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H(D) H(D) H(D) Кодер Источник Кодер Модулятор Среда канала информации источника распространения сигналов или Получатель Декодер Декодер Демодулятор хранения информации источника канала информаци • • • • • X x X p(x) µ(x) x X X = {x, p(x)} • • µ(x) µ(p(x)) • A = {0, 1} µ(pm) = mµ(p) • x1,..., xn X {p(x)} µ(x1,..., xn) = µ(x1) +... + µ(xn) x1,..., xn I(x) x X = {x, p(x)} I(x) = - log p(x).

• I(x) 0, x X.

• x1, x2 X p(x1) p(x2) I(x1) I(x2) • x1,..., xn n I(x1,..., xn) = I(xi).

i=1 X = {a, b, c, d} p(a) = 1/2 p(b) = 1/4 p(c) = p(d) = 1/8 I(a) = 1 I(b) = 2 I(c) = I(d) = 3 X = {a, b} p(a) = 0.05 p(b) = 0.95 I(a) = 4, 322 I(b) = 0, 216 x X = {x, p(x)} X = {x, p(x)} X = {x, p(x)} H(X) = M [- log p(x)] = - p(x) log p(x).

xX H(X) 0.

H(X) log |X|.

X X Y H(X) = H(Y ).

X Y H(XY ) = H(X) + H(Y ).

X X = {x, p(x)} A X. X = {x, p (x)} p (x) P (A), x A, |A| p (x) = p(x), x A.

/ H(X ) H(X).

X g(x). Y = {y = g(x)} H(Y ) H(X) g(x) (a) H(X) - log |X| = - p(x) log p(x) - p(x) log |X| = xX xX (b) 1 = p(x) log p(x)|X| xX (c) 1 log e p(x) - 1 = p(x)|X| xX 1 = log e - p(x) = 0.

|X| xX xX ln x x - 1, log x (x - 1) log e.

x-1 ln(x) 0 1 -1 ln(x) x - 1 ln x (x-1) x = 1 p(x) = 1/|X| x X X = {0, 1} p(1) = p p(0) = 1 - p = q H(X) = -p log p - q log q (p).

(p) p (p) p p [0, 1] p = 0 p = 1 (p) p 0 p 1 (0) = (1) = 0 (p) = (1 - p) p = 1/2 (p) (p) = - log p + log(1 - p).

p = 1/2 (p) = - log e/p - log e/(1 - p) < 0, p (p) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 p (p) f(x) R = {x} R m R x, x R [0, 1] y = x + (1 - )x R M X = {1, 2,..., M} M p = (p1,..., pM) p = (p 1,..., p M) [0, 1] q = p + (1 - )p.

q = (q1,..., qM) q M M M qi = pi + (1 - ) p i = + 1 - = 1.

i=1 i=1 i= f(x) x R x, x R [0, 1] f(x + (1 - )x ) f(x) + (1 - )f(x ).

(0, 1) f (x) f (x2) f ( x1 +(1-)x2) f (x1)+ (1-) f (x2) f (x1) x xx1 x1 + (1- )x (x1, f(x1)) (x2, f(x2)) f(x1 + (1 - )x2) f(x1) + (1 - )f(x2), f(x) x R M 1,..., M [0, 1] m = 1 x1,..., xM m=R M M f mxm mf(xm).

m=1 m= m xm x xm m f (M [x]) M [f(x)].

H(p) p p M M H(p) = - pm log pm = fm(p).

m=1 m= fm(p) -(log e)/pm pm (0, 1) fm(p) X |X| = M p = (p1,..., pM) H(p) X A X p = ((p1 + p2)/2, (p1 + p2)/2, p3,..., pM).

H(p) H(p).

p = p = (p1, p2, p3,..., pM), p = (p2, p1, p3,..., pM).

H(p ) = H(p ) = H(p) p = (p + p )/p + p H(p) = H 1 H(p ) + H(p ) = H(p).

2 X = {x} Y = {y} XY = {(x, y), p(x, y)} y Y p(x|y) X x X I(x|y) = - log p(x|y), x y X x X I(x|y) x X H(X|y) = - p(x|y) log p(x|y), xX X y Y p(x|y) = 0 z log z z x p(x) = p(x|y) = H(X|y) y y H(X|Y ) = - p(x, y) log p(x|y) xX yY X Y H(X|Y ) 0.

