, . 2 , . , , [27] 29- . . 6.5.
6.5
, , .
. , , ( ), . , , .
(. 6.6).
6.6
, . , (7, 4), (15, 11), n, k, , , , .
, .
, . , , . , 1, 3, 5 7- ;
, .
:
, . , . . , . , .
, (7, 4), , , . . , , :
|
|
dmin = =3, (6.15) . . 6.6, , , .
6.5. 15 , .
(6.17) , 4. . , . (7, 4). , .
, (8, 4). :
(7, 4) Si,
3 , :
S1 = 0 S2 = 0 ;
S1=0 S2 = 1 ;
S1=0 S2 = 0 ( , ),
S1=0 S2=1 ( ).
6.6. . 6.6, (8,2), .
. 6.5 , :
, dmin=5, , , 1 4.
, , . , , , . .
. , , , . . , , .
. , .
ai, d ( ) . i = i. . , d , , , , . , , . d2s+1 s s , . , aj(j i) . .
|
|
6.7. (8,2).
, 5. (6.26 ) (6.26 ) a8, (6.26) (6.26), 5 (6.26 ) (6.26) a5 = a5, 5:
8
. lxn lxn , l n :
, , :
nxn n. , , , I. = |ij| = |bij| lxn lxn:
Α = |ij| lxn lxn:
== |ij| lxn = |bij| nxm , Cik lxm l- k-ro :
GF(P), . GF(P).
6.8. GF(2):
Cik =12 :
,
, . , , η- () GF(P). (n,k)- n 2-1 ( ). , (8,2), , :
n k , , . , , , , .
V1, V2, V3,..., Vn , 1...n ( ),
, , .
. .
2k 1 k. , (8,2)
, , , () .
|
|
k n GF(q), (n, k)-. (n, k)- k n k . ( ) Q == qk-1.
, , ki k . ki n,k:
, , (8,2), 5=1, 8=1:
: 1) ; 2) ; 3) - , .
M2 Μ1 , . . . .
(n, k)-, , (Mn,k) : kXk - ( ) k*(n-k), n k :
, k , (n k) .
, - ki = (a1 a2...ai...an)
:
, , . .
(6.27), (6.28), -. ( ) , - , .
6.9. Ik, Pk,n-k Mn,k (7,4)
, (6.25)
(7,4)
, - Pk,n-k , . , . . .
- , -.
d , - k , : - , , d-l .
|
|
, , l , d , l .
(7,4) d = 3.
- - ( l= 1) . : 011, 110, 101, 111.
. , , :
, (l>1) -, d = 3 .
Μn,k = [Ik Pk,n-k], () Η (n---k)Xn:
ni , , , :
S, :
, Ani = (a1, a2,...ai,...,ak, ak+1,...aj,...,an) ), (Ani + ξni) t :
, ι , , S = (Sk+1, Sk+2,..., Sj,...Sn) ().
( )
6.10. Η (7,4) :
1100011.
4,3
:
t 1100011 :
, , 4 (1101011)
, - 111 () . . (7,4), -, () .
6.5.
, . n-k .
( ) () .
, 2.
.
6.11. (7,4), .
6.7.
, , n- ( 6.8 1. (1-3) .
k , , - n . , , . 6.7.
6.7.
1,2 , 0 1 . , C1 , 3, 5 7 , , T1 1, 2 0, 4 1.
(. 6.8) , , . .
|
|
6.8.
( 6.8), . (7 4)
0 1 . , 0, . . DC . , , , . . , , , . , . (7,4) , ,
, , . 6.9
6.9
1 2 3
, , , 101 5. T5, .
6.12. (8,2), (. 6.4).
, .
. 6.9. , , (5- 8-). , .
6.6.
. (n,k) , k n , , k n- . , .
, . ,
, .
, . , , , 001011: