, , , , . , .
mod q 㳺 , , q - - , mod q. q mod q.
R = 1 / ( k + 1) , ( ) mod q .
2.
(. 7), . :
bi = ai, i [ 1, k ],
ai , i - - ; bi , i - ; k .
R = 0,5. , .
3. -
- :
- , ;
- m q;
- ;
- , .
, , .
:
- m q ;
- ;
- n , n ( ), ;
- n : .
|
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:
- ;
- v , qi .
, , , m , , , m , .
ʳ N 0
, n
N 0 = ( / n ) n ;
,
N 0= ( Q) n Q ´ ( Q + n) Q / n n ,
Q / n, .
v - l = q / v , m , (v ³ m)
N 0 = ( l m / n ) n ;
m (l ³ m, v = n),
N 0 = () n .
() , , . , . , .
. , ̳ EAN. EAN (European Article Numbering) UPC (Uniform Product Code).
EAN 10 ( 0...9 ) (, ). 4, 5, 6, 7, 8, 10, 12, 13 14 . EAN: EAN-13 EAN-8, ( ).
, . 5, 4 .
:
1- : , ( ), 3;
2 - : , ;
3 - : , 1- 2- ;
4 - : , , 9, , , 3- , 10 .
EAN : A, B, C D (. 1) , , ( 1, 2, 3), - ( 4 5). . 7 (7- ), 3, 5 6 . 1 3, 1 2 (. 2, 0 , 1 ).
1
- | D | |||
1 | ||||
2 | ||||
3 | ||||
4 | ||||
5 |
|
|
: , . 1, : 0 , 00 , 000 0000 , 1 (), 11 (), 111 ( ) 1111 ( ).
2
1 | 0.......... 0 | 1 |
3 | 1...........1 | 2 |
1 | 0...........1 | 2 |
3 | 1...........0 | 1 |
EAN 4, 5, 6, 7 (. 1), 1 () 2 (). 8, 10, 12 14 - , 4. , D (. . 1). - 1.
EAN-13 , 4, 1 ( ). (12- ) , . , 1 4 (. 3). ˳ . 3 . 1, . , 4 1, (. . 1). .
, 7 . ϳ , EAN-13, 8 , EAN-8. EAN-8 1, , 4, , , 1. EAN-8 u 7 , u 6, u 5, u 4 (. .1).
3
u 12 | u 11 u 10 u 9 u 8 u 7 u 6 |
UPC 10 . UPC EAN , (. 1).
UPC, UPC- UPC-. UPC- 12 (12- ), EAN-13. , ( ) . ³ UPC- EAN-13 :
(u 11 ) ;
UPC- ( 1), 6 , ( 4), 5 , ( 1) (. . 1);
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u 11 u , 1 4, ;
0, - UPC-.
, EAN-13.
UPC- 6 3 . () , () ( ) (. . 1).
EAN UPC , . , , , 10. . 10, . - , , EAN-13 , (. . 1) .
16. .
1. , .
2. .
3. .
4.