1. GF(2n)
n
X 2 1 + 1.
2. () n,
n
GF(2) X 2 1 + 1,
n
X 2 1 + 1, GF(2), , n .
3. GF(2 n) 10000 (),
n
(X 2 1 + 1).
n
(X 2 1 + 1), .. . (2 n 1).
4. GF(2 n) a, , GF(2 n) a, .. GF(2 n) .
. ai GF(2 4).
1 GF(2 4)
4
(X 2 1 + 1) = ( 15 + 1). ( 15 + 1) :
( 15 + 1) = ( 1) ( 2) 1 ( 4) 2 ( 4) 3 ( 4),
:
( 1) = + 1,
( 2) = 2 + + 1,
1 ( 4) = 4 + + 1,
2 ( 4) = 4 + 3 + 1,
3 ( 4) = 4 + 3 + 2 + + 1.
3 ai GF(2 4) 10000 1 ( 4) = 4 + + 1.
.
1 ( 4) = 4 + + 1 Û 10011 , .. j, j = 0, 1, 2, 3, , 4 + + 1. 0, 1, 2, 3 1 ( 4), 1 ( 4) , .. 0, 1, 2, 3.
4 Û 10000 :
Å | |||
Û 4 |
5 Û100000 :
Å | |||
Û 5 |
, :
Å Å Å Å Å Å Å Å | ||
Û 4 | ||
Û 7 | ||
Û 8 | ||
Û 10 | ||
Û 12 | ||
Û 13 | ||
Û 14 | ||
Û 15 = 0 |
1.9.1.
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1.9.1 a0 - a14 GF(2 4)
j | ai | |
0 | a0 | |
1 | a1 | |
2 | a2 | |
3 | a3 | |
4 | a4 | |
5 | a5 | |
6 | a6 | |
7 | a7 | |
8 | a8 | |
9 | a9 | |
10 | a10 | |
11 | a11 | |
12 | a12 | |
13 | a13 | |
14 | a14 |
GF(24) 0 1
1 ( 4) = 4 + + 1 Û 10011.
GF(2 n) .
2.
(), .. , GF(2 n).
. a5 = 0110, a6 = 1100, a5 + a6 = 1010,
Å | |
. a14 = 1001, a14 a14 = a13 mod P1 (X4) Û 10011.
Å 1001 |
Å | ||
Û a13 |
b GF(2 n) (), 1 (mod P (X)),
b a 1 (mod P (X)).
n, 0, ()
(). , (),
j (()) = 2 n 1 ( ).
n
a 1 = a j (()) 1 mod P (X) = a 2 2 mod P (X).
. = 100 () = 1011 GF(2 3).
3
a 1 = 100 2 2 (mod 1011) = 100 6 (mod 1011) = 100 2 100 4 (mod 1011).
100 2 (mod 1011) = 10000 Å 10110 = 110
Å | ||
100 4 (mod 1011) = 110 2 (mod 1011) = 010
Å 000 |
Å | ||
100 2 100 4 (mod 1011) = 110 010 (mod 1011) = 1100 (mod 1011) = 111
Å | ||
, 1 = 111.
: = 100, 1 = 111, () = 1011, 1 = 110 100 = 11100.
Å Å | ||
.. 1(mod 1011) = 1.
1.10 GF(2 n)
1. , - .
2. GF(2 n) .
3. GF(2 n) .
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4. GF(2 n) ( n) = n + X + 1.
n X ( ). ( n) .
( n) = n + X + 1 n (n < 1000):
1, 3, 4, 6, 9, 15, 22, 28, 30, 46, 60, 63, 127, 153, 172, 303, 471, 532, 865, 900.
m1 m2, .
, , , ().
:
1.
,
.
2.
,
, .
:
, ,
, , .
.
I. - M :
1) M ;
2) , ;
3) 0, 1;
4) , ;
5) 0, - 1, . ( ) . . , .
2.1.1 ,
: : array [1.. n ] of real - , ; 1 ³ [1] ³ ³ [ n ] > 0,
[1] + + [ n ] = 1.
: : array [1.. n, 1.. L ] of 0..1 - .
Fano (1, n, 0) { Fano}
Fano.
: b - P,
e - P,
k - C.
: .
if e > b then
k: = k + 1 { }
m: = Med (b, e) { }
for i from b to e do
C [ i, k ]: = i > m { 0, 1}
End for
Fano (b, m, k) { }
Fano (m + 1, , k) { }
End if
Med [ b..e ], .. m (b ≤ m < e), [ b..m ] [ m + 1.. e ].
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. , . , . , 1, . , , , .. .
: , .
. p1 = = p2 =...= p8 = 2-3 , , 2.1.1. . 0, - 1. . . 0, - 1 . 0, - 1. .
2.1.1
1 2 3 4 5 6 7 8 |
() .
H = log2 N = log2 8 = 3 /
li - i - ; pi - i - li.
, H = L, . . .
: . , H = L.
. 8 : a1 = 0,5; a2 = 0 ,25; a3 = 0,098; a4 = = 0,052; a5 = 0,04; a6 = 0,03; a7 = 0,019; a8 = 0,011. - .
2.1.2.
2.1.2
ai | pi | lipi | |||||||
0,5 | - | - | - | - | - | 0,5 | |||
0,25 | - | - | - | - | 0,5 | ||||
0,098 | - | - | 0,392 | ||||||
0,052 | - | - | 0,208 | ||||||
0,04 | - | - | 0,16 | ||||||
0,03 | - | 0,15 | |||||||
0,019 | 0,114 | ||||||||
0,011 | 0,066 |
. .
. :
1) .
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2) n0 , 2 £ n0 £ m N - n0 / m - 1 - , , .
3) , , .
4) . . , m N - n0/m - 1- 1.
, , m = 2 , , , , , , , 1.
.
: A = 0,5; B = 0,15; C = 0,12; D = 0,1;
E = 0,04; F = 0,04; G = 0,03;
H = 0,02.
.
0,02 (H) 0,03 (G), , G 0,05. 0,04 (F) 0,04 (E) 0,08.
, 0,05 0,08. , 0,13.
, , , 1.
(. 2.1.1) 1, - 0. , , (1,00).
2.1.3.
2.1.3
pili | ||||
A B C D E F G H | 0,50 0,15 0,12 0,10 0,04 0,04 0,03 0,02 | 0 0 1 0 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 | 0,50 0,45 0,36 0,30 0,20 0,20 0,15 0,10 |
.
f1, f2, f3, : = 0,24; = 0,18; = 0,38;
= 0,1; = 0,06; = 0,02; = 0,02.
.
2.1.4
0,38 0,24 0,18 0,1 0,06 0,02 0,02 | f1 f2 f3f1 f3f2 f3f3f1 f3f3f2 f3f3f3 |
, f1 f2, .
2.1.1 Huffman
: n , : array [1.. n ] of real - , ; 1 ³ [1] ³ ³ [ n ] > 0, [1] + + [ n ] = 1.
: : array [1.. n, 1.. L ] of 0..1 - .
l: array [1.. n ] of 1.. L .
if n = 2 then
C [1, 1]: = 0; l [1]: = 1 { }
C [2, 1]: = 1; l [2]: = 1 { }
Else
q: = P [ n 1]+ P [ n ] { }
j: = Up (n, q) { }
Huffman (P, n 1) { }
Down (n, j) { }
End if
. Up , q , . Down n n 1 . j , j, , j 0 1, .
: 0, 1. if Huffman. Up () , , ( ) . j. , , . , .. . , . j, Huffman.
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. , 256 ASCII , , , *.txt *.exe .
:
1. ,
.
2. .