
, m
1
, . . . , m
2u−r
) ∈ F
q
[x
1
, . . . , x
r
] r u−r+1
• B ⊆ N
u,r
r− C = {β
j
1
, . . . , β
j
r
} ⊆ B
B O
u
(x
1
, . . . , x
r
) O
u,r
(β
j
1
, . . . , β
j
r
) = 0
•
C = {β
j
1
, . . . , β
j
r
} r− F
q
O
u
(x
1
, . . . , x
r
)
O
u
(β
j
1
, . . . , β
j
r−1
, x) u − r + 1
O
u,r
(β
j
1
, . . . , β
j
r−1
, x) β
j
1
, . . . , β
j
r−1
u− B ∈ N
u,r
B = C ∪ D
O
D
O
O
u,r
(β
j
1
, . . . , β
j
r−1
, x)
• C = {β
j
1
, . . . , β
j
r
} O
u,r
(x
1
, . . . , x
r
)
O
u,r
(β
j
1
, . . . , β
j
r−1
, x) D
u −r B = C ∪D
u i = 0, . . . , 2u−r
• C = {β
j
1
, . . . , β
j
r
} O
u,r
(x
1
, . . . , x
r
)
O
u,r
(β
j
1
, . . . , β
j
r−1
, x) u−r +1 B
i = 0, . . . , 2u −r C r {β
j
r
}
D
u,r
(x
1
, . . . , x
r
, m
0
, m
1
, . . . , m
2u−r
) =
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
1 ··· 1 m
0
··· m
u−r
x
1
··· x
r
m
1
··· m
u−r+1
x
2
1
··· x
2
r
m
2
··· m
u−r+2
··· ···
x
u
1
··· x
u
r
m
u
··· m
2u−r
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
D
u,r
(x
1
, . . . , x
r
, m
0
, m
1
, . . . , m
2u−r
) u
x
j
O
u,r
(x
1
, . . . , x
r
, m
0
, m
1
, . . . , m
2u−r
)
O
u,r
(x
1
, . . . , x
r
, m
0
, m
1
, . . . , m
2u−r
) =
Ã
Y
i<j
(x
i
− x
j
)
!
−1
D
u,r
(x, m
0
, m
1
, . . . , m
2u−r
).
O
u,r
(x
1
, . . . , x
r
, m
0
, m
1
, . . . , m
2u−r
)