By J. Wolfowitz (auth.)

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This process is repeated for each letter. Thus, let the channel be in state Ci when the i th letter Xi is to be sent, i = 1, ... , n. The state c. is known to both sender and receiver. (uo),whoseconditionaldistribution, given Uo, cI> YI(UO)'···' Yi-I(UO)' is w( . 1 Ci)' If i< n the channel moves into state CHI = 4> (c" Xi)' The state CHI is also known to the receiver; if the mechanism through which it is known is the function 4>' then Ci+! = 4>'(c;. (uo))' Thus the state of the channel for any letter transmitted is determined, through the function 4>, by the previous letter transmitted and the state for that letter, and is therefore "calculable" by the sender.

It remains only to evaluate S = ~ II [C (i, j)]R(i,jlc). 12) c ,i,j Suppose C = (Cl' ... , Cn). Then n-I JI [C(i,j)]R(i,jlC) = II k=I i,j C(C k, CHI). From the definition of matrix multiplication it follows readily that S is the sum of the elements of the matrix Ln-I. The sum of the elements of M is t, and so is the sum of the elements of Mn-I. Define (X . ] z, . 14) from which the theorem readily follows. 2. Let A, 0 < I. < 1, be arbitrary. 1. 1, might, because of the coding scheme, give information about transmitted "words" in the same n-sequence other than their own, which would enable codes to be longer.

The theorem for Ä = 0 will then be true a fortiori. A code (n, N, Ä) for 5 can be divided into (not necessarily disjoint) subcodes such that in each subcode all the sequences u. are :n-sequences with the same :n, all ofwhose components are multiples of ~ . Hence fewer than (n + l)a subcodes will suffice. We will say that each sub code "belongs" to its corresponding :n. Let :n be any :n-vector, and 5 such that se H(:n' 1 s) - f:niH(w(. 1 i 1 s)) < C + V~ . c. f. w(. 1 . 1 s). 's in 5. 3) (n + l t · eXP2 {ne + [1 + K~ a3 (2a + <5)J} V; which proves the theorem.

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