n -
100 .
. 6.20 .
. 6.18 1000011 ( 4- ).
6.18
, k , , .
6.19. . (7,3) g(x) = (x+1)(x3 + x2+1).
k , :
n-
n -
. . 6.21.
1000010 4- 5- . 6.19.
9- II 1100. , 5- II, 4- (1000) 11 , . 4- II
6.19
. , , , . , , , , , , .
, . . , , :
() m>=, = 2m-1; b .
|
|
h(x) f(x) , ():
1 () . f(x) g1(x) g2(x) , i*(). , m = b , (). n- ( ) h(x) . , h(x) , .
n- , , h(x) m.
, , . 6.22.
h(x) I [ ()] (), , (2m-1)- . , n- , 2m-1.
h(x) 2b-1 + 1, (2b 1), (2b 1) . ()b-1 . (2b 1)- (2b 1)
, . . n- , n (2b1). , 2m-1 2b 1 , n .
() , (b 1) II ( b2 + 1) .
, , Κι , K2 . ( ) () , II. () h(x) .
, () , j-, , n- , x () (x2b-1 + 1) .
j- , (n-j) , , h(x) m*j 1 n.
(n j) , .
h(x) , .
|
|
6.20.
, (b = 3). n = (231)(2 3-1) = 35. , () = 101.
() g1(x) = 3 + x2+ 1 () 7 . ,
()^b-1 () g2(x) = x5 + 1 5 .
I II 101 . 6.20. k b 1 .
6.20
35- . .
. , ( i) , , , , , . , i+k k .
. .
:
, - , , . , :
1. , , ( );
2. - ( );
3. i, j, .
λ-, ai(i j) λ aj (i j) . i , , , [16].
. , .
.
6.21. (7,3) g(x) = (x+1)(x3 + x + 1), (d = 4)
a0 a0 = 0, :
:
. 6.23.
. n M (6.47).
|
|
, 0. .
. 6.21 , 4- 1001110.
6.21
, 1, 0, .
6.9.
. , - , , s. , , dmin = 2s + 1.
- . GF(q), q . , q = 2.
(. . 6.10). g(x) , n(n = 2m1), , , 2^m.
:
(ri()) g(x) ri(x).
. 2m , GF(2m) m GF(2). (1) (0).
, (a()) (ra()),
6.22. 24, g(x) = x4 + x + 1
, (). 16 : (0), (1), (x), (+1), (2), (2+1), (2+), (2++1), (3), (3+1), (3+), (3+2), (3++1), (3+2+1), (3+2+), (3+2++1).
, . α (), , , . 6.22.
6.22
, , , . .
, α, , αi , . n=2m-1 , αn = 1 .
|
|
α, . αm-1, αm-2,..., α, 1 , ( ).
6.23. , g(x) = 4 + +1 (. 6.23).
6.23
α α15 = 1.
, 2m
α , , . . β = αi.
, , n + 1 = 0. GF(2) , xn +1. . , 6.13 , Gf(24):
, xn+1 ( , ) . . β m, β, :
, .
g(x) , , g(β)=0, β. gβ(x). GF(2) , GF(2m).
6.24.
, g(x) = x4 + x + 1, , α GF(24) g(x). α2, α4 α8.
g3(x) α3.
g5(x) g7(x) α5 α7. g5(x) α5 α10 g7(x) α7, α14, α13, α11.
,
, 14 x15 + 1. α0 1 x + 1.
. , , . . GF(2m), g(x) m, n = 2m1. , , n, , .
s
ξ, ξ, i j.
. ξ. , .
,
,
3- 4- . . , . .
.
, , ξi ξj :
|
|
,
, , , s1=s3 = 0.
, , z= =ξ-1.
, , 5 . g(x) α3 ( ), α5 ( ) . . , .
()
, n =15, ,
6.25. (15, 11) g(x) = x4 + x + 1.
. 6.23, GF(24) . 6.24.
6.24
6.24. l+s1z = 0. n = 15 h(x) g(x) I -j. II , α g(x), α4 α + 1.
(n + 1)- h(x) . α-j α, 1+si*α. 1 +s1 α 0, s1 α-1 , , .
, . j- ( ), 1+ +s1αj = 0 s1 = α-j. .
6.26. (15, 7) g(x) = (x4 + x + 1)(x4 + x3 + x2 + x + 1).
:
1) s1 h(x) 4 + x + 1;
2) s3 α9 α6 α1 h(x) 4 + 3 + x2 + x + 1;
3) s3 α3 α2 α1;
4) s12, s3/s1 s12 + s3/s1,
5) q(z) = 1+s1z+(s12 + + s3/s1)z2 s1 α (s12 + s3/s1) α2.
6.10.
, , , q, . , . q-
mγ γ. - . q .
. . .
γ, (, ). .
() , . . 6.25.
, ( ) ,
, , . , .
(6.55) , . , () , (). . , , , . , (. 6.26).