Сидельников В.М. Теория кодирования. Справочник по принципам и методам кодирования
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f
0
(x) = x −1 f
j
(x), j = 1, . . . , m, F
r
n
j
, n
j
|l , q = r
l
θ
s
j
{θ
s
1
, . . . , θ
s
m
}
F
r
F
q
θ
s
, s =
0, 1, . . . , d − 2 f(x) F
r
θ
s
, s = 0, 1, . . . , d − 2
¤
d
0
k
0
F
r
BCH
r,q
(n, d) n ≤ q − 1, q = r
l
BCH
r
(n, d), n|q −1, B
(d)
A
θ
d
0
≥ d, k
0
≥ n − 1 −
µ
d − 2 −
·
d − 2
r
¸¶
l,
[x] x
d
0
≥ d BCH
r,q
(n, d)
RS
q
(n, d)
k
0
BCH
r,q
(n, d) F
n
r
B
(d)
A
BCH
r,q
(n, d)
B
(d)
A,r
F
n
r
k
0
= n − m m
B
(d)
A,r
k
0
B
(d)
A
B
(d)
A,r
¡
d − 1 −
£
d−1
r
¤¢
l B
(d)
A,r
ω = {ω
1
, . . . , ω
l
} F
q
F
r
α ∈ F
q
α =
P
l
j=1
a
j
ω
j
α = (a
1
, . . . , a
l
)
α F
q
ω
α
j
i
B
(d)
A
α
j
i
T
B
(d)
A,r
F
r
l(d − 1)
n B
(d)
A,r
α
r
α
x → x
r
F
r
α → α
r
F
l
r
α
r
= αU
U ∈ M
l
(F
r
)
B
(d),i
A
i− B
(d)
A
B
(d),i
A,r
l ×n B
(d),i
A
B
(d),ri
A,r
= B
(d),i
A,r
· U.
B
(d),ri
A,r
B
(d),i
A,r
ri ≤ d − 2 B
(d),ri
A,r
B
(d)
A,r
BCH
r
(n, d)
B
(d),j
A
, j > 0, B
(d)
A
j r
£
d−2
r
¤
B
(d),0
A
F
r
1 F
r
B
(d),0
A,r
l(d − 2 −
£
d−2
r
¤
) + 1 ¤
\
BCH
r,q
(n, d) B
(d)
A
b
B =
b
B
(d)
A
=
α
1+s
1
α
1+s
2
··· α
1+s
n
α
2+s
1
α
2+s
2
··· α
2+s
n
α
3+s
1
α
3+s
2
2 ··· α
3+s
n
2
···
α
d−1+s
1
α
d−1+s
2
··· α
d−1+s
n
, d > 2, n ≤ q, −n ≤ s ≤ n.
r−
\
BCH
r,q
(n, d)
b
B
(d)
A
F
r
RS
q
(n, d)
B
(d)
A
b
B
(d)
A
d
\
BCH
r,q
(n, d) s = 0
BCH
r,q
(n, d) r = 2
d
\
BCH
2,q
(n, d) 1
BCH
2,q
(n, d)
d
0
k
0
F
r
\
BCH
r,q
(n, d) n ≤ q − 1
BCH
r
(n, d)
d
0
≥ d, k
0
≥ n −
µ
d − 1 −
·
d − 1
r
¸¶
l
l
0
,
[x] x
¤
RS
r
(n, d) BCH
r,q
(n, d)
\
BCH
r,q
(n, d) r < q
F
q
F
r
K(x) ∈ F
r
[x]/x
n
− 1 K(B
(d)
A
θ
)
I R
(n)
q
= F
r
[x]/ mod x
n
−1
f(x) ∈ F
r
[x], f(x)|x
n
−1, I
a ∈ K(B
(d)
A
θ
) a(θ
j
) = 0, j = 0, . . . , d−2
f(x) F
r
θ
j
∈ F
q
, j = 0, . . . , d − 2
f(x) ∈ F
r
[x] f(θ
j
) = 0, j = 0, . . . , d −2
θ
j
f(x) θ
jr
f(x) f
r
(x) = f(x
r
)
f(x) F
r
f
j
(x) θ
j
f
j
(x)
S
d
= {θ
j
0
, . . . , θ
j
m
}
F
r
{θ
0
, . . . , θ
d−2
}
r− K(B
(d)
A
θ
)
F
r
f(x) = f
0
(x) ···f
m
(x),
f
i
(x) F
r
θ
j
i
f(x) F
r
θ
j
∈ F
q
, j = 0, . . . , d −2 r−
K(B
(d)
A
θ
) ¤
r
i
F
r
θ
j
i
r
i
= deg f
i
(x)
deg f(x) = r
0
+ ··· + r
m
,
r
i
≤ l, θ
j
i
6∈ F
r
r
i
= 1, θ
j
i
∈ F
r
deg f(x) ≤ 1 +
·
d − 2
r
¸
l .
dim BCH
r,q
(n, d) = deg
x
n
−1
f(x)
≥ n −1 −
£
d−2
r
¤
l
k
0
BCH
r,q
(n, d)
f(x)
\
BCH
r,q
(n, d) f(x) deg f(x) ≤
£
d−1
r
¤
l
d
r q = r
l
h, 0 ≤ h ≤ q − 1,
xvy v mod r
l
−1 = q−1
C
h
= {xvr
j
y|j = 0, . . . , l − 1}
h C
h
h
min
h = (h
0
, . . . , h
l−1
) l−
[0, r − 1] r− h
h = h
0
+ h
1
r + ···h
l−1
r
l−1
xvry xvy
xvry = xvy
(1)
.
