Сидельников В.М. Теория кодирования. Справочник по принципам и методам кодирования
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a
(1)
a
(2)
O(d
3
+ dn)
z
j
h
Λ a ∈ F
q
\ {0}
h a
−1
B
0
= h ·B ·Λ
z
1
= 1
c
1
, c
2
, . . . , c
d
d
X
s=1
c
s
z
s
f
j
(ω
s
) = 0, j = 0, . . . , d −2.
c
1
, c
2
, . . . , c
d
B
0
d
B
00
d
· diag(z
1
, z
2
, . . . , z
d
)(c
1
, c
2
, . . . , c
d
)
T
= 0,
B
00
d
= (f
i
(ω
j
)), i = 0, 1, . . . , d − 2, j = 1, 2, . . . , d d − 1 × d
B
00
d
· diag(z
1
, z
2
, . . . , z
d
) d
B
0
B
00
d
= h · B
d
B
d
=
ω
0
1
ω
0
2
··· ω
0
d
ω
1
ω
2
··· ω
d
ω
2
1
ω
2
2
··· ω
2
d
···
ω
d−2
1
ω
d−2
2
··· ω
d−2
d
h · B
d
· ·diag(z
1
, z
2
, . . . , z
d
)(c
1
, c
2
, . . . , c
d
)
T
= 0,
B
d
· diag(c
1
, c
2
, . . . , c
d
) · (z
1
, z
2
, . . . , z
d
)
T
= 0.
z
2
, z
3
, . . . , z
d
c
1
, c
2
, . . . , c
d
ω
1
, ω
2
, . . . , ω
d
z
1
= 1
ω
0
2
ω
0
3
··· ω
0
d
ω
2
ω
3
··· ω
d
ω
2
2
ω
2
3
··· ω
2
d
···
ω
d−2
2
ω
d−2
3
··· ω
d−2
d
· diag(c
2
, c
3
, . . . , c
d
)
z
1
, z
2
, . . . , z
d
h = (h
i,j
), i, j = 0, . . . , d −2
z
j
d−2
X
s=0
h
i,s
ω
s
j
= z
j
f
i
(ω
j
).
i, 0 ≤ i ≤ d − 2 j 1 d − 1
h
i,0
, h
i,1
, . . . , h
i,d−2
h
i,0
, h
i,1
, . . . , h
i,d−2
i h
h ω
1
, ω
2
, . . . , ω
d
z
1
, z
2
, . . . , z
d
z
d+1
, z
d+2
, . . . , z
n
B
0
h
−1
h
−1
· B
0
=
z
1
ω
0
1
z
2
ω
0
2
··· z
n
ω
0
n
z
1
ω
1
z
2
ω
2
··· z
n
ω
n
z
1
ω
2
1
z
2
ω
2
2
··· z
n
ω
2
n
···
z
1
ω
d−2
1
z
2
ω
d−2
2
··· z
n
ω
d−2
n
,
z
d+1
, z
d+2
, . . . , z
n
h z
j
O(d
4
+ dn)
O(d
4
+ dn)
n
n = q+1 F
r
r
q F
(i)
(x)
h F
r
z
j
f
j
(x)
h Λ
B
0
N t
t
w (t, w)−
t = 2 (2, w)−
(t, w)−
T, |T | = t,
t
R R
W, |W | = w, w
R
S
N
N, |N| = N,
Q = {a
1
, . . . , a
N
} N = {N
a
1
, . . . , N
a
N
}
Q
a ∈ Q N
a
K
F
q
K
L(K) F
q
K
K
a
⊂ L(K
a
) a ∈ Q
K
a
a a
K
a
a
A T = { a
1
, . . . , a
t
}
Q t
A a ∈ T
k
a
1
,...,a
t
T K
a
a
1
, . . . , a
t
T
k
a
1
,...,a
t
T
T
k
a
1
,...,a
t
(t, w)−
(t, w)− S D
0
(S) = |M|
k(t,w)
M
k(t, w)
(t, w)−
D
0
(S)
(t, w)−
D(S) = k(t, w) log
2
|M|
M n D(S) = k(t, w)n
n
(2, w)−
S
B
F
2
m
F
2
m
s D(S) = ms(w + 1)
S
F
n
q
n− F
q
q = p
l
Q Q =
{a
1
, . . . , a
N
} ⊆ F
n
q
Q
k
a
1
,...,a
t
T = {a
1
, . . . , a
t
}
T K =
{k
1
, . . . , k
D
} K
K
F
u
q
u K ⊂ F
u
q
K
F
q
K F
u
q
K = {ξ
1
, . . . , ξ
D
}
ξ
j
F
u
q
K
K
F
u
q
k
a
1
,...,a
t
F
q
K k
a
1
,...,a
t
L(K)
K K
F
u
q
L(K) |K| K
k
1
, . . . , k
j
S
F
q
k ∈ K k
1
, . . . , k
j
k k
1
, . . . , k
j
k F
u
q
k
c
1
, . . . , k
c
j
k ∈ K k
1
, . . . , k
j
P (k = x/k
1
, . . . , k
j
) = P (k = x).
¤
K
k
a
1
,...,a
t
F
u
q
k
1
, . . . , k
j
k k
k
1
, . . . , k
j
a ∈ Q S
K
a
= {k
a,1
(K), . . . , k
a,n
(K)} k
a,j
(K), j = 1, . . . , n,
K
k
a,j
(K)
k ∈ L(K)
G ⊆ L(K) L(G)
F
q
G
k
k G
k
G
G
Q
S a F
n
q
S
T
w
= {a
1
, . . . , a
w
} ⊂ Q
k
a
1
,...,a
t
T = {a
1
, . . . , a
t
} T ∩ T
w
= ∅
T
w
K(T
w
) =
[
a∈T
w
K
a
L(K(T
w
))
S
w− T
w
= {a
1
, . . . , a
w
} ⊂ Q T = {a
1
, . . . , a
t
}
T ∩ T
w
= ∅
k
a
1
,...,a
t
6∈ L(K(T
l
)).
k
a
1
,...,a
t
T
w
S w
(t, w)− w−
T
w
= {a
1
, . . . , a
w
} ⊂ Q (t, w)− R
t,w
(Q)
N = |Q| t w
R
t,w
(Q)
D(R
t,w
(Q))
(t, w)
Q
F
n
q
Q ⊂ F
n
q
K K = {ξ
i
1
,...,i
t
| 1 ≤ i
1
≤ i
2
≤
··· ≤ i
t−1
≤ n} ξ
j
1
,...,j
t
= ξ
i
1
,...,i
t
j
1
, . . . , j
t
i
1
, . . . , i
t
1 ≤ i
1
≤ i
2
≤ ··· ≤ i
t−1
≤ n
K
ξ
j
1
,...,j
t
1 ≤ j
1
≤ ··· ≤ j
t
≤ n K
¡
n+t−1
t
¢