Сидельников В.М. Теория кодирования. Справочник по принципам и методам кодирования
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K ⊆ J
w,n
j, 2j = d, d = d(K) K
A(n, j, w)
A(n, j, w)
M
2
(n, d)
d
A(n, d, w) ≤
jn
jn −w(n −w)
,
jn −w(n −w) > 0
K J
w,n
j(x, y) =
d(x,y)
2
π
x, y ∈ J
w,n
(π(x), π(y)) = 1 −
j(x, y)n
w(n − w)
≤ 1 −
jn
w(n − w)
.
j(x, y) ≥ j 1 −
jn
w(n−w)
< 0
X
x,y∈K
(π(x), π(y)) =
n
X
s=1
X
x,y∈K
bx
s
by
s
=
Ã
X
x∈K
bx
s
!
2
≥ 0,
bx
s
by
s
s− π(x) π(y)
0 ≤
X
x,y∈K
(π(x), π(y)) ≤ |K| + |K|(|K| −1)(1 −
jn
w(n − w)
),
(π(x), π(y)) = 1 x = y (π(x), π(y)) ≥ 1 −
jn
w(n−w)
x 6= y
|K| ≤
jn
jn −w(n −w)
.
¤
M
2
(n, d) ≤
2
n
A(n, d, w)
¡
n
w
¢
.
K ⊂ F
n
2
d
K w χ(x)
x ∈ F
n
2
χ(x)
y ∈ K w x
χ(x) ≤ A(n, d, w) |K|
¡
n
w
¢
=
P
y∈K
¡
n
w
¢
=
P
x∈F
n
2
χ(x) ≤ A(n, d, w)2
n
¤
d ≥
n
2
d ≥
n
2
M
2
(n, d) ≤
(
2d
2d−n
, 2d −n > 0
4d, n −2d = 0
.
K
d(x, y), x, y ∈ K, x 6= y n d =
n
2
φ :
x → φ(x) x ∈ F
n
2
−
1
√
n
1
√
n
φ F
n
2
φ(F
n
2
) S
n−1
R
n
(φ(x), φ(x)) = |φ(x)|
2
= 1
x, y ∈ F
n
2
(φ(x), φ(y)) =
1
n
(n − 2d(x, y)).
K ⊆ F
n
2
d ≥
n
2
X
x,y∈K
(φ(x), φ(y)) =
n
X
s=1
X
x,y∈K
bx
s
by
s
=
n
X
s=1
Ã
X
x∈K
bx
s
!
2
≥ 0,
bx
s
s− φ(x)
X
x,y∈K
(φ(x), φ(y)) = |K| +
X
x,y∈K,x6=y
(φ(x), φ(y)) ≤ |K| + |K|(|K| −1)
n − 2d
n
,
(φ(x), φ(y)) = 1 x = y (φ(x), φ(y)) =
1
n
(n −
2d(x, y)) ≤
1
n
(n − 2d) x 6= y
|K| + |K|(|K| − 1)
n − 2d
n
≥ 0.
2d−n > 0
n−2d = 0 K
0
K
K 1
|K|
2
K K
0
K
K
0
K
00
K
00
d
n
00
= n − 1 2d − n
00
> 0
K
00
|K| ≤ 2|K
00
|
K
K
00
n
00
= n − 1
|K
00
| − |K
00
|(|K
00
| − 1)
n
00
− 2d
n
00
= 0,
|K
00
| =
|K|
2
= n
X
x,y∈K
00
(φ(x), φ(y)) = n +
X
x,y∈K
00
,x6=y
(φ(x), φ(y)) = n −n(n −1)
1
n − 1
= 0.