H(X|Y ) H(X), X Y H(XY ) = H(X) + H(Y |X) = H(Y ) + H(X|Y ).

H(X1...Xn) = H(X1) + H(X2|X1) + + H(X3|X1X2) +.... + + H(Xn|X1,..., Xn-1).

H(X|Y Z) H(X|Y ) X Z y Y n H(X1...Xn) H(Xi), i= X1, Xn H(X|Y ) - H(X) = - p(x, y) log p(x|y) + xX yY + p(x, y) log p(x) = xX yY p(x) = p(x, y) log p(x|y) xX yY p(x) p(x, y) - 1 log e = p(x|y) xX yY = p(y)p(x)- p(x, y) log e = xX yY xX yY = 0.

p(x, y) = p(x)p(y|x) = p(y)p(x|y), p(x1,..., xn) = p(x1)p(x2|x1)...p(xn|x1,..., xn-1).

XY Z = {(x, y, z), p(x, y, z)} y Y p(x, z|y) p(x|y) H(X|y, Z) = MXZ|y[- log p(x|yz)], H(X|y) = MX|y[- log p(x|y)].

H(X|y, Z) H(X|y).

y X = {x, p(x)} g(x) Y = {y = g(x), x X}.

H(Y ) H(X).

H(XY ) = H(X|Y ) +H(Y ) = H(Y |X ) +H(X).

0 = g(x) x H(Y |X) = 0 H(X|Y ) (x1, x2,..., xt,... ) xt Xt = X xt t t H(Xt) = H(X) H1(X) H1(X) n x = (x1,..., xn) X1X2...Xn = Xn.

H(X1...Xn) = H(Xn) n H(Xn) n H(Xn) Hn(X) =, n n Hn(X) n xn x1,..., xn- xn H(Xn|X1,..., Xn-1) H(Xn|X1,..., Xn-1) = H(X|Xn-1).

Hn(X) H(X|Xn-1) H(X|Xn) n Hn(X) n Hn(X) H(X/Xn-1) lim Hn(X) = lim H(X|Xn) n n H(Xn) = H(X) + H(X|X1) +... + H(X|Xn-1).

n n (a) H(Xn+1) = H(X1...XnXn+1) = (b) = H(X1...Xn) + H(Xn+1|X1,..., Xn) = (c) = H(Xn) + H(X|Xn) (d) H(Xn) + H(X|Xn-1) (e) H(Xn) + Hn(X) = (f) = (n + 1)Hn(X).

Hn(X) (n+1) Hn(X) H(X|Xn) lim Hn(X) lim H(X|Xn).

n n m < n H(Xn) = H(X1... Xn) = (a) = H(X1... Xm) + H(Xm+1... Xn|X1,..., Xm) = (b) = mHm(X) + H(Xm+1|X1,..., Xm) +...

+H(Xn|X1,..., Xn-1) (c) mHm(X) + (n - m)H(X|Xm).

H(X|Xm) m n n.

lim Hn(X) H(X|Xm), n m. m lim Hn(X) lim H(X|Xm).

n m H(X) = lim Hn(X), H(X|X) = lim H(X|Xn).

n n H(X) = H(X|X) H(X1...Xn) = H(X1) +... + H(Xn).

H(Xn) = nH(X).

n n Hn(X) = H(X), H(X) = H(X).

H(X|Xn) = H(Xn+1|X1,..., Xn) = H(X), H(X|X) = H(X).

s H(X|Xn) = H(Xn+1|X1,..., Xn) = = H(Xn+1|Xn-s+1,..., Xn) = H(X|Xs).

n H(X|X) = H(X|Xs).

H(Xn) = H(X1... XsXs+1... Xn) = = H(X1... Xs) + H(Xs+1... Xn|X1,..., Xs).

H(Xs+1... Xn|X1,..., Xs) = H(Xs+1|X1,..., Xs) + + H(Xs+2|X1,..., Xs+1) +...

+ H(Xn|X1,..., Xn-1), H(Xs+1...Xn|X1,..., Xs) = (n - s)H(X|Xs).

