Coding Theorems of Information Theory: Reihe: by Jacob Wolfowitz

By Jacob Wolfowitz

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Let w (. I . , not necessarily in 5, and let w (. I . f. Let U o be any n-sequence and A a set of n-sequences such that P 8 {V(U o) e A} > b. 3. 2) n where an -+ 0 as n -+ 00, and an depends only on band n and not on u o, A, orw('I'1 s). 3) • i,j Then the probability, when (any given) sequence U o is sent, that any specified sequence Vo will be received, is not less than n- 2n under w ( '1'1 s), and not less than (n- 2- a . 2- Vri)n under w (. 1 • 1 S/). 4) as n -+ 00. 4). 3) holds. If w (io 1 io 1 s) ~ n- 2 then, for any tt o, p s {N (io, /01 u o, v (u o)) ~ I} ~ ~ and, for large n, ') Ps,{N(io,joluo,v(uo))~I}~: .

1. Let n be any n-vector. 1). 2). 3) holds, and the theorem is proved. f. w (. / . / s) being used for any word; call this channel 52' In this channel the decoding system cannot be changed but the sequence sent can be made to depend on s. c. f. w (. 2) Here U j (s) depends upon s but Aj is the same for all s. ,A N are disjoint. The practical use of this system is obvious. When the sender knows that w (. / . f. being used by the system and he wishes to send the ith word he sends uj(s). When the receiver receives a sequence in A; he concludes that the ph word has been sent.

2-Vn for i,j = 1, .. " a. 's of 5, the canonical channel, and let 5* also denote its index set. The following lemma is very simple. 1. There exists a constant K6 > 0 with the following property: Let w (. I . , not necessarily in 5, and w (. I . f. in the sense of the preceding paragraph. I:niH(W(·1 i I s')1 < K Vl! 6• 2- T . Proof: It is enough to prove the result for large n. In the interval o ~ x ~ 2-Vn , -x logx < 2- Vn 2 . In the interval 2-Vii ~ x ~ 1, a' 2-Vn times the maximum value of the derivative of -x logx is also less than Vn 2-- 2 .

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