(φ(x), φ(y)) ≤ 0 (φ(x), φ(y)) = 0 x, y ∈
K
00
, x 6= y. ¤
K m n = 2
m
− 1
B
H
2
m−1
K
n = 4t, d =
n
2
, M = 2n
n ×n− A ±1 ∈ R
ˆ
K n A
A −1
ˆ
x,
ˆ
y ∈
ˆ
K,
ˆ
x 6=
ˆ
y, (
ˆ
x,
ˆ
y) 0
ˆ
x 6= −
ˆ
y
−n
ˆ
x = −
ˆ
y
ˆ
K d =
n
2
2n
n × n−
A n d =
n
2
K n d =
n
2
n × n− A
ˆ
K = {φ(x)|x ∈ K}, |
ˆ
K| = 2 n, n
d =
n
2
2n
A(K)
ˆ
K
00
(φ(x), φ(y)) =
2−2d(x,y)
n
= 0, x 6= y.
n > 2 n = 4t
n = 4t+ 2 > 2
a, b, c A a, b
a + b
n
2
= 2t + 1
±2
(a + b, c) = (a, c) + (b, c) = 0 (a + b, c) 6= 0
a + b
¤
n = 4t
t
t
n = 4t = p + 1
p p = 4t − 1 n = 2
m
A ⊗ B A = (a
i,j
)
i,j=1,...,n
B
n m
A ⊗ B =
a
1,1
B a
1,2
B ··· a
1,n
B
a
2,1
B a
2,2
B ··· a
2,n
B
a
n,1
B a
n,2
B ··· a
n,n
B
.
A A
0
n n
0
A ⊗ A
0
n · n
0
A
⊗n
1
= A
1
⊗ ··· ⊗ A
1
n
2
n
2 × 2−
A
1
=
µ
1 1
1 −1
¶
.
hx, yi x = (x
1
, . . . , x
m
) y = (y
1
, . . . , y
m
)
F
m
2
hx, yi =
m
X
i=1
x
i
y
i
,
F
2
B
F
m
2
v
x,y
, x, y ∈ F
m
2
B = kv
x,y
k x
y
A
m
= k(−1)
hx,yi
k.
2
m
× 2
m
−
2
m
×
2
m
F
m
2
F
m
2
= {(x
1
, . . . , x
m−1
, 0)|(x
1
, . . . , x
m−1
) ∈
F
m−1
2
} ∪ {(x
1
, . . . , x
m−1
, 1)|(x
1
, . . . , x
m−1
) ∈ F
m−1
2
}
A
m
=
µ
A
m−1
A
m−1
A
m−1
−A
m−1
¶
.
µ
A
m−1
A
m−1
A
m−1
−A
m−1
¶
=
µ
A
m−1
0
0 A
m−1
¶
·
µ
I
m−1
I
m−1
I
m−1
−I
m−1
¶
,
I
j
2
j
× 2
j
−
A
m
= I
1
⊗ A
m−1
· A
1
⊗ I
m−1
⊗
A
m−1
A
m
= I
2
⊗ A
m−2
· I
1
⊗ A
1
⊗ I
m−2
· A
1
⊗ I
m−1
A
m
= I
m−1
⊗ A
1
· I
m−2
⊗ A
1
⊗ I
1
···I
m−j
⊗ A
1
⊗ I
j−1
···I
1
⊗ A
1
⊗ I
m−2
· A
1
⊗ I
m−1
m
Y
j=1
I
m−j
⊗ A
1
⊗ I
j−1
.
B
j
= I
m−j
⊗A
1
⊗I
j−1
±1
A
m
B
j
±1
a
A
m
a A
m
n
2
= 2
2m
n = 2
m
A
m
aA
m
m aB
m
·B
m−1
···B
1
m2
m
= log
2
n · n B
j
2
m
2
m
− 1
K ⊆ F
n
q
n
k d
d ≤ n − k + 1.
A K k = dim K
A k
k A
k A
a A k
k−
a wt(a) ≤ 1 + n − k
¤
k
q
q−
d q
n ≤ q + 1 2 ≤ k ≤ q q = 2
l
d = 2, n − 2 n = q + 2
d
N(k, d)
k d
N(k, d) ≥
¼
d
2
k−1
»
+ ··· +
¼
d
2
»
+ d,
exd x ≤exd
K n
d
n ≥ d + N
µ
k − 1,
¼
d
2
Ȧ
.