H(Xn) = sHs(X) + (n - s)H(X|Xs).

n n H(X) = H(X|Xs).

s Hn(X) = H(X|Xs) + (Hs(X) - H(X|Xs)) = n s = H(X|Xn) + (Hs(X) - H(X|Xs)).

n Hn(X) H(X|Xn) n H(X|Xs) s X = {x, p(x)} n Tn() H(X) Tn() = x : I(x) - H(X), n Tn() > lim P (Tn()) = 1.

n n |Tn()| 2n(H(X)+).



> 0 n0 n n|Tn()| (1 - )2n(H(X)-).

x Tn() 2-n(H(X)+) p(x) 2-n(H(X)-).

n i, i = 1,..., n m n 1 P i - m.

n ni=n T 1 P (Tn()) = p(x) |Tn()| min p(x) |Tn()|2-n(H+ ).

xTn() xTn() > 0 n0 n > nP (Tn()) 1 -.

Tn() P (Tn()) |Tn()| max p(x) |Tn()| 2-n(H(X)-).

xTn() Tn() 2nH(X) nH(X) H(X) H(X) M Tn() M2-nH(X) 1 - M2-nH(X) M 2nH(X) H(X)) M2-nH(X) x(x) x x (x) = {x(x), x X} x x = (x1,..., xn) n x p(x) = p(xi) = p(x) (x).

i=1 xX 1 x(x) I(x) = - log p(x).

n n xX H(X) x(x) p(x) n x x X {p(x), x X} A = {a} A = {0, 1} C = {c} |C| = M A A A X = {0, 1, 2, 3} C1 = {00, 01, 10, 11} C2 = {1, 01, 001, 000} C3 = {1, 10, 100, 000} C4 = {0, 1, 10, 01} C C C C C C Cc3=0 c2=c1=c0= X = {1,..., M} {p1,..., pM} C = {c1,..., cM} l1,..., lM M = M[li] = pili.

l i= M l1,..., lM M i 2-l 1.

i= M l1,..., lM L L maxi li L 2L li li +1 li +2 L i 2L-l L M i 2L-l 2L.

i= 2L l1 = 2 l2 = l3 = 3 l4 = L = L li 23-2 = 2 вершины 23-3 = 1 вершина 23-3 = 1 вершина 23-1 = 4 вершины 2l l l 2l -l l2 2l - 2l -l1 2l 3 3 2l - 2l -l2 - 2l -l1 lM M M M M 2l - 2l -lM-1 - 2l -lM-2 -... - 2l -l l1 = 1 l2 = l3 = l4 = l1 = 1. Из 2l = 2 вершин, выбираем одну для слова длины l1 = 1.

Имеем 22 - 22-l = 2 вершины. Этого достаточно для построения слова длины l2 Имеем 23 - 23-l - 23-l = 2. Этого достаточно для завершения построения кода.

M X = {x, p(x)} H < H + l X = {1,..., M} p1,..., pM xm lm = - log pm m = 1,..., M.

a a a a M M M M m 2-l = 2- - log pm 2log pm = pm = 1.

m=1 m=1 m=1 m=M l = pmlm = m=M = pm - log pm < m=M < pm (- log pm + 1) = m=M = H + pm = m== H + 1, X = {0, 1} {, 1 - }.

1 + () X = {x, p(x)} H l H.

l H l(x) x 2-l(x) = H - l - p(x) log p(x) - p(x)l(x) = p(x) log.

p(x) xX xX xX log x (x - 1) log e, 2-l(x) H - l log e p(x) - 1 = p(x) xX = log e 2-l(x) - p(x) xX xX log e 1 - p(x) = 0.

xX x p(x) = 2-l(x).

l(x) = - log p(x) = - log p(x) = H l l = H x X p(x) = 2-l(x) {l(x)} X = {1,..., M} {p1,..., pM} p1 p2... pM C = {c1,..., cM} {l1,..., lM} pi < pj li lj li < lj C xi cj xj ci C C C lM = maxm lm lM = maxm lm lM lM X = {1,..., M} C X = {1,..., M - 1} {p 1,..., p M-1} p 1 = p1,..., p M-2 = pM-2, p M-1 = pM-1 + pM.

C C X x 1,...x M-2 C c i = ci i = 1,..., M- x M-1 c M- cM-1 cM C X C X C C lm m M - 2, lm = lM-1 + 1 m = M - 1, M.