0 < h < q
1
2
= r
l
2
|C
h
| = l
j , 0 <
j < l h hr
j
mod r
l
− 1 = q − 1
C
h
= {h
0
|h
0
∈ C
h
} = {h
0
(j)
|j = 0, . . . , l − 1},
h
0
(j)
h
0
j h = (h
0
, . . . , h
l−1
) ∈
C
h
h
j
0
h j
0
j
0
= 0 h
0
6= 0 h l−j
0
h
0
= h
(l−j
0
)
C
h
h
0
∈ C
h
h
0
≤ h < r
l
2
C
h
= C
h
0
h h
0
l −
l
2
h
j <
l
2
h
(j)
h
0
j ≥
l
2
h
(j)
h
j ≥
l
2
j ≥
l
2
¤
θ F
q
τ = θ
s
, s|q−1,
F
q
n =
q−1
s
deg f
j
(x) = l
0 < js < r
l
2
f
j
(x) F
r
τ
j
deg f
j
(x) = |C
sj
|
θ F
q
τ = θ
s
, s|q−1,
F
q
q−1
s
(d − 2)s < r
l
2
BCH
r,q
(n, d) (d − 1)s < r
l
2
BCH
r,q
(n, d)
dim BCH
r,q
(n, d) = n −1 − l ·
µ
d − 2 −
·
d − 2
r
¸¶
dim
\
BCH
r,q
(n, d) = n − l ·
µ
d − 1 −
·
d − 1
r
¸¶
.
BCH
r,q
(n, d)
deg f(x) = 1 + ml.
m = d − 2 −
£
d−2
r
¤
m
{τ, τ
2
, . . . , τ
d−2
}
£
d−2
r
¤
j [1, d − 2]
r [1, d −2] r
d − 2 −
£
d−2
r
¤
C
j
l
m = d −2 −
£
d−2
r
¤
¤
RS
q
(n, d)
C
(d)
A
=
z
1
α
0
1
z
2
α
0
2
··· z
n
α
0
n
z
1
α
1
z
2
α
2
··· z
n
α
n
z
1
α
2
1
z
2
α
2
2
··· z
n
α
2
n
···
z
1
α
d−2
1
z
2
α
d−2
2
··· z
n
α
d−2
n
, d > 3, n ≤ q + 1,
z
j
∈ F
∗
q
= F
q
\ {0} A = {α
1
, . . . , α
n
} ⊆ F
q,∞
, α
j
6= α
i
j 6= i α
j
= ∞
C
(d)
A
(0, . . . , 0, z
j
)
T
n ≤ q + 1
d n − d + 1
RS
q
(n, d)
C
(d)
A
C
(d)
A
= B
(d)
A
· D D =
diag(z
1
, z
2
, . . . , z
n
), z
j
∈ F
q
\{0}, B
(d)
A
C
(d)
A
B
(d)
A
BCH
r
(n, d)
BCH
r
(n, d) = RS
q
(n, d) ∩ F
n
r
BCH
r
(n, d)
RS
q
(n, d) F
r
F
q
A ⊆ F
q,∞
z = (z
1
, z
2
, . . . , z
n
) ∈ F
∗n
q
k
0
F
r
z
n = q + 1
q + 1 F
q
K(x)
K hf(x)R
(q+1)
q
i R
(q+1)
q
mod x
q+1
− 1
f(x)
G x
q+1
− 1 x
q+1
− 1
|G| = q+1 G ⊂ F
q
2
G
θ G
f(x)
F
q
2
±1
t <
q+1
2
Θ
t
= {θ, . . . , θ
t
}
F
q
F
q
2
(θ
j
)
q
= θ
−j
=
θ
q+1−j
θ
j
θ
q+1−j
j ≤ t <
q+1
2
Θ
t
f(x) Θ
t
f(x) = f
1
(x) ···f
t
(x),
f
j
(x) θ
j
∈ F
q
2
F
q
deg f
j
(x) = 2 f(x) 2t
K
f
f(x) q +1 −
2t d 2t + 1
f(x) θ
1
, . . . , θ
t
,
θ
−1
, . . . , θ
−t
K
f
B
f
=
θ
1·0
θ
1·1
θ
1·2
··· θ
1·(n−1)
θ
2·0
θ
2·1
θ
2·2
··· θ
2·(n−1)
···
θ
t·0
θ
t·1
θ
t·2
··· θ
t·(n−1)
θ
−1·0
θ
−1·1
θ
−1·2
··· θ
−1·(n−1)
θ
−2·0
θ
−2·1
θ
−2·2
··· θ
−2·(n−1)
···
θ
−t·0
θ
−t·1
θ
−t·2
··· θ
−t·(n−1)
.
