By Jacob Wolfowitz

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F. w (. I . I s') such that w (j Ii I s') is an integral multiple of 2- Vn for i = 1, ... , a,j = 1, ... , a-I, and Iw (j I i I s) - w (j I i I s') I ~ a . 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.

7) First suppose n = kl, with k an integer. c. of which C (I) is the capacity. 1 that, for 0 ~ A < 1 and all k sufficiently large, a code (k, N, A) for this channel satisfies N< 2k [C{IHi] <2n[CY> +i] <2n(c+i-). 6. c. (but not necessarily vice versa). Hence the theorem follows for n of the form k l. Suppose now n = kl + k', 1 ~ k' < l. Let n' = (k + 1) l. 9) for k sufficiently large. 2. 2 was trivial because of the way we defined C. 1 will also be very easy. 7. Let 1 be a positive integer to be chosen later, and suppose n = kl, 0 < A ~ 1.

Cn) be any channel sequence, which will be considered fixed in the argument to follow. Let u o= (Xl' . . , xn) be any n-sequence. , (n-l), such that ck = i, Ck+l= j, and X k = i'; here i,j = 1, ... , t, and i' is any element of B (i,j). Let n={n(i,j,i')}, i,j=I, ... ,t; be any matrix such that n (i, j, i') ~ i'sB(i,j) 0, and, for i, j, = 1, ... , t, }; n(i,j, i') = 1. i) Such matrices will now take the place of n-vectors n. The n-sequence U o will be called a n-sequence (always with respect to the fixed c) if i'l uo) -R(i,j I c)· n(i,j, i') I ~ ;;::; 2t VaR(i, j I c) .