M l = pmlm = m=M-= pmlm + pM-1lM-1 + pMlM = m=M-= pmlm + (pM-1 + pM)(lM-1 + 1) = m=M-= p mlm + p M-1lM-1 + pM-1 + pM = m=M- = p mlm + pM-1 + pM = l + pM-1 + pM.

m= M M0 = M M0 > M0 M0 - M- l = p mlm C m= l l M M = M - X = {a, b, c, d, e, f} H = 2, = 2(0, 35 + 0, 2) + 3(0, 15 + 3 0, 1) = 2, l X 0,a 0,b Буква Кодовое 0,слово a 0,c b 0,c 0,d d 0,e f 0,e 0,0,f X = {1,..., M} {p1,..., pM} p1 p2... pM M-q1 = 0, q2 = p1,..., qM = pi.

i= m lm = - log pm qm l = 2, b d e f i j i < j ci i cj j a b c d e f x pm qm lm qm cm a b c d e f li j-1 j-i-qj - qi = pk - pk = pk pi.

k=1 k=1 k=i li = - log pi - log pi.

i pi 2-l.

i qj - qi 2-l.

li - li li ci cj ci cj < H + 1.

l p1,..., pM.

M = 3 p1 = 0, 6 p2 = 0, 3 p3 = 0, q1 = 0 q2 = 0, 6 q3 = 0, а) p2 ppq2 > 1/ 2, поэтому кодовый символ = qq2 q1 = 1/ б) q2 < 3 / 4, поэтому кодовый символ = 1/ 2 q2 q3/ Длина отрезка =1/4, а вероятность символа в) p2 = 0,3 >1/ 4.

1/ 2 Построение кодовоq2 3/ 5/го слова закончено.

m = 2 qm lm m 2-l m 2-l pm lm - log pm qm m pm pm- pm pm- qm) qm + pm/2) lm pm/2. lm X = {1,..., M} {p1,..., pM} m = m-1,..., M qm = pi i= m pm m = qm +.

xm lm = - log(pm/2) m p1 = 0, 1 p2 = 0, 6 p3 = 0, [a] a cm xm pm qm m lm cm cm i j i < j j > i ci cj min{li, lj} j-pj i-1 pi j - i = ph + - ph - = 2 h=1 h=j-pj - pi = ph + h=i pj - pi pi + = pj + pi max {pi, pj} =.

2 pm pm lm = - log - log.

2 max{pi, pj} j - i 2- min{li,lj}, ci cj min{li, lj} < H + 2.

l p2/ p2/ppp < 1/ 2, поэтому q1 = кодовый символ = q2 q3 1/ >1/ 4, поэтому q1 = 0 кодовый символ = 1/ q1 1/ Длина отрезка = 1/4 < p2 / 2 = 0,3.

1/ 1/ 3 / Построение кодово- го слова закончено.

x1, x2,... xi X = {x} H = H(X) n x Xn c(x) l(x) n Rn = M [l(x)] n R = inf Rn.

n inf n n H H R H.

Xn M [l(x)] H(Xn) = nHn(X) nH(X) = nH.

Hn(X) n Rn H n R = inf Rn H, n H > R H +.

Xn M [l(x)] H(Xn) + 1 = nHn(X) + 1.

n1 n n |Hn(X) - H|, Hn(X) H +, n > n1.

n2 n n.

n n max{n1, n2} R = inf Rm m Rn = M [l(x)] = n Hn(X) + n H + + = 2 = H +, n H R H + o(n), o(n) 0 n n |X| = 28 = 256. n = |X2| = 216 = 65536 n = 224 = 16777216 232 = 4294967296,...

n 2/n n X = {1,..., M} {p1,..., pM} {q1,..., qM} Xn = {x} xj i (xi,..., xj) x = (x1,..., xn) Xn = {x} |Xn| x c(x) x x c(x) n p(x) q(x) l(x) = - log p(x) + (x) = q(x) + p(x)/ q(x) Xn y x y x X) X x x x < x x, x X x = (x1,..., xn) y = (y1,..., yn) i xi = yi y x yi xi q(x) = p(y), yx p(x) n p(x) = p(xi).

i= q(x) q(xn) q(xn-1) 1 q(xn) = p(yn) = 1 ynxn 1 = p(yn-1yn) + p(yn-1yn) = 1 yn-1xn-1 yn yn-1=xn-1 ynxn 1 1 1 = p(yn-1) + p(yn-1) p(yn) = 1 ynxn yn-1xn-1 yn-1=xn-1 1 1 = q(xn-1) + p(xn-1)q(xn), 1 q(xn) xn q(xn) = q(xn-1) + p(xn-1)q(xn);





1 1 p(xn) = p(xn-1)p(xn).