2t B
f
F
q
2
∆ =
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
β
1
β
2
β
3
··· β
2t
β
2
1
β
2
2
β
2
3
··· β
2
2t
···
β
t
1
β
t
2
β
t
3
··· β
t
t
β
−1
1
β
−1
2
β
−1
3
··· β
−1
2t
β
−2
1
β
−2
2
β
−2
3
··· β
−2
2t
···
β
−t
1
β
−t
2
β
−t
3
··· β
−t
2t
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
2t B
f
j− ∆ β
t
j
∆
0
β
j
, j = 1, . . . , 2t
F
q
2
∆
0
= β
t
1
···β
t
2t
∆ ∆ 6= 0
2t B
f
K f(x)
2t ¤
Θ
t
1 K
0
(x − 1)f(x)
K K
j− B
f
θ
t
j
K
f
r− n = q, q = r
l
z
j
K
G
a = (a
1
, . . . , a
n
) ∈ F
n
r
R
a
(x) =
a
1
x − α
1
+ ··· +
a
n
x − α
n
,
α
j
∈ F
q
α
i
6= α
j
i 6= j
a
G(x) m F
q
F
q
G
G
a
R
a
(x) ≡ 0 mod G(x).
R
a
(x)
R
a
(x) =
F
1
(x)
F
2
(x)
,
F
1
(x), F
2
(x) F
1
(x) ≡ 0 mod G (x)
F
1
(x) G(x)
a R
a
(x)
w(a) = deg F
2
(x)
d(K
G
) = min
a∈K
G
,a6=0
deg F
2
(x) ≥ F
1
(x) + 1.
a ∈ K
G
G(x)|F
1
(x)
d(G
G
) ≥ deg G(x) + 1 = m + 1.
ϑ
1
, . . . , ϑ
u
G(x)
R
a
(ϑ
j
) = 0, j = 1, . . . , u
B(K
G
) =
1
ϑ
1
−α
1
1
ϑ
1
−α
2
···
1
ϑ
1
−α
n
1
ϑ
2
−α
1
1
ϑ
2
−α
2
···
1
ϑ
2
−α
n
···
1
ϑ
u
−α
1
1
ϑ
u
−α
2
···
1
ϑ
u
−α
n
K
G
F
r
B(K
G
) F
q
F
q
m
i
F
q
ϑ
j
G(x) ∈ F
q
[x] m
1
+···+ m
u
= m
1
ϑ
i
−α
j
∈ F
q
m
i
m
i
F
m
i
F
q
B
0
(K
G
)
F
q
m
B
0
(G
G
)
B
00
(G
G
) F
r
l · m
k
k = n − B
00
(G
G
) ≥ n − l · m.
B(K
G
) K
G
⊂ F
n
r
B(K
G
) = H · C
(m−1)
A
, A = F
q
F
q
C
(d)
A
H
α
j
G(x)
modG(x) x − α
j
f
j
(x)
m − 1 (x − α
j
)f
j
(x) ≡ 1 mod G(x) (x − α
j
)f
j
(x) ≡
1
x−α
j
mod G(x)
f
j
(x)
f
j
(x) ≡
1
x − α
j
≡ A
j
G(x) − G(α
j
)
x − α
j
mod G(x),
A
j
A
j
= −G
−1
(α
j
)
G(x) =
P
i
= 0
m
g
i
x
i
x
i
−α
i
j
x−α
j
= x
i−1
+ x
i−2
α
j
+ ··· + xα
i−2
j
+ α
i−1
j
x
i−1
f
j
(x) = g
i
+ g
i+1
α
j
+ ··· + g
m
α
m−i
j
.
a K
G
F (x) = a
1
f
1
(x) + ··· +
a
n
f
n
(x) ≡ modG(x) F (x) m a
K
G
F (x)
n
X
j=1
a
j
G
−1
(α
j
)(g
i
+ g
i+1
α
j
+ ··· + g
m
α
m−i
j
) = 0, i = 0, . . . , m − 1.
H · B
(m−1)
A
· Da
T
= 0,
B
(m−1)
A
D = (G
−1
(α
1
), . . . , G
−1
(α
n
))
H ·B
(m−1)
A
·D K
G
H
d ≥ deg G(x) + 1
K
G
d(K
G
) K
G
d(K
G
) ≥ 2m + 1,
m = deg G(x)
R
a
R
a
(x) =
F
0
(x)
F (x)
.
2
F
0
(x) F
0
(x) =
b
F
2
(x)
b
F (x)
a K
G
F
0
(x) =
b
F
2
(
x
)
≡
0 mod
G
(
x
) deg
F
0
(
x
)
≥
2
m w
(
a
) = deg
F
(
x
)
≥
2m + 1 ¤
K
G
B
(d)
A
, A = F
q
d = 2m
K ⊆ F
q
d(a, b) a, b ∈ K