АРИФМЕТИЧЕСКОЕ КОДИРОВАНИЕ СООБЩЕНИЙ С ИСПОЛЬЗОВАНИЕМ СЛУЧАЙНЫХ ПОСЛЕДОВАТЕЛЬНОСТЕЙ

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• 2008 2(2)

519.72

1

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. .. , .

E-mail: vpotapov@math.nsc.ru

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. , -

, -

.

: , , , -

.

, . - A . - f A . - .

, , . .. [1, 2] - . , [3, 4]. [5].

- . , , - (0,1) . -, , , .. [6, 7].

A P A* A. - P(wa)=P(w), wa, aA, wA*, P(w) = 1, . - ( ), , . .P(a1, , an) = P(a1)P(an). I(w)= log P(w) wA

* - (A, P). (., , [8]) , I(w) .

. wAn - L(w)= P(x), xAn, , w, R(w) = L(w) + P(w). [0,1) - i(w) = [L(w), R(w)). i(w) P(w) - q(w) , , 2log P(w), . f(w) wAn

1 ( 08-01-00671).

• 132 ..

q(w) log P(w), - . f An , |f(w)| I(w) + 1, . . . , f A*, w , P(w)/2. f , Cod(n) f(w) wAn, Cod , - |Cod(n)| = log n(1 + o(1)). -, , [8, 9].

. x1, , xn, - (0,1). wAn. m -, i(w), .. xmi(w). w gx(w)=Cod(m). , gx A

n. 1. 1, , , (0,1)

. E|g(w)| I(w)(1+ o(1)) |w|.. A (0, 1). P(iA) = P(A) A

i1, , n P(1, , n1A, nA) = (1 P(A))n1P(A). m

A (0, 1), i , , iA.

E(m) = 1

1

(1 ( )) ( ) 1/ ( ).n

n

n P A P A P A

=

= log t , E(log m) log(E(m)).

A = i(w). E|g(w)| = E|Cod(m)| = (1+o(1))E(log m) (1+o(1)) log (E(m)) = I(w)(1+o(1)) |w|. .

f, g . , 1 , - I(w).

- . -. B = 2A A. w = a1an W = b1bn, aibi i = 1 n. - F B* , : () Fo , A, Fo(F(W)) WB*. , , . - (B, P), P B.

[7] ( ) -. , , ( )( )min ( ) log ( )

b B a b

H P b Q a

= , -

Q A, -, . . WB*. - bB (B, P) ( ) log ( )

a b

I b Q a

= , I(W)

W = b1bn I(bi). Q , H. wAn

i(w) Q. WB* i(W) = i(w), wA*, - WB*. x1, , xn, (0,1). m , i(W), .. xmi(W). W - Gx(W) = Cod(m). , Gx B

n. 2. 1,, , (0,1)

. E|G(W)| I(W)(1+ o(1)) |W|. 2 1. [7] , -

.

• 133

1. Rissanen J. Generalized Kraft inequality and arithmetic coding // IBM J. Res. Develop. 1976. V. 20. No. 3. P. 198 203. 2. Rissanen J., Langdon G. Universal modelling and coding // IEEE Trans. Inform. Theory. 1981. V. IT-27. No. 1. P. 12 23. 3. .., .. // . 1999. . 35. . 4. C. 95 108.

4. Witten I.H., Neal R.M., Cleary J.G. Arithmetic coding for data compression // Commun. ACM. 1987. V. 30. No. 6. P. 520 540.

5. .. // :. II. .: - -

, 2001. . 59 70.

6. .. // . . 2006. . 140. 1. . 321 325.

7. .. // . . 4. .: -, 2004. . 385 399.

8. .. . . : . , 1999.

9. .. // -. .: , 1968. . 20. . 173 179.