1 q(xi ) p(xi )) 1 q(xi+1) p(xi+1)) 1 q(xi ) i = 1, 2,... F p(xi ) i = 1, 2,...

G X = {a, b, c} pa = 0, 1 pb = 0, 6 pc = 0, x = (bcbab) n = F (F + G/2) - log G + 1 = F G M pi, i = 1,..., M n (x1,..., xn) q1 = i = 2 M qi = qi-1 + pi- i = 1 n F F + q(xi)G G p(xi)G c - log G + F + G/ i xi p(xi) q(xi) F G b c b a b - log G + 1 = F + G/2 = 0, 5423...

F = 0, 541 pc = 0,pa = 0,1 pb = 0,а) F = 0;G = qb qc qa = F = 0,x1 = b paG pbG pcG F = 0,1;G = 0,б) F F = 0,1 F + G = 0,x2 = c paG pbG pcG в) F = 0,52;G = 0, F = 0,52 F + G = 0,F x3 = b paG pbG pcG F = 0,538;G = 0,г) F + G = 0, F F = 0,x4 = a paG pbG pcG F = 0,538;G = 0,0108;

д) x5 = b 0,0,F = 0,F + G = 0,0,5391+0,F = 0, x1 = b) x2 = c F = 0, 5391. G/ F = 0, 541.

F 1/29 = 1/ n n p1,..., pM d F G 2d n nd n(n + 1) d + 2d +... + nd = d.

n n X = {1,..., M} {p1,..., pM} {q1,..., qM} n F x M {p1,..., pM} qi, i = 1,..., M n F (x1,..., xn) qM+1 = 1 S = 0 G = i = 1 n j = S + qj+1G < F j j + S S + qjG G pjG xi = j (x1,..., xn) i G p(xi ) i S q(xi ) X = {a, b, c} pa = 0, 1 pb = 0, 6 pc = 0, S G q(x) S + qG p(x) x xi F = a < F b < F b c > F a < F b < F c c < F a < F b < F b c > F a < F a b > F a < F b < F b c > F x = (x1,..., xn) X n log |X| = {} X H n Rn(). rn() = Rn() - H rn() = sup Rn() - H.

rn() n lim rn() = 0, n X = {0,..., M - 1} x = (x1,..., xn) X n(a) a X x n X c(x) = (c1(x), c2(x)) c1(x) c2(x) c1(x) c2(x) |X| = X l1 l ll2 = 6 + 6 + 12 2 + 5 3 +... + 6 = 178.

ll1 = 29 + 8 15 = 149.

l = l1 + l2 = 149 + 178 = 327.

50 lmax lmax I _ F C E 0 D L W T 1 1 0 A H N 0 O S U ni ni M M - 2M - 2M - _ O W A D E N C F S L H U I T M H R H + 1 + (M log M + 3M - 1).

n X = {0,..., M - 1} x = (x1,..., xn) (x) = (0(x),..., M-1(x)) n n X x (x) n n + M - N(n, M) = M - N( ) n! N( ) =.

0!...M-1! x l1(x) l2(x) l(x) l(x) = l1(x) + l2(x) = = log N(M) + log N( ) = n + M - 1 n! = log + log.

M-M - i(x)! i=50 + l1 = log = 190, 50! l2 = log = 150.

12!5!24!33!22! l l = (0,..., M-1) 1 0 n, j+1 j, j 0.

log n j.

j:j> = (0, 1,...) log n j + log M! = 25 + 108 = 133.

l1 = j! j:j> j l = l1 + l2 = n n >> M N(n, M) (n + 1)M-1.