A K
A =
µ
0 0 ··· 0 1 1 ··· 1
A
1
B
1
¶
,
A
1
n −d k − 1 B
1
d
k − 1
A
1
k − 1 A
1
A
d
d
A
1
K
1
n − d
d
1
v ∈ K
1
(v|u) ∈ K (v|u)
v u ∈ F
d
2
(v|u) = (v|u)+a u u
a A
d
1
+ wt(u) ≥ d
d
1
+ u = d
1
+ d − wt(u) ≥ d.
2d
1
≥ d d
1
≥
¨
d
2
§
n − d ≥ N
µ
k − 1,
¼
d
2
Ȧ
,
¼
e
d
2
s
d
2
»
=
¨
d
2
s+1
§
¤
m n = 2
m
− 1
2
m−1
N(m, 2
m−1
) ≥ 2
m−1
+ 2
m−2
+ ··· + 2 + 1 = 2
m
− 1,
K
x ∈ K x = x+(1, 1, . . . , 1)
K
n
2
x, y
d [d, n −d] n
y = x)
n d M
A
(n, d)
M
A
(n, d) ≤
2n
3
− 2n(n − 2d)
2
3n − (n − 2d)
2
− 2
,
3n − (n − 2d)
2
− 2 > 0 n ≥ 4
K d x, y ∈ K
φ : K → S
n−1
|(π(x), π(y))| = |n −2d(x, y)| ≤ n − 2d,
x 6= y; y 6= x x = y y = x |(π(x), π(y))| = n
K
X
x,y∈K
(π(x), π(y))
4
= 2n
4
|K| +
X
x,y∈K;x6=y;y6=x
(π(x), π(y))
4
≤
2n
4
|K| + (n − 2d)
2
X
x,y∈K;x6=y;y6=x
(π(x), π(y))
2
=
2n
4
|K| − 2(n − 2d)
2
n
2
|K| + (n − 2d)
2
X
x,y∈K
(π(x), π(y))
2
.
n ≥ 4 P
4
(x) 4
p = 2 P
4
(x)
K
(2,n)
4
(x)
P
4
(x) =
1
24
((n − 2x)
4
+ (8 − 6n)(n − 2x)
2
+ 3n(n − 2))
1
|K|
2
X
x,y∈K
P
4
(x)(d(x, y)) =
1
|K|
2
X
x,y∈K
P
4
(x)
µ
n − (π(x), π(y))
2
¶
=
1
|K|
2
X
x,y∈K
1
24
¡
(π(x), π(y))
4
+ (8 − 6n)(π(x), π(y))
2
+ 3n(n − 2)
¢
≥ 0,
K ⊂ F
n
p
T (K) =
1
|K|
2
P
x,y∈K
(π(x), π(y))
2
K
(2,n)
2
(x) =
1
2
((n − 2x)
2
− n)
T (K) ≥ n K
T (K) ≥ n
0 ≤
2n
4
|K|
−
2n
2
(n − 2d)
2
|K|
+ (n − 2d)
2
T (K) + (8 − 6n)T (K) + 3n(n −2) ≤
2n
4
|K|
−
2n
2
(n − 2d)
2
|K|
+ (n − 2d)
2
n + 2n − 3n
2
(n − 2d)
2
+ (8 − 6n) ≤ 0. ¤
K
n = 2
m
m
2
2m+1
= 2n
2
d = 2
m−1
− 2
1
2
(m+1)
=
1
2
(n −
√
2n)
K
Ker
n = 2
2m
, |K
Ker
| = n
2
= 2
4m
d =
1
2
(n −
√
n) = 2
2m−1
− 2
m−1
K d(x, y), x, y ∈ K,
0,
n
2
,
1
2
(n ±
√
rn), 0 < r < 3 r = 1
r = 2
n, d, M
F
q
n
d k = n − r n, d, r
d−1
X
s=1
(q − 1)
s
µ
n − 1
s
¶
≤ q
r
− 1.
B r
n d−1
B
0
r n
0
d − 1
B
0
B
d − 1 B
0
B q
r
− 1 F
r
q
\ B
B
0
B
s, 1 ≤ s ≤ d−1
(q −1)
s
B
B
|B| ≤
d−1
X
s=1
(q − 1)
s
µ
n
0
s
¶
.