(n + 1) n M - l(x) = l1(x) + l2(x) = = log N(M) + log N( ) M - nH( - log(2n) p) - log(pi) + (M - 1) log(n + 1) + 1.

i n x M H M - 1 log(n + 1) + K R H +, 2 n K n n • • • xn+1 = a x1,..., xn n(a) pn(a) =, n n(a) a n n(a) + pn(a) =.

n + M n n M G G = G = 1/1 1 1 1 1 G = 1 · · · · · · ·... = 256 257 258 259 260 12!(5!)2(4!)3(3!)2(2!)=.

256 ·... · l = - log G + 1 = 342.

n(a) + 1/2 2n(a) + pn(a) = =.

n + M/2 2n + M 1 1 1 1 1 G = 1 · · · · · · ·... = 256 258 260 262 264 23!!(9!!)2(7!!)3(5!!)2(3!!)=.

256 ·... · (2n - 1)!! = 1 3... (2n - 1), (2n)!! = 2 4... (2n).

l = - log G + 1 = 323.

n n(a) pn(a) =, n(a) > 0.

n + n + pn(esc) =, n + n(a), n(a) > 0;

n+pn(a) = 1, n(a) = 0, n+1 M-Mn Mn n 1 1 1 G = 1 · · · ·...

2 256 3 1/11!(4!)2(3!)3(2!)2 G = ·.

50! 256 · 255 ·... · l = - log G + 1 = 291.

x = (x1,..., xn) (1,..., M) Mn (i - 1)! (M - Mn)! i=G = · = n! M! Mn i! (M - Mn)! i== · Mn n! M! i i=Mn i! i= M-M(n + 1)-M.

n! M H M log(n + 1) + K R H +, 2 n K n n(a)-1/, n(a) > 0;

n pn(a) = Mn, n(a) = 0.

2n M-Mn 1 1 2 1 3 1 4 1 1 5 G = 1 · · · · · · · · ·...

2 256 4 255 6 254 8 253 10 12 (2 4... 100) (256 255... 242).

(2 - 3) (2 12 - 3)!!((2 5 - 3)!!)2((2 4 - 3)!!)3((2 3 - 3)!!)G = 100!! 14!.

256 · 255 ·... · l = - log G + 1 = 283.

n M Ki i = 1,..., n 1 + K1/n M log n+K2n M log n+K2n M log n+K2n N i, j N i < j, li lj li lj 0m 1m m i 1i- i li = i i pi = (1 - )i-1, i = 1, 2,...

= 1/2 pi = 2-i i = 1, 2,... > 1/ T = 2m i i/T + m i T i li = i/T + 1 + m m = 3 i = i/T + 1 = T i/T + log T i T m = log T m + T = T = T T + T +1 1 < T -1 + T.

T li i i = 2 log i (i) i i (i) = (i) + 1, (i).

li = 2 log i + 1.

(21) (4) (2) (21) = (1110)(0)(00)(0101) = 11100000101.

i = 1 i > i) | | i)|)| | i)|) i) = (1110)(00)(0101) = 1110000101.

i 1, i = 1, li = log i + 2 log log i + 2, i > 1, li log i + 2 log(1 + log i) + 2, i = 1, 2,....

i = i i i pi pj i > j l H(1 + o(H)), l H i o(H) H i i li i li i li 215 1150015 140111111015 21023 11023001023 14010011901023 1 pi, i, i = 1, 2,....

i pi l = lipi i= pi(log i + 2 log log i + 2) i= 1 pi log +2 pi log 1 + log +pi i=1 pi i= H + 2 log(1 + H) + 2.

log x i log i + 2 log log i 2 log H + H X = {0, 1,..., M - 1} x1, x2,...

xn-1 = (x1,..., xn-1) xn rn = r xn-r = xn x1, x2,...

r1, r2,...

xn- xn-1 = (x1,..., xn-1) xn rn = r xn-r = xn dn xn-n-r+ x1, x2,...

d1, d2,...

xn- X = {a, b, c} cabbbabbac abc d1, d2,...

H(X) R H(X)(1 + o(H(X)), o(H) 0 H ri(a) i (i-1) x = a li(a) li(a) log ri(a) + 2 log(1 + log ri(a)) + 2.

l(a) r(a) a a l(a) log r(a) + 2 log(1 + log r(a)) + 2.

r(a).

p(a) R = p(a) H(X) + 2 log (1 + H(X)) + 